# Yamabe type equations with a sign-changing nonlinearity, and the prescribed
curvature problem

Bruno Bianchini and Luciano Mari and Marco Rigoli  Bruno Bianchini
Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova
Via Trieste 63, I-35121 Padova (Italy<EMAIL_ADDRESS>Luciano Mari
Departamento de Matemática, Universidade Federal do Ceará
Av. Humberto Monte s/n, Bloco 914, 60455-760 Fortaleza (Brazil)
<EMAIL_ADDRESS>Marco Rigoli
Dipartimento di Matematica, Università degli studi di Milano
Via Saldini 50, I-20133 Milano (Italy<EMAIL_ADDRESS>
(Date: August 29, 2024)

###### Abstract.

In this paper, we investigate the prescribed scalar curvature problem on a
non-compact Riemannian manifold $(M,\langle\,,\,\rangle)$, namely the
existence of a conformal deformation of the metric $\langle\,,\,\rangle$
realizing a given function $\widetilde{s}(x)$ as its scalar curvature. In
particular, the work focuses on the case when $\widetilde{s}(x)$ changes sign.
Our main achievement are two new existence results requiring minimal
assumptions on the underlying manifold, and ensuring a control on the
stretching factor of the conformal deformation in such a way that the
conformally deformed metric be bi-Lipschitz equivalent to the original one.
The topological-geometrical requirements we need are all encoded in the
spectral properties of the standard and conformal Laplacians of $M$. Our
techniques can be extended to investigate the existence of entire positive
solutions of quasilinear equations of the type

$\Delta_{p}u+a(x)u^{p-1}-b(x)u^{\sigma}=0$

where $\Delta_{p}$ is the $p$-Laplacian, $\sigma>p-1>0$, $a,b\in
L^{\infty}_{\mathrm{loc}}(M)$ and $b$ changes sign, and in the process of
collecting the material for the proof of our theorems, we have the opportunity
to give some new insight on the subcriticality theory for the Schrödinger type
operator

$Q_{V}^{\prime}\ :\
\varphi\longmapsto-\Delta_{p}\varphi-a(x)|\varphi|^{p-2}\varphi.$

In particular, we prove sharp Hardy-type inequalities in some geometrically
relevant cases, notably for minimal submanifolds of the hyperbolic space.

###### Key words and phrases:

Yamabe equation, Schrödinger operator, subcriticality, p-Laplacian, spectrum,
prescribed curvature

###### 2010 Mathematics Subject Classification:

primary 58J05, 35B40; secondary 53C21, 34C11, 35B09.

###### Contents

  1. 1 Introduction, I: existence for the generalized Yamabe problem
  2. 2 Introduction, II: our main results in their general setting
  3. 3 Preliminaries
  4. 4 Criticality theory for $Q_{V}$, capacity and Hardy weights
  5. 5 Hardy weights and comparison geometry
    1. 5.1 Hardy weights on manifolds with a pole
    2. 5.2 Hardy weights on minimally immersed submanifolds
    3. 5.3 Specializing our main theorems: an example
  6. 6 Proofs of Theorems 2.1 and 2.2
  7. 7 Proofs of our geometric corollaries, and concluding comments

##  1\. Introduction, I: existence for the generalized Yamabe problem

Generalizations of the classical Yamabe problem on a Riemannian manifold have
been the focus of an active area of research over the past 30 years. Among
these, the prescribed scalar curvature problem over non-compact manifolds
appears to be challenging: briefly, given a non-compact Riemannian manifold
$(M^{m},\langle\,,\,\rangle)$ with scalar curvature $s(x)$ and a smooth
function $\widetilde{s}\in C^{\infty}(M)$, the problem asks under which
conditions there exists a conformal deformation of $\langle\,,\,\rangle$,

(1.1) $\widetilde{\langle\,,\,\rangle}=\varphi^{2}\langle\,,\,\rangle,\qquad
0<\varphi\in C^{\infty}(M),$

realizing $\widetilde{s}(x)$ as its scalar curvature. When the dimension $m$
of $M$ is at least $3$, writing $\varphi=u^{\frac{2}{m-2}}$, the problem
becomes equivalent to determining a positive solution $u\in C^{\infty}(M)$ of
the Yamabe equation

(1.2) $\displaystyle\Delta
u-\frac{s(x)}{c_{m}}u+\frac{\widetilde{s}(x)}{c_{m}}u^{\frac{m+2}{m-2}}=0,\qquad
c_{m}=\frac{4(m-1)}{m-2}.$

Here, $\Delta$ is the Laplace-Beltrami operator of the background metric
$\langle\,,\,\rangle$. For $m=2$, setting $\varphi=e^{u}$ one substitutes
(1.2) with

$2\Delta u-s(x)+\widetilde{s}(x)e^{2u}=0$

where now $u\in C^{\infty}(M)$ may change sign (see [43]). Hereafter, we will
confine ourselves to dimension $m\geq 3$, and $M$ will always be assumed to be
connected. Agreeing with the literature, we will call the linear operator in
(1.2):

(1.3)
$L_{\langle\,,\,\rangle}\doteq-\Delta-\frac{s(x)}{c_{m}}=-\Delta+\frac{m-2}{4(m-1)}s(x)$

the conformal Laplacian of $(M,\langle\,,\,\rangle)$.

The original Yamabe problem is a special case of the prescribed scalar
curvature problem, namely that when $\widetilde{s}(x)$ is a constant, and for
this reason, in the literature, the prescribed scalar curvature problem is
often called the generalized Yamabe problem. Besides establishing existence of
a positive solution $u$ of (1.2), it is also useful to investigate its
qualitative behaviour since this reflects into properties of
$\widetilde{\langle}\,,\,\rangle$. For instance, $u\in L^{\frac{2m}{m-2}}(M)$
is equivalent to the fact that $\widetilde{\langle}\,,\,\rangle$ has finite
volume. Also, if $u$ is bounded between two positive constants, the identity
map

(1.4) $i\ \ :\ \
(M,\langle\,,\,\rangle)\longrightarrow(M,\widetilde{\langle}\,,\,\rangle)$

is globally bi-Lipschitz, and thus $\widetilde{\langle}\,,\,\rangle$ inherits
some fundamental properties of $\langle\,,\,\rangle$. For instance, geodesic
completeness, parabolicity, Gromov-hyperbolicity, etc. (see [36, 35]).
Agreeing with the literature, when
$C^{-1}\langle\,,\,\rangle\leq\widetilde{\langle\,,\,\rangle}\leq
C\langle\,,\,\rangle$ for some constant $C>0$ we will say that
$\widetilde{\langle}\,,\,\rangle$ and $\langle\,,\,\rangle$ are uniformly
equivalent.

Given the generality of the geometrical setting, it is reasonable to expect
that existence or non-existence of the desired conformal deformation heavily
depends on the topological and metric properties of $M$ and their relations
with $\widetilde{s}(x)$. As we shall explain in awhile, a particularly
intriguing (and difficult) case is when $\widetilde{s}(x)$ is allowed to
change sign. In this situation, with the exception of a few special cases, a
satisfactory answer to the prescribed curvature problem is still missing. To
properly put our results into perspective, first we describe some of the main
technical problems that arise when looking for solutions of (1.2) for sign-
changing $\widetilde{s}(x)$. Then, we briefly comment on some classical and
more recent approaches. In particular, we pause to describe in detail four
results that allow us to grasp the situation in the relevant examples of
Euclidean and hyperbolic spaces and to underline the key features of our new
achievements. We stress that, when $\widetilde{s}(x)\leq 0$, there is a vast
literature and the interaction between topology and geometry is better
understood. Among the various references on the existence problem, we refer
the reader to [8, 72, 71, 16, 48].

If $\widetilde{s}(x)$ is positive somewhere, basic tools to produce solutions
are in general missing. More precisely, uniform $L^{\infty}$-estimates fail to
hold on regions where $\widetilde{s}(x)$ is non-negative, and comparison
theorems are not valid where $\widetilde{s}(x)$ is positive. This suggests
why, in the literature, equation (1.2) in a non-compact ambient space has
mainly been studied via variational and concentration-compactness techniques
([37, 80]) or radialization techniques ([57, 56, 42, 8]). We also quote the
interesting method developed in [72, 71, 9].

To the best of our knowledge, up to now there have been few attempts to adapt
the variational approach to (non-compact, of course) spaces other than
$\mathbb{R}^{m}$, [37, 80]. In this respect, a particularly interesting result
is the next one due to Q.S. Zhang [80].

######  Theorem 1.1 ([80], Thm. 1.1).

Let $(M^{m},\langle\,,\,\rangle)$ be a complete manifold with dimension $m\geq
3$ and scalar curvature $s(x)\geq 0$. Suppose that $\mathrm{vol}(B_{r}(x))\leq
Cr^{m}$ for some uniform $C$ independent of $x$, and that $M$ has positive
Yamabe invariant $Y(M)$:

(1.5)
$Y(M)=\inf\left\\{\int_{M}\Big{[}|\nabla\phi|^{2}+\frac{s(x)}{c_{m}}\phi^{2}\Big{]}\
\ :\ \ \phi\in\mathrm{Lip}_{c}(M),\ \int_{M}\phi^{\frac{2m}{m-2}}=1\right\\},$

$c_{m}$ as in (1.2). Assume further that

  * -

$\widetilde{s}(x)\geq 0,\not\equiv 0$ and $\widetilde{s}(x)\rightarrow 0$ as
$r(x)\rightarrow+\infty$,

  * -

$\widetilde{s}(x)$ is sufficiently flat around at least one of its maximum
points.

Then, there exists a solution $u\in L^{\frac{2m}{m-2}}(M)$ of (1.2) such that

(1.6) $u\leq C\big{(}1+r(x)\big{)}^{-\frac{m-2}{2}},$

for some $C>0$. In particular,
$\widetilde{\langle\,,\,\rangle}=u^{\frac{4}{m-2}}\langle\,,\,\rangle$ has
finite volume and is geodesically incomplete.

######  Remark 1.1.

_The flatness condition above is the one usually required for the compact
Yamabe problem, see[28, 29]. _

The above theorem is not, indeed, the most general statement of Zhang’s
result, but however the version here is a good compromise between generality
and simplicity, and it is enough for the sake of comparison with our main
theorems. On the positive side, topological conditions on $M$ are not so
demanding. However, we underline that the polynomial volume growth assumption
is essential for Zhang’s method to work, hence this excludes the case of
negatively curved manifolds like the hyperbolic space
$\mathbb{H}^{m}_{\kappa}$ of sectional curvature $-\kappa^{2}$. On the
contrary, as the recent [13] highlights, the radialization methods developed
by W.M. Ni, M. Naito and N. Kawano in [57, 56, 42] on $\mathbb{R}^{m}$, and by
P. Aviles and R. McOwen in [8] for $\mathbb{H}^{m}$ are very flexible with
respect to curvature control on $M$, but on the other hand they require $M$ to
possess a pole (that is, a point $o$ for which the exponential map $\exp_{o}$
is a diffeomorphism), a quite restrictive topological assumption. We quote the
two results, starting from Ni-Naito-Kawano’s theorem.

######  Theorem 1.2 ([57, 56, 42]).

Let $\widetilde{s}(x)\in C^{\infty}(\mathbb{R}^{m})$, $m\geq 3$, and suppose
that there exists $B\in C^{0}(\mathbb{R})$ such that

(1.7) $|\widetilde{s}(x)|\leq B\big{(}r(x)\big{)}\qquad\text{and}\qquad
rB(r)\in L^{1}(+\infty).$

Then, there exists a small $\gamma_{0}>0$ such that, for each
$\gamma\in(0,\gamma_{0})$, there exists a conformal deformation
$\widetilde{\langle}\,,\,\rangle$ of the flat metric $\langle\,,\,\rangle$
such that

(1.8)
$\widetilde{\langle\,,\,\rangle}_{x}\rightarrow\gamma\langle\,,\,\rangle_{x}\qquad\text{as
}\,r(x)\rightarrow+\infty.$

######  Theorem 1.3 ([8], Thm 4).

Let $(M^{m},\langle\,,\,\rangle)$ be a complete manifold with a pole and
dimension $m\geq 3$, and suppose that there exist constants
$\bar{\kappa}\geq\kappa>0$ such that sectional curvature $K$ of $M$ be pinched
as follows:

(1.9) $-\bar{\kappa}^{2}\leq K\leq-\kappa^{2}<0,\qquad\text{with
}\quad\bar{\kappa}^{2}<\frac{(m-1)^{2}}{m(m-2)}\kappa^{2}.$

Suppose also that $\widetilde{s}(x)\in C^{\infty}(M)$ satisfies

(1.10) $-C_{1}\leq\widetilde{s}(x)\leq-C_{2}<0\qquad\text{outside of a compact
set,}$

for some constants $C_{1},C_{2}>0$. Then, there exists $\delta>0$ sufficiently
small such that, if

(1.11) $\widetilde{s}(x)\leq\delta\qquad\text{on }\,M,$

there exists a conformal deformation $\widetilde{\langle\,,\,\rangle}$
realizing $\widetilde{s}(x)$ and satisfying

(1.12) $C^{-1}\langle\,,\,\rangle\leq\widetilde{\langle\,,\,\rangle}\leq
C\langle\,,\,\rangle\qquad\text{on }\,M,$

for some positive constant $C$.

######  Remark 1.2.

_Theorem 1.3 has later been improved in [72] with a different technique:
however, the main Theorem 0.1 in [72] still requires (1.11) and a couple of
conditions on the curvatures of $M$ that, though more general than (1.9),
nevertheless are more demanding than (1.23), (1.24) appearing in our Corollary
1.2 below. _

######  Remark 1.3.

_For the special case of the Hyperbolic space, in[71] (see Theorem 1.1
therein) the authors were able to guarantee the existence of a solution for
the Yamabe equation giving rise to a complete metric even when (1.10) is
replaced by the weaker_

$-Cr(x)^{2}\leq\widetilde{s}(x)<0\qquad\text{outside of a compact set.}$

_The counterpart of this improvement is that a control of the type ( 1.12) is
no longer available. We remark that, in Theorem 1.1 of [71], condition (1.11)
still appears. _

Inspired by Ni-Naito-Kawano’s approach, in [13] we have obtained sharp
existence theorems for (1.2) (and, more generally, for (2.1) below) on
manifolds possessing a pole $o$ via mild assumptions on the radial sectional
curvature $K_{\mathrm{rad}}$ (the sectional curvature restricted to $2$-planes
containing $\nabla r$, with $r(\cdot)=\mathrm{dist}(\cdot,o)$). In the
particular case of manifolds close to the hyperbolic space, our outcome has
been the following result. Observe that condition 1.15 below guarantees the
existence of solutions even when $\widetilde{s}(x)$ is strongly oscillating.
On the other hand, (1.16) implies that the conformally deformed metric is
incomplete and has finite volume.

######  Theorem 1.4 ([13], Thm 2).

Let $(M,\langle\,,\,\rangle)$ be a complete manifold of dimension $m\geq 3$,
with a pole $o$ and sectional curvature $K$ satisfying

(1.13) $-\kappa^{2}-\mathcal{K}\big{(}r(x)\big{)}\leq K(x)\leq-\kappa^{2},$

for some constant $\kappa>0$ and some non-negative $\mathcal{K}\in
C^{0}(\mathbb{R}^{+}_{0})\cap L^{1}(\mathbb{R}^{+})$. Suppose that the scalar
curvature $s(x)$ of $M$ is such that

(1.14) $s(x)\geq-\frac{(m-1)^{3}\kappa^{2}}{m-2}\qquad\text{on }M.$

Then, for each $\widetilde{s}(x)\in C^{\infty}(M)$ satisfying, for some $B\in
C^{0}(\mathbb{R}^{+}_{0})$,

(1.15) $|\widetilde{s}(x)|\leq B(r(x)),\qquad e^{-2\kappa r}B(r)\in
L^{1}(+\infty),$

the metric $\langle\,,\,\rangle$ can be conformally deformed to a smooth
metric $\widetilde{\langle\,,\,\rangle}$ of scalar curvature
$\widetilde{s}(x)$, satisfying

(1.16) $\Gamma_{1}e^{-2\kappa
r(x)}\langle\,,\,\rangle_{x}\leq\widetilde{\langle\,,\,\rangle}_{x}\leq\Gamma_{2}e^{-2\kappa
r(x)}\langle\,,\,\rangle_{x}\qquad\forall\,x\in M,$

for some $0<\Gamma_{1}\leq\Gamma_{2}$. In particular,
$\widetilde{\langle}\,,\,\rangle$ is incomplete and has finite volume.
Furthermore, $\Gamma_{2}$ and consequently $\Gamma_{1}$ can be chosen to be as
small as we wish.

######  Remark 1.4.

_The growth conditions ( 1.7) and (1.15) are sharp: it is proved in [21] (for
$\mathbb{R}^{m}$) and [15] (for $\mathbb{H}^{m}_{\kappa}$) that no conformal
deformation exists whenever $\widetilde{s}(x)\leq 0$ on $\mathbb{R}^{m}$
(respectively, on $\mathbb{H}^{m}_{\kappa}$) and_

$\widetilde{s}(x)\leq-\frac{C}{r(x)^{2}\log
r(x)}\qquad\left(\text{respectively, }\ \
\widetilde{s}(x)\leq-\frac{Ce^{2\kappa r(x)}}{r(x)\log r(x)}\right)$

_for some $C>0$ and large $r(x)$. _

The above four results are, to the best of our knowledge, an up-to-date
account of what is known on the prescribed scalar curvature problem, in
dimension $m\geq 3$ and with sign-changing $\widetilde{s}(x)$, on non-compact
manifolds close to $\mathbb{R}^{m}$ and $\mathbb{H}^{m}_{\kappa}$. Figures 1
and 2 below summarize Theorems 1.1 to (1.4) when assumptions overlap.

Figure 1. Euclidean space, (1.7) and Zhang’s assumptions on $\widetilde{s}(x)$
in force. Figure 2. Manifolds close to $\mathbb{H}^{m}_{\kappa}$,
$-C_{1}\leq\widetilde{s}(x)\leq-C_{2}<0$ for large $x$.

A first step in the direction of removing the pole requirement has been taken
in [14] by adapting some ideas of [13] via the use of Green functions.
Unfortunately, even though the requirements on $s(x)$ and $\widetilde{s}(x)$
in Theorem 5 of [14] are sharp, they express in a form that is generally
difficult to check. In summary, the task of obtaining results of the type
above but with a substantial weakening of the geometric assumptions calls for
new ideas, and this is the objective of the present work. More precisely, we
have a twofold concern in this paper. First, we aim to produce an existence
theorem for sign-changing $\widetilde{s}(x)$ where topological and geometrical
conditions are confined to a minimum. Second, we also want to keep control on
the conformally deformed metric, in particular in such a way that
$\langle\,,\,\rangle$ and $\widetilde{\langle\,,\,\rangle}$ are uniformly
equivalent. Our contributions are Theorems 1.5, 1.6 and 2.3 below, a special
case of Theorems 2.1, 2.2 which we are going to describe in awhile.

######  Notation.

_Hereafter, given $b\in L^{\infty}_{\mathrm{loc}}(M)$, we respectively denote
with $b_{+}$ and $b_{-}$ its positive and negative parts, so that
$b=b_{+}-b_{-}$. For $a\in L^{\infty}_{\mathrm{loc}}(M)$, we will write_

$a(x)=O\big{(}b(x)\big{)}\qquad\big{(}\text{respectively,}\quad a(x)\asymp
b(x)\ \big{)}\quad\text{as $x$ diverges}$

_to indicate that there exists a constant $C>0$ and a compact set $\Omega$
such that_

$a(x)\leq Cb(x)\qquad\big{(}\text{respectively,}\quad C^{-1}b(x)\leq a(x)\leq
Cb(x)\ \big{)}$

_on $M\backslash\Omega$. _

Our first result deals with the case of non-parabolic manifolds with non-
negative scalar curvature. We recall that a manifold $M$ is said to be non-
parabolic if it admits a positive, non-constant solution of $\Delta u\leq 0$.
The notion of non-parabolicity will be recalled later in a more general
setting (see Proposition 2.1 and the subsequent discussion), here we limit to
refer the interested reader to [34] for deepening. The following theorem shall
be compared to Theorems 1.1 and 1.2. In particular, to compare Theorem 1.5
below with Zhang’s Theorem 1.1 we need some tools that will be defined in the
next introduction, and therefore we postpone the analysis to Remark 2.3.

######  Theorem 1.5.

Let $(M,\langle\,,\,\rangle)$ be a non-parabolic manifold of dimension $m\geq
3$ and scalar curvature $s(x)$ satisfying

(1.17) $s(x)\geq 0\text{ on M,}\qquad s\in L^{1}(M),$

and let $\widetilde{s}\in C^{\infty}(M)$ with the following properties:

(1.18) $\text{$\widetilde{s}_{+}$ has compact support,}\qquad\widetilde{s}\in
L^{1}(M).$

Then, for each constant $C>0$, $\langle\,,\,\rangle$ can be pointwise
conformally deformed to a new metric $\widetilde{\langle\,,\,\rangle}$ of
scalar curvature $\widetilde{s}(x)$ such that
$\widetilde{\langle\,,\,\rangle}\leq C\langle\,,\,\rangle$. Moreover, if $s$
and $\widetilde{s}$ have compact support, each such
$\widetilde{\langle\,,\,\rangle}$ can be chosen to be uniformly equivalent to
$\langle\,,\,\rangle$.

######  Remark 1.5.

_Non-parabolicity is a very mild requirement, and it is necessary to guarantee
existence in all the cases investigated in Theorem 1.5. In fact, if $M$ is
scalar flat and taking $\widetilde{s}(x)$ to be compactly supported, non-
negative and not identically zero, any eventual solution $u$ of the Yamabe
equation (1.2) would be a (non-constant) positive solution of $\Delta
u=-\widetilde{s}(x)/c_{m}u^{\frac{m+2}{m-2}}\leq 0$, showing that $M$ must be
non-parabolic. _

Theorem 1.5 applies, for instance, to the physically relevant setting of
asymptotically flat spaces. According to [47], $(M^{m},\langle\,,\,\rangle)$
is called asymptotically flat if

  * -

its scalar curvature $s(x)$ satisfies (1.17),

  * -

there exists a compact set $K\subset M$ such that each connected component
$U_{j}$ of $M\backslash K$ has a global chart
$\Psi_{j}:(\mathbb{R}^{m}\backslash
B_{R}(0),\langle\,,\,\rangle_{\mathrm{can}})\rightarrow U_{j}$ for which the
local expression $g_{ij}$ of $\langle\,,\,\rangle$ satisfies

(1.19)
$|g_{ij}-\delta_{ij}|=O\big{(}r^{-p}\big{)},\quad|\partial_{k}\,g_{ij}|=O\big{(}r^{-p-1}\big{)},\quad|\partial^{2}_{kl}\,g_{ij}|=O\big{(}r^{-p-2}\big{)}$

as $r(x)=|x|\rightarrow+\infty$, for some $p>(m-2)/2$ and for each $1\leq
i,j,k,l\leq m$.

######  Corollary 1.1.

Let $(M,\langle\,,\,\rangle)$ be an asymptotically flat manifold of dimension
$m\geq 3$. Then, for each smooth function $\widetilde{s}(x)$ satisfying

(1.20) $\text{$\widetilde{s}_{+}$ has compact
support}\qquad\text{and}\qquad\widetilde{s}\in L^{1}(M),$

and for each constant $C>0$, $\widetilde{s}(x)$ is realizable via a conformal
deformation $\widetilde{\langle}\,,\,\rangle$ of $\langle\,,\,\rangle$
satisfying $\widetilde{\langle}\,,\,\rangle\leq C\langle\,,\,\rangle$.
Furthermore, if $\widetilde{s}\equiv 0$ outside some compact set, then
$\widetilde{\langle}\,,\,\rangle$ can be chosen to be uniformly equivalent to
$\langle\,,\,\rangle$.

Now, we deal with manifolds whose original scalar curvature can be somewhere
negative.

######  Theorem 1.6.

Let $(M^{m},\langle\,,\,\rangle)$ be a non-parabolic Riemannian manifold of
dimension $m\geq 3$ with scalar curvature $s(x)$. Suppose that the conformal
Laplacian (1.3) admits a positive Green function on $M$.

Let $\widetilde{s}(x)\in C^{\infty}(M)$ be such that

(1.21) $\text{$\widetilde{s}_{+}$ has compact
support,}\qquad\widetilde{s}(x)\asymp s(x)\,\text{ as $x$ diverges.}$

Then, there exists $\delta>0$ such that if

(1.22) $\widetilde{s}(x)\leq\delta\qquad\text{on }M,$

then $\widetilde{s}$ is realizable via a uniformly equivalent, conformal
deformation $\widetilde{\langle}\,,\,\rangle$ of $\langle\,,\,\rangle$.

######  Remark 1.6.

_Observe that ( 1.21) implies that the original scalar curvature $s(x)$ is
non-positive outside a compact set. _

Note that Theorems 1.5 and 1.6 seem to be new even in the simpler case
$\widetilde{s}(x)\leq 0$ on $M$. In this respect, these are skew with the main
theorem in [48] and with Theorem 2.30 of [12].

The requirement (1.3) shows the central role played by the conformal Laplacian
$L_{\langle}\,,\,\rangle$ for the Yamabe equation. The relevance of
$L_{\langle\,,\,\rangle}$ for the original Yamabe problem is well-known and
highlighted, for instance, in the comprehensive [47]. We will spend a
considerable part of the paper to discuss on assumptions like (1.3). Clearly,
Theorem 1.6 is tightly related to Aviles-McOwen’s Theorem 1.3, a parallel
which is even more evident in view of the next

######  Corollary 1.2.

Let $(M,\langle\,,\,\rangle)$ be a complete manifold of dimension $m\geq 3$
with a pole $o$ and sectional curvature $K$ satisfying

(1.23) $K\leq-\kappa^{2},$

for some constant $H>0$. Suppose further that

(1.24) $s(x)\geq-\frac{(m-1)^{3}}{(m-2)}\kappa^{2}$

on $M$. Let $\widetilde{s}(x)\in C^{\infty}(M)$ satisfying

(1.25) $\displaystyle-C_{1}\leq\widetilde{s}(x)\leq-C_{2}<0\qquad\text{outside
some compact set,}$

for some constants $0<C_{2}<C_{1}$. Then, there exists $\delta>0$ such that if

(1.26) $\widetilde{s}(x)\leq\delta\qquad\text{on }M,$

then $\widetilde{s}(x)$ is realizable by a conformal deformation
$\widetilde{\langle\,,\,\rangle}$ of $\langle\,,\,\rangle$ which is uniformly
equivalent to $\langle\,,\,\rangle$.

######  Remark 1.7.

_Corollary 1.2 improves on Theorem 1.3, since requirement (1.9) in Theorem 1.3
implies (1.23), (1.24). In particular, the hyperbolic space
$\mathbb{H}^{m}_{\kappa}$ of sectional curvature $-\kappa^{2}$ satisfies all
the assumptions of Corollary 1.2, being $s(x)=-m(m-1)\kappa^{2}$. Moreover, as
the proof in [8] shows, (1.9) is essential to ensure the existence of
$\delta>0$ in (1.11); the case of equality in (1.24) seems, therefore, hardly
obtainable with the approach described in [8]. It is worth to observe that the
existence of a pole and the pinching assumption (1.9) on the sectional
curvature are needed in [8] to apply both the Laplacian comparison theorems
(from above and below) for the distance function $r(x)=\mathrm{dist}(x,o)$ in
order to find suitable radial sub- and supersolutions. On the contrary, here
the weaker (1.23) and (1.24) are just used to ensure that the conformal
Laplacian has a positive Green function. _

We pause for a moment to comment on assumption (1.22). Both Theorems 1.5 and
1.6 will be consequences of Theorem 2.3 below, and thus they will be proved
via a common technique. The reason why assumption (1.22) is required in
Theorem 1.6 but not in Theorem 1.5 can be summarized in the existence, in the
second case, of a global, positive supersolution for the conformal Laplacian
(that is, a solution $w$ of $L_{\langle\,,\,\rangle}w\geq 0$ on $M$) which is
bounded both from below and from above by positive constants; one can take,
for instance, $u\equiv 1$. Such a function is not possible to construct in the
general setting of Theorem 1.6 (see Remark 6.2 below for deepening). We stress
that, unfortunately, the value of $\delta$ in Corollary 1.2 is not explicit:
indeed, it depends on a uniform $L^{\infty}$ bound for solutions of some
suitable PDEs, which is shown to exist via an indirect method.
The need of (1.22) to obtain existence for $\widetilde{\langle\,,\,\rangle}$
is investigated in Remark 6.4. It is very interesting that the same condition
(1.22) appears both in our theorem and in Aviles-McOwen’s one, as well as in
Theorems 0.1 in [72] and 1.1 in [71], although the techniques to prove them
are different. This may suggest that, in general, (1.22) could not be
removable. However, at present we still have no counterexample showing that
(1.22) is necessary. For future work, we thus feel interesting to investigate
the next

######  Question.

Can assumption (1.22) in Theorem 1.6 be removed, even without expecting the
new metric to be uniformly equivalent to $\langle\,,\,\rangle$?

##  2\. Introduction, II: our main results in their general setting

Although the prescribed scalar curvature problem is the main focus of our
investigation, the techniques developed here allow us to study more general
classes of PDEs, namely nonlinear extensions (described in (2.4)) of the
equation

(2.1) $\Delta u+a(x)u-b(x)u^{\sigma}=0\qquad\text{on }M,\qquad u>0\ \text{ on
}M$

with $\sigma>1$, $a,b\in C^{\infty}(M)$ and sign-changing $b$. Note that the
signs of $a,b$ are reversed with respect to those of $s,\widetilde{s}$ in
(1.2), and that $\sigma$ can be greater than $\frac{m+2}{m-2}$, preventing a
direct use of variational techniques. However, when
$\sigma\leq\frac{m+2}{m-2}$ and $b(x)<0$ on $M$, the investigation of (2.1) on
Euclidean space is still the core of a very active area of research. In this
respect, we quote the seminal [17] and, for sign-changing $b(x)$ (and singular
$a(x)$), the recent [30].

As a matter of fact, even for (2.1) the spectral properties of the linear part
$L=-\Delta-a(x)$ play a prominent role, in particular the analysis of the
fundamental tone $\lambda_{1}^{L}(M)$ of the Friedrichs extension of
$\left(L,C^{\infty}_{c}(M)\right)$. We recall that $\lambda^{L}_{1}(M)$ is
characterized via the Rayleigh quotient as follows:

(2.2)
$\lambda_{1}^{L}(M)=\inf_{0\not\equiv\varphi\in\mathrm{Lip}_{c}(M)}\frac{\int_{M}\big{[}|\nabla\varphi|^{2}-a(x)\varphi^{2}\big{]}\mathrm{d}x}{\|\varphi\|^{2}_{L^{2}(M)}}.$

For instance, if $\lambda_{1}^{L}(M)<0$, the situation is somewhat rigid:

  * (a)

if $b(x)\leq 0$, then (2.1) has no positive solutions. This follows from a
direct spectral argument111Indeed, assume the contrary and let $u>0$ solves
(2.1); then, $u$ is a positive solution of $Lu\geq 0$, and a result of [32,
54, 3] implies that $\lambda_{1}^{L}(M)\geq 0$, contradicting our assumption.;

  * (b)

if $b(x)\geq 0$ and the zero set of $b$ is small in a suitable spectral sense,
then there always exist a minimal and a maximal (possibly coinciding) positive
solutions of (2.1); see [48, 63] and Section 2.4 in [12].

It is important to underline that, in both cases, the geometry of $M$ only
reveals via the spectral properties of $L$. In other words, no _a-priori_
assumptions of completeness of $M$, nor curvature nor topological requests are
made. As suggested by $(a)$ and $(b)$ above, it seems that the subtler case is
that of investigating existence under the assumption $\lambda_{1}^{L}(M)\geq
0$. This condition is often implicitly met in the literature and it is
automatically satisfied in many geometric situations. This happens, for
instance, for Theorems 1.1 to 1.4.

There is another aspect of the above picture which is worth mentioning.
Partial differential equations similar to (2.1) are of interest even for
quasilinear operators more general than the Laplacian. Just to give an
example, of a certain importance in Physics, we can consider the general
equation for radiative cooling

(2.3) $\kappa^{-1}\mathrm{div}\big{(}\kappa|\nabla u|^{p-2}\nabla
u\big{)}-\big{(}\tau\kappa^{-1}\big{)}u^{4}=0\qquad\text{on }\mathbb{R}^{m},$

where $\kappa>0$ is the coefficient of the heat conduction and $\tau$ is a
function describing the radiation (see [70], p.9). The existence problem for
this type of quasilinear PDEs when the coefficient of the nonlinearity changes
sign seems to be quite open. This suggests to extend our investigation to the
existence of positive solutions to the quasilinear, Yamabe-type equation

(2.4) $\Delta_{p,f}u+a(x)u^{p-1}-b(x)F(u)=0\qquad\text{on }M,$

where $f\in C^{\infty}(M)$,

(2.5) $\Delta_{p,f}u=e^{f}\mathrm{div}\big{(}e^{-f}|\nabla u|^{p-2}\nabla
u\big{)}$

and $F(u)$ is a nonlinearity satisfying the following assumptions:

(2.6) $\left\\{\begin{array}[]{l}F\in C^{0}(\mathbb{R}),\quad F(0)=0,\quad
F>0\ \text{ on }\mathbb{R}^{+},\\\\[5.69046pt]
\displaystyle\frac{F(t)}{t^{p-1}}\ \text{ is strictly increasing on
$\mathbb{R}^{+}$},\\\\[8.5359pt] \displaystyle\lim_{t\rightarrow
0^{+}}\frac{F(t)}{t^{p-1}}=0,\qquad\lim_{t\rightarrow+\infty}\frac{F(t)}{t^{p-1}}=+\infty\end{array}\right.$

The prototype example of $F(t)$ is $F(t)=t^{\sigma}$ for $\sigma>p-1$ and
$t\geq 0$. Of course (2.1) is recovered by choosing $p=2$ and $f$ constant. We
underline that, even for (2.4) in the Euclidean space and with
$F(t)=t^{\sigma}$, there seems to be no result covering the cases described in
Theorems 2.1 and 2.2. The family of operators in (2.5) above encompasses two
relevant geometrical cases: $\Delta_{p,0}$ the standard $p$-Laplacian
$\Delta_{p}$, and $\Delta_{2,f}$ the drifted Laplacian, $\Delta_{f}$,
appearing, for instance, in the analysis of Ricci solitons and quasi-Einstein
manifolds. Note that the radiative cooling equation is of type (2.4) provided
$1<p\leq 5$. Note also that, since the definition of $\Delta_{p,f}$ is
intended in the weak sense, solutions will be, in general, only of class
$C^{1,\mu}_{\mathrm{loc}}(M)$ by [78].

######  Notation.

_Hereafter, with a slight abuse of notation, with
$C^{1,\mu}_{\mathrm{loc}}(M)$ we mean that for each relatively compact open
set $\Omega\subset M$, there exists $\mu=\mu(\Omega)\in(0,1)$ such that $u\in
C^{1,\mu}(\overline{\Omega})$. _

######  Remark 2.1.

_We stress that, with possibly the exception of Theorem 1.1 when
$F(t)=t^{\sigma}$ and $\sigma\leq p^{*}-1$, the techniques used to prove
Theorems 1.1 to 1.4 seem hard to extend to deal with (2.5) for nonradial $f$,
even for $p=2$. For constant $f$, it seems also very difficult to adapt them
to investigate (2.5) when $p\neq 2$. In particular, the transformation
performed in [13] for $p=2$ to absorb the linear term $a(x)u$ can only be
applied when the driving operator is linear. _

To state our main result, Theorem 2.1 below, we need to introduce some
terminology. Let $\mathrm{d}\mu_{f}$ be the weighted measure
$e^{-f}\mathrm{d}x$, with $\mathrm{d}x$ the Riemannian volume element on $M$.
For $V\in L^{\infty}_{\mathrm{loc}}(M)$, we consider the functional $Q_{V}$
defined on $\mathrm{Lip}_{c}(M)$ by

(2.7)
$Q_{V}(\varphi)=\frac{1}{p}\left[\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}-\int_{M}V|\varphi|^{p}\mathrm{d}\mu_{f}\right].$

Its Gateaux derivative $Q^{\prime}_{V}$ is given by

(2.8) $Q_{V}^{\prime}(w)=-\Delta_{p,f}w-V|w|^{p-2}w\qquad\text{for }w\in
W^{1,p}_{\mathrm{loc}}(M).$

When $M=\mathbb{R}^{m}$, the spectral properties of $Q_{V}$ have been
investigated in [5, 4, 33] and in a series of papers by Y. Pinchover and K.
Tintarev (see in particular [67, 68]). From now on, we follow the notation and
terminology in [68]. In the linear case $p=2$, $f\equiv 0$, that is, for the
Schrödinger operator $Q^{\prime}_{V}=-\Delta-V$, we refer the reader to [54,
3, 60, 59, 55].

To begin with, and according to [55, 67], we recall the following:

######  Definition 2.1.

For $V\in L^{\infty}_{\mathrm{loc}}(M)$, define $Q_{V}$ as in (2.7) and let
$\Omega\subseteq M$ be an open set.

  * i)

$Q_{V}$ is said to be _non-negative_ on $\Omega$ (shortly, $Q_{V}\geq 0$) if
and only if $Q_{V}(\varphi)\geq 0$ for each
$\varphi\in\mathrm{Lip}_{c}(\Omega)$, that is, if and only if the Hardy type
inequality

(2.9)
$\int_{M}V(x)|\varphi|^{p}\mathrm{d}\mu_{f}\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\qquad\forall\
\varphi\in\mathrm{Lip}_{c}(\Omega).$

holds.

  * ii)

$Q_{V}$ is said to be _subcritical_ (or _non-parabolic_) on $\Omega$ if and
only if $Q_{V}\geq 0$ there, and there exists $w\in
L^{1}_{\mathrm{loc}}(\Omega)$, $w\geq 0$, $w\not\equiv 0$ on $\Omega$, such
that

(2.10) $\int_{M}w(x)|\varphi|^{p}\mathrm{d}\mu_{f}\leq
Q_{V}(\varphi)\qquad\forall\ \varphi\in\mathrm{Lip}_{c}(\Omega).$

Sometimes, especially in dealing with the prescribed scalar curvature problem
and when no possible confusion arises, we also say that $Q_{V}^{\prime}$, and
not $Q_{V}$, is non-negative (or subcritical). The term “non-parabolic" is
justified by the following statement for $Q_{0}$ (that is, $Q_{V}$ with
$V(x)\equiv 0$), which is part of Proposition 4.4 below:

######  Proposition 2.1.

Let $(M,\langle\,,\,\rangle)$ be Riemannian, $f\in C^{0}(M)$ and $p>1$. Then,
$Q_{0}$ is subcritical on $M$ if and only if there exists a non-constant,
positive weak solution $g\in C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ of
$\Delta_{p,f}g\leq 0$.

According to the literature, the existence of such $g$ is one of the
equivalent conditions that characterize $M$ as being not $p$-parabolic; there
are various other characterizations of $p$-parabolicity, given in terms of
Green kernels, $p$-capacity of compact sets, Ahlfors’ type maximum principles,
and so on. We refer to the survey [34] for deepening in the linear case $p=2$,
and to (see [79, 64, 41, 40, 38]) for $p\neq 2$. The equivalence in
Proposition 2.1 has been observed, in the linear setting, by [7, 19, 49], and
for $p\neq 2$ it has also recently been proved in [22] with a technique
different from our.

In fact, all of these characterizations of the non-parabolicity of
$-\Delta_{p,f}$ can be seen as a special case of a theory developed in [55,
66] (when $p=2$) and in [67, 68] for operators $Q_{V}$ with potential. In
Section 4, we recall the main result in [67, 68], the ground state
alternative, and we give a proof of it by including a further equivalent
condition, see Theorem 4.1 below; as a corollary, we prove Propositions 4.4
and 2.1.

######  Remark 2.2.

_In the prescribed scalar curvature problem, the role of $\langle\,,\,\rangle$
and $\widetilde{\langle}\,,\,\rangle$ can be exchanged. Such a symmetry
suggests that those geometric conditions which are invariant with respect to a
conformal change of the metric turn out to be more appropriate to deal with
the Yamabe equation. This is the case for the non-negativity and the
subcriticality of the conformal Laplacian $L_{\langle\,,\,\rangle}$ of
$(M,\langle\,,\,\rangle)$ in (1.3). In fact, the covariance of $L$ with
respect to the conformal deformation
$\displaystyle\widetilde{\langle\,,\,\rangle}=u^{\frac{4}{m-2}}\langle\,,\,\rangle$
of the metric:_

$L_{\widetilde{\langle}\,,\,\rangle}(\cdot)=u^{-\frac{m+2}{m-2}}L_{\langle\,,\,\rangle}(u\cdot)$

_implies that, for each $0\leq w\in L^{1}_{\mathrm{loc}}(M)$ and
$\varphi\in\mathrm{Lip}_{c}(M)$,_

$\displaystyle\int_{M}\Big{[}\|\widetilde{\nabla}\varphi\|^{2}+\frac{\widetilde{s}}{c_{m}}\varphi^{2}\Big{]}\mathrm{d}\widetilde{x}-\int_{M}\widetilde{w}\varphi^{2}\mathrm{d}\widetilde{x}=\int_{M}\Big{[}|\nabla(u\varphi)|^{2}+\frac{s}{c_{m}}(u\varphi)^{2}\Big{]}\mathrm{d}x-\int_{M}w(u\varphi)^{2}\mathrm{d}x,$

_where $\sim$ superscript indicates quantities referred to
$\widetilde{\langle}\,,\,\rangle$, $\|\cdot\|$ is its induced norm, and_

$\widetilde{w}=wu^{-\frac{4}{m-2}}\in
L^{1}_{\mathrm{loc}}(M),\quad\widetilde{w}\geq 0.$

_Consequently, $L_{\langle\,,\,\rangle}$ is non-negative (resp. subcritical)
if and only if so is $L_{\widetilde{\langle}\,,\,\rangle}$. _

######  Remark 2.3.

_As a direct consequence of the ground state alternative, the positivity of
the Yamabe invariant $Y(M)$ in Zhang’s Theorem 1.1 implies that
$L_{\langle\,,\,\rangle}$ is subcritical (see Remark 4.3). On the other hand,
in our Theorem 1.5 the subcriticality of $L_{\langle\,,\,\rangle}$ follows
combining the non-parabolicity of $M$ and $s(x)\geq 0$. Although, in general,
the positivity of $Y(M)$ might not imply the non-parabolicity of $M$, this is
so if $M$ is scalar flat outside a compact set and $\mathrm{vol}(M)=+\infty$.
Indeed, if $s(x)\equiv 0$ outside a compact set $K$, then $Y(M)>0$ gives the
validity of an $L^{2}$-Sobolev inequality on $M\backslash K$, and coupling
with $\mathrm{vol}(M)=+\infty$ the non-parabolicity of $M$ follows by a result
in [20, 64]. We underline that, in the same assumptions, again by [20, 64]
property $\mathrm{vol}(M)=+\infty$ is automatic when $M$ is geodesically
complete. Summarizing, if the manifold in Zhang’s Theorem 1.1 is scalar flat
near infinity, the geometric requirements there properly contain those of our
Theorem 1.5. _

We are now ready to state

######  Theorem 2.1.

Let $M^{m}$ be a Riemannian manifold, $f\in C^{\infty}(M)$ and
$p\in(1,+\infty)$. Suppose that $Q_{0}$ is subcritical on $M$, and let $a\in
L^{\infty}_{\mathrm{loc}}(M)$ be such that $Q_{a}$ is subcritical on $M$.
Consider $b\in L^{\infty}_{\mathrm{loc}}(M)$, and assume

  * $i)$

$b_{-}(x)$ has compact support;

  * $ii)$

$a(x)=O\big{(}b(x)\big{)}$ as $x$ diverges;

  * $iii)$

for some $\theta>0$, $\big{(}a(x)-\theta b_{+}(x)\big{)}_{-}\in
L^{1}(M,\mathrm{d}\mu_{f})$.

Fix a nonlinearity $F(t)$ satisfying (2.6). Then, there exists $\delta>0$ such
that if

(2.11) $b(x)\geq-\delta\qquad\text{on }M,$

there exists a weak solution $u\in C^{1,\mu}_{\mathrm{loc}}(M)$ of

(2.12)
$\left\\{\begin{array}[]{l}\displaystyle\Delta_{p,f}u+a(x)u^{p-1}-b(x)F(u)=0\qquad\text{
on M}\\\\[5.69046pt] 0<u\leq\|u\|_{L^{\infty}(M)}<+\infty.\end{array}\right.$

If we replace $ii)$ and $iii)$ by the stronger condition

$iv)\qquad b_{+}(x)\asymp a(x)\qquad\text{as }x\text{ diverges,}$

and we keep the validity of (2.11), then $u$ can also be chosen to satisfy

(2.13) $\inf_{M}u>0.$

In the next theorem, we remove requirement (2.11); see also Remark 6.4 for a
related discussion.

######  Theorem 2.2.

Let $M^{m}$ be a Riemannian manifold, $f\in C^{\infty}(M)$ and
$p\in(1,+\infty)$. Suppose that $Q_{0}$ is subcritical on $M$ and let $a\in
L^{\infty}_{\mathrm{loc}}(M)$ be such that $Q_{a}$ is subcritical on $M$.
Consider $b\in L^{\infty}_{\mathrm{loc}}(M)$, and assume

  * $i)$

$b_{-}(x)$ has compact support;

  * $ii^{\prime})$

$a(x)\leq 0$ outside a compact set;

  * $iii^{\prime})$

$a(x),b(x)\in L^{1}(M,\mathrm{d}\mu_{f})$.

Fix a nonlinearity $F(t)$ satisfying (2.6). Then, there exists a sequence
$\\{u_{k}\\}\subset C^{1,\mu}_{\mathrm{loc}}(M)$ of distinct weak solutions of

(2.14)
$\left\\{\begin{array}[]{l}\displaystyle\Delta_{p,f}u_{k}+a(x)u_{k}^{p-1}-b(x)F(u_{k})=0\qquad\text{
on M}\\\\[5.69046pt]
0<u_{k}\leq\|u_{k}\|_{L^{\infty}(M)}<+\infty,\end{array}\right.$

such that $\|u_{k}\|_{L^{\infty}(M)}\rightarrow 0$ as $k\rightarrow+\infty$.
If we replace $ii^{\prime})$ and $iii^{\prime})$ by the stronger condition

$iv^{\prime})\qquad a(x),b(x)\quad\text{have compact support,}$

then each $u_{k}$ also satisfies $\inf_{M}u_{k}>0$.

One of the main features in the proof of Theorems 2.1 and 2.2 above is a new
flexible technique, which is based on a direct use of the non-negativity and
subcriticality assumptions on $Q_{a}$ and $Q_{0}$. Consequently, all the
geometric information needed on $M$ is encoded in the spectral behaviour of
$Q_{0}$ and $Q_{a}$. For this reason, in Sections 4 and 5 we concentrate on
operators $Q_{V}$ to show that the assumption on $Q_{0},Q_{a}$ in Theorems 2.1
and 2.2 can be made explicit and easily verifiable in various relevant cases.

We now come to the strategy to prove Theorems 2.1 and 2.2. The lack of tools
to produce solutions in the present generality forces us to proceed along very
simple, general schemes. In particular, the argument can be roughly divided
into three parts:

  * (1)

For a big relatively compact domain $\Omega$, we solve locally (2.4) with
boundary condition 1 and $b(x)$ replaced by $b_{+}(x)$. Call $z_{\Omega}$ the
solution. This is the easiest part, and is addressed in Lemma 6.1.

  * (2)

We find uniform $L^{\infty}$ estimates from below and above for $z_{\Omega}$,
independent of $\Omega$. According to our geometric assumptions, these
estimates can be on the whole $M$ or on a relatively compact set $\Lambda$.
The proof of this step combines Lemmas 6.2 and 6.3, and Proposition 6.1.

  * (3)

Making use of the results in Step (2), we “place" $b_{-}$ in the Dirichlet
problem for (2.4) on a domain $\Omega$ via an iterative procedure, to produce
a local solution of (2.4) that possesses uniform upper and lower bounds. The
desired global solution is then obtained by passing to the limit along an
exhaustion $\\{\Omega_{j}\\}$. Note that this is the point where a distinction
between Theorems 2.1 and 2.2 appears.

Among the lemmas, which are of independent interest, we underline and briefly
comment on the next uniform $L^{\infty}$-estimate, Lemma 6.3. This result is a
cornerstone both for steps (2) and (3).

######  Lemma 2.1 (Uniform $L^{\infty}$-estimate).

Let $M$ be a Riemannian manifold, $f\in C^{\infty}(M)$, $p\in(1,+\infty)$. Let
$A,B\in L^{\infty}_{\mathrm{loc}}(M)$ with $B\geq 0$ a.e. on $M$. Assume that
either

  * $(i)$

$B\equiv 0$ and $Q_{A}$ is subcritical, or

  * $(ii)$

$B\not\equiv 0$ and $Q_{A}$ is non-negative.

Suppose that there exist a smooth, relatively compact open set $\Lambda\Subset
M$ and a constant $c>0$ such that

(2.15) $A\leq cB\qquad\text{a.e. on }M\backslash\Lambda,$

and fix a smooth, relatively compact open set $\Lambda^{\prime}$ such that
$\Lambda\Subset\Lambda^{\prime}$, and a nonlinearity $F(t)$ satisfying (2.6).
Then, there exists a constant $C_{\Lambda}>0$ such that, for each smooth,
relatively compact open set $\Omega$ with $\Lambda^{\prime}\Subset\Omega$, the
solution $0<z\in C^{1,\mu}(\overline{\Omega})$ of

(2.16)
$\left\\{\begin{array}[]{ll}\Delta_{p,f}z+A(x)z^{p-1}-B(x)F(z)=0&\quad\text{on
}\Omega,\\\\[5.69046pt] z=1&\quad\text{on }\partial\Omega.\end{array}\right.$

satisfies

(2.17) $z\leq C_{\Lambda}\qquad\text{on }\Omega.$

The proof of the above estimate is accomplished by using non-negativity (resp.
subcriticality) of $Q_{A}$ alone. As far as we know, the argument in the proof
seems to be new and applicable beyond the present setting. Clearly, when $z$
is $C^{2}$ and $B\in C^{0}(M)$, $B>0$ on $M$, possibly evaluating (2.16) at a
interior maximum point $x_{0}$ we get

(2.18) $\frac{F(z)}{z^{p-1}}(x_{0})\leq\sup_{M}\left(\frac{A_{+}}{B}\right),$

whence by (2.6) $z(x_{0})$ is uniformly bounded from above. Taking into
account the boundary condition for $z$, in this case the uniform $L^{\infty}$
estimate is trivial with no assumption on $Q_{A}$. On the other hand, even a
single point at which $B(x)=0$ makes this simple argument to fail, and
actually Lemma 2.1 will be applied in cases when we have no control at all on
the zero-set of $B$. Observe that (2.15) is just assumed to hold outside of a
compact set, hence $A$ is not required to be non-positive on the set where
$B=0$. This suggests that the validity of (2.18) cannot be recovered “in the
limit" by using approximating positive functions $B_{\varepsilon}$ for $B$ and
related solutions $z_{\varepsilon}$ for $z$. Note also that, when $B\equiv 0$,
Proposition 3.4 below shows that $Q_{A}$ is necessarily non-negative, for
otherwise $z$ might not exist for sufficiently large $\Omega$’s. Therefore, in
the present generality at least the non-negativity of $Q_{A}$ on the whole $M$
needs to be assumed in any case.

We pause for a moment to comment on the subcriticality of $Q_{V}$. By its very
definition, a sufficient condition for $Q_{V}$ to be subcritical is the
coupling of the following two:

  * -

$Q_{0}$ is subcritical, thus there exists $w\in L^{1}_{\mathrm{loc}}(M)$,
$w\geq 0$, $w\not\equiv 0$ such that

(2.19)
$\int_{M}w(x)|\varphi|^{p}\mathrm{d}\mu_{f}\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\qquad\forall\
\varphi\in\mathrm{Lip}_{c}(M),\ \text{ and}$

  * -

$V\leq w$, $V\not\equiv w$.

Therefore, when $Q_{0}$ is subcritical, we can state simple, explicit
conditions guaranteeing the subcriticality of $Q_{V}$ provided that we know
explicit $w\in L^{1}_{\mathrm{loc}}(M)$, $w\geq 0$, $w\not\equiv 0$ satisfying
(2.19). We define each of these $w$ a Hardy weight for $\Delta_{p,f}$.

In the literature, there are conditions to imply the subcriticality of $Q_{0}$
that involve curvature bounds, volume growths, doubling properties and Sobolev
type inequalities. For example, when $f\equiv 0$, in [41] it is proved that a
complete, non-compact manifold $M$ with non-negative Ricci curvature outside a
compact set is not $p$-parabolic (i.e. $Q_{0}$ is subcritical) if and only if
$p<m$. The interested reader can also consult [40, 73]. However, it seems
challenging to obtain explicit Hardy weights in the setting of [41, 40, 73].
Nevertheless, Hardy weights have been found in some interesting cases,
starting with the famous Hardy type inequality for Euclidean space

(2.20)
$\left(\frac{m-p}{p}\right)^{p}\int_{\mathbb{R}^{m}}\frac{|\varphi|^{p}}{r^{p}}\mathrm{d}x\leq\int_{\mathbb{R}^{m}}|\nabla\varphi|^{p}\mathrm{d}x\qquad\forall\
\varphi\in\mathrm{Lip}_{c}(\mathbb{R}^{m}),$

where $r(x)=|x|$ and $m>p$. In recent years ([19, 49, 12, 10, 1, 22, 23, 24])
it has been observed how Hardy weights are related to positive Green kernels
for $\Delta_{p,f}$. By exploiting the link established in Proposition 4.4
below, we will devote Section 5 to produce explicit Hardy weights in various
geometrically relevant cases, see Theorems 5.1, 5.2, 5.3 below: in fact, a
typical construction of Hardy weights via the Green kernel is compatible with
comparison results for the Laplacian of the distance function, and thus Hardy
weights can be transplanted from model manifolds to general manifolds, as
observed in [12], Theorem 4.15 and subsequent discussion. Moreover, the set of
Hardy weights is convex in $L^{1}_{\mathrm{loc}}(M)$, thus via simple
procedures one can produce new weights, such as multipole Hardy weights or
weights blowing up along a fixed submanifold of $M$. Hardy weights can also be
transplanted to submanifolds, but this procedure is more delicate and requires
extra care. Let $N^{n}$ be a Cartan-Hadamard manifold (i.e. a simply
connected, complete manifold of non-positive sectional curvature), and let
$M^{m}$ be a minimal submanifold of $N^{n}$. Suppose that the sectional
curvature $\bar{K}$ of $N$ satisfies $\bar{K}\leq-\kappa^{2}$, for some
constant $\kappa\geq 0$. In [19, 49], the authors proved the following Hardy
type inequality:

(2.21)
$\left(\frac{m-2}{2}\right)^{2}\int_{M}\frac{\varphi^{2}}{\rho^{2}}\mathrm{d}x\leq\int_{M}|\nabla\varphi|^{2}\mathrm{d}x\qquad\forall\
\varphi\in\mathrm{Lip}_{c}(M),$

$\rho(x)$ being the extrinsic distance in $N$ from a fixed origin $o\in N$.
The Hardy weight in (2.21) is sharp if $\kappa=0$ (in particular, if $N$ is
the Euclidean space), but not if $\kappa>0$. Here, we will prove (2.21) as a
particular case of Theorem 5.3 below, which also strengthen (2.21) to a sharp
inequality when $\kappa>0$, in particular for minimal submanifolds of
hyperbolic spaces. We stress that our Hardy weight for $\kappa>0$ is skew with
the one found in [49].

Using the Hardy inequalities mentioned before, we can rewrite the
subcriticality assumption for $Q_{0}$ and $Q_{V}$ in Theorems 2.1, 2.2 in
simple form for a wide class of manifolds; by a way of example, see Corollary
5.1 in Section 7. We conclude by rephrasing Theorems 2.1, 2.2 in the setting
of the generalized Yamabe problem.

######  Theorem 2.3.

Let $(M,\langle\,,\,\rangle)$ be a non-parabolic Riemannian manifold of
dimension $m\geq 3$ and scalar curvature $s(x)$. Suppose that the conformal
Laplacian $L_{\langle\,,\,\rangle}$ in (1.3) is subcritical, and let
$\widetilde{s}\in C^{\infty}(M)$.

  * $(I)$

Assume that

    * $i)$

$\widetilde{s}_{+}$ has compact support;

    * $ii)$

$s_{-}(x)=O\big{(}\widetilde{s}_{-}(x)\big{)}$ as $x$ diverges;

    * $iii)$

for some $\theta>0$, $\big{(}\theta\widetilde{s}_{-}(x)-s_{-}(x)\big{)}_{+}\in
L^{1}(M)$.

Then, there exists $\delta>0$ such that if

(2.22) $\widetilde{s}(x)\leq\delta,$

the metric $\langle\,,\,\rangle$ can be pointwise conformally deformed to a
new metric $\widetilde{\langle\,,\,\rangle}$ with scalar curvature
$\widetilde{s}(x)$ and satisfying

(2.23) $\widetilde{\langle\,,\,\rangle}\leq C\langle\,,\,\rangle\qquad\text{on
}M,$

for some constant $C>0$. Moreover, if $ii)$ and $iii)$ are replaced by the
stronger

$iv)\qquad s(x)\asymp\widetilde{s}(x)\qquad\text{as $x$ diverges,}$

then (under the validity of (2.22)) there exists a pointwise conformal
deformation $\widetilde{\langle\,,\,\rangle}$ of $\langle\,,\,\rangle$ as
above and satisfying

(2.24) $C_{1}\langle\,,\,\rangle\leq\widetilde{\langle\,,\,\rangle}\leq
C_{2}\langle\,,\,\rangle\qquad\text{on }M,$

for some constants $0<C_{1}\leq C_{2}$. In particular,
$\widetilde{\langle}\,,\,\rangle$ is non-parabolic, and it is complete
whenever $\langle\,,\,\rangle$ is complete.

  * $(II)$

If $ii)$ and $iii)$ are replaced with

    * $ii^{\prime})$

$s(x)\geq 0$ outside a compact set;

    * $iii^{\prime})$

$s(x),\widetilde{s}(x)\in L^{1}(M)$,

then the existence of the desired conformal deformation is guaranteed without
the requirement (2.22), and moreover the constant $C$ in (2.23) can be chosen
as small as we wish (so that, indeed, there exist infinitely many conformal
deformations realizing $\widetilde{s}$). If $ii^{\prime})$ and $iii^{\prime})$
are replaced with

$iv^{\prime})\qquad s(x),\widetilde{s}(x)\qquad\text{have compact support,}$

each of these conformally deformed metrics $\widetilde{\langle}\,,\,\rangle$
satisfies (2.24).

The paper is organized as follows. In Section 3 we collect some basic material
on $Q_{V}$ and $Q_{V}^{\prime}$. Section 4 will then be devoted to the
criticality theory for $Q_{V}$, its link with Hardy weights and with a
$Q_{V}$-capacity theory. In Section 5, we use comparison geometry to produce
sharp Hardy inequalities. Section 6 contains the proof of Lemma 2.1 and of our
main Theorems 2.1, 2.2. Then, in Section 7 we derive our geometric
corollaries, and we place them among the existing literature. Finally, in the
Appendix we give a full proof of the pasting lemma, an important technical
result for the $Q_{V}$-capacity theory. Besides the presence of new results, a
major concern of Sections 3 to 5 is to help the reader to get familiar with
various aspects of the theory of Schrödinger type operators $Q^{\prime}_{V}$.
For this reason, the experienced reader may possibly skip them and go directly
to Section 6.

##  3\. Preliminaries

In this section we recall some general facts for the operators
$\Delta_{p,f},Q_{V},Q^{\prime}_{V}$ that are extensions, to a quasilinear
setting, of some classical results of spectral theory (see [55, 66, 54, 3]).
The interested reader may consult [5, 4, 33, 67, 68] for further information.

######  Notation.

_Hereafter, given two open subsets $\Omega,U$, with $\Omega\Subset U$ we
indicate that $\Omega$ has compact closure contained in $U$. We say that
$\\{\Omega_{j}\\}$ is an exhaustion of $M$ if it is a sequence of relatively
compact, connected open sets $\Omega_{j}$ with smooth boundary and such that
$\Omega_{j}\Subset\Omega_{j+1}\Subset M$, $M=\bigcup_{j}\Omega_{j}$. The
symbol $1_{U}$ denotes the characteristic function of a set $U$, and the
symbol $\doteq$ is used to define an object. _

######  Definition 3.1.

Let $\Omega\subset M$ be an open set and, for $V\in
L^{\infty}_{\mathrm{loc}}(\Omega)$, let $Q_{V},Q_{V}^{\prime}$ be as in (2.7),
(2.8). We say that $w\in W^{1,p}_{\mathrm{loc}}(\Omega)$ is a supersolution on
$\Omega$ (respectively, subsolution, solution) if $Q_{V}^{\prime}(w)\geq 0$
weakly on $\Omega$ (resp. $\leq 0$, $=0$) that is, if

$\int_{\Omega}|\nabla w|^{p-2}\langle\nabla
w,\nabla\varphi\rangle\mathrm{d}\mu_{f}-\int_{\Omega}V|w|^{p-2}w\varphi\geq
0\qquad\text{(resp. }\leq 0,\,=0)$

for each non-negative $\varphi\in\mathrm{Lip}_{c}(\Omega)$.

The basic technical material that is necessary for our purposes is summarized
in the following

######  Theorem 3.1.

Let $\Omega\Subset M$ be a relatively compact, open domain with $C^{1,\alpha}$
boundary for some $0<\alpha<1$. Let $f\in C^{\infty}(M)$, $p\in(1,+\infty)$,
$V\in L^{\infty}_{\mathrm{loc}}(M)$ and define $Q_{V},Q^{\prime}_{V}$ as in
(2.7), (2.8). Let $g\in L^{\infty}(\Omega)$, $\xi\in
C^{1,\alpha}(\partial\Omega)$ and suppose that $u\in W^{1,p}(\Omega)$ is a
solution of

(3.1) $\left\\{\begin{array}[]{ll}Q^{\prime}_{V}(u)=g&\quad\text{on
}\Omega,\\\\[5.69046pt] u=\xi&\quad\text{on
}\partial\Omega.\end{array}\right.$

Then,

  * $(1)$

_[Boundedness]_ $u\in L^{\infty}_{\mathrm{loc}}(\Omega)$, and for any
relatively compact, open domains $B\Subset B^{\prime}\Subset\Omega$ there
exists a positive constant
$C=C(p,f,m,g,\xi,\Omega,\|u\|_{L^{p}(B^{\prime},\mathrm{d}\mu_{f})})$ such
that

$\|u\|_{L^{\infty}(B)}\leq C.$

If $\xi\in C^{2,\alpha}(\partial\Omega)$, $C$ can be chosen globally on
$\Omega$, and thus $u\in L^{\infty}(\Omega)$.

  * $(2)$

_[ $C^{1,\mu}$-regularity]_ When $u\in L^{\infty}(\Omega)$, there exists
$\mu\in(0,1)$ depending on $p,f,m,g,\alpha$ and on upper bounds for
$\|u\|_{L^{\infty}},\|g\|_{L^{\infty}},\|\xi\|_{C^{1,\alpha}},\|V\|_{L^{\infty}}$
on $\Omega$ such that

$\|u\|_{C^{1,\mu}(\overline{\Omega})}\leq C$

for some constant $C$ depending on $\alpha,p$, the geometry of $\Omega$ and
upper bounds for $\|u\|_{L^{\infty}}$,
$\|g\|_{L^{\infty}},\|\xi\|_{C^{1,\alpha}},\|V\|_{L^{\infty}}$ on $\Omega$.

  * $(3)$

_[Harnack inequality]_. For any relatively compact open sets $B\Subset
B^{\prime}\Subset\Omega$ there exists $C=C(f,p,m,B,B^{\prime})>0$ such that,
for each $u\geq 0$, $u\in W^{1,p}(\Omega)$ solution of $Q^{\prime}_{V}(u)=0$
on $\Omega$,

(3.2) $\sup_{B}u\leq C\inf_{B^{\prime}}u.$

In particular, either $u>0$ on $\Omega$ or $u\equiv 0$ on $\Omega$.

  * $(3a)$

_[Half-Harnack inequalities]_ For any relatively compact, open sets $B\Subset
B^{\prime}\Subset\Omega$ the following holds:

    * _(Subsolutions)_ for each $s>p-1$, there exists $C=C(f,p,m,B,B^{\prime},V,s)>0$ such that for each $u\geq 0$ , $u\in W^{1,p}(\Omega)$ solution of $Q^{\prime}_{V}(u)\leq 0$ on $\Omega$

(3.3) $\sup_{B}u\leq C\|u\|_{L^{s}(B^{\prime})};$

    * _(Supersolutions)_ for each

$s\in\left(0,\frac{(p-1)m}{m-p}\right)\quad\text{if }p<m,\qquad
s\in(0,+\infty)\quad\text{if }p\geq m,$

there exists $C=C(f,p,m,B,B^{\prime},V,s)>0$ such that for each $u\geq 0$ ,
$u\in W^{1,p}(\Omega)$ solution of $Q^{\prime}_{V}(u)\geq 0$ on $\Omega$

(3.4) $\|u\|_{L^{s}(B^{\prime})}\leq C\inf_{B}u.$

  * $(4)$

_[Hopf lemma]_ Suppose that $\xi\geq 0,g\geq 0$ and let $u\in
C^{1}(\overline{\Omega})$ be a solution of (3.1) with $u\geq 0$, $u\not\equiv
0$. If $x\in\partial\Omega$ is such that $u(x)=\xi(x)=0$, then, indicating
with $\nu$ the inward unit normal vector to $\partial\Omega$ at $x$ we have
$\langle\nabla u,\nu\rangle(x)>0$.

######  Remark 3.1.

__

  * (1)

The local boundedness of $u$ is a particular case of Serrin’s theorem, see
[70], Theorem 7.1.1, and does not need the boundary condition. When $\xi\in
C^{2,\alpha}(\partial\Omega)$, global boundedness can be reached via a
reflection technique described at page 54 of [33], see also [77, 11].

  * (2)

is a global version, Theorem 1 of [50], of a local regularity result in [78]
and [27].

  * (3)

is due to J. Serrin, see Theorem 7.2.1 in [70] for $p<m$, the discussion at
the beginning of Section 7.4 therein for $p=m$, and Theorem 7.4.1 for $p>m$.

  * (3a)

The half-Harnack for subsolutions can be found in Theorem 7.1.1 of [70], the
one for supersolutions in the subsequent Theorems 7.1.2 (case $p<m$) and 7.4.1
(case $p>m$) of [70]. Again, see the discussion at the beginning of Section
7.4 of [70].

  * (4)

The Hopf lemma can be found in Corollary 5.5 of [69].

An important tool for our investigation is the following Lagrangian
representation in [67].

######  Proposition 3.1.

For each $\varphi,g\in W^{1,p}_{\mathrm{loc}}(M)$, with $\varphi/g$ a.e.
finite on $M$, the Lagrangian

(3.5)
$\mathcal{L}(\varphi,g)=|\nabla\varphi|^{p}+(p-1)\left(\frac{\varphi}{g}\right)^{p}|\nabla
g|^{p}-p\left(\frac{\varphi}{g}\right)^{p-1}|\nabla g|^{p-2}\langle\nabla
g,\nabla\varphi\rangle$

satisfies $\mathcal{L}(\varphi,g)\geq 0$ on $M$, and
$\mathcal{L}(\varphi,g)\equiv 0$ on some connected open set $U$ if and only if
$\varphi$ is a constant multiple of $g$ on $U$.
Moreover, suppose that $g\in W^{1,p}_{\mathrm{loc}}(M)$ is a positive solution
of $Q_{V}^{\prime}(g)=0$ (resp. $Q_{V}^{\prime}(g)\geq 0$) on $M$. Then, for
$\varphi\in L^{\infty}_{c}(M)\cap W^{1,p}(M)$, $\varphi\geq 0$ it holds

(3.6)
$Q_{V}(\varphi)=\int_{M}\mathcal{L}(\varphi,g)\mathrm{d}\mu_{f}\qquad(\text{resp,
}\geq).$

###### Proof.

The non-negativity of $\mathcal{L}(\varphi,g)$ follows by applying Cauchy-
Schwarz and Young inequalities on the third addendum in (3.5), and analyzing
the equality case, $\mathcal{L}(\varphi,g)\equiv 0$ if and only if
$\varphi=cg$ on $M$, for some constant $c\in\mathbb{R}$.
We now prove the integral (in)equality in (3.6). By Harnack inequality, $g$ is
locally essentially bounded from below on $M$. This, combined with our
regularity requirement on $\varphi$, guarantees that $\varphi^{p}/g^{p-1}\in
W^{1,p}(M)$ and is compactly supported. Thus, we integrate on $M$ the
pointwise identity

$|\nabla\varphi|^{p}-|\nabla g|^{p-2}\langle\nabla
g,\nabla\left(\frac{\varphi^{p}}{g^{p-1}}\right)\rangle=\mathcal{L}(\varphi,g),$

and couple with the weak definition of $Q_{V}^{\prime}(g)=0$ (resp. $\geq 0$)
applied to the test function $\varphi^{p}/g^{p-1}$:

$0=\int_{M}|\nabla g|^{p-2}\langle\nabla
g,\nabla\left(\frac{\varphi^{p}}{g^{p-1}}\right)\rangle\mathrm{d}\mu_{f}-\int_{M}Vg^{p-1}\left(\frac{\varphi^{p}}{g^{p-1}}\right)\mathrm{d}\mu_{f}\qquad\text{(resp.
}\leq\text{)}$

to deduce (3.6). ∎

Rewriting the expression of $\mathcal{L}(\varphi,g)$ we deduce the next useful
Picone type inequality due to [4, 26, 6].

######  Proposition 3.2.

Let $M$ be a Riemannian manifold, and let $\Omega\Subset M$ be a relatively
compact, connected open set. Then, the functional

(3.7) $I(w,z)=\int_{\Omega}|\nabla w|^{p-2}\left\langle\nabla
w,\nabla\frac{w^{p}-z^{p}}{w^{p-1}}\right\rangle\mathrm{d}\mu_{f}-\int_{\Omega}|\nabla
z|^{p-2}\left\langle\nabla
z,\nabla\frac{w^{p}-z^{p}}{z^{p-1}}\right\rangle\mathrm{d}\mu_{f}$

is non-negative on the set

${\cal D}_{\Omega}=\left\\{(w,z)\in W^{1,p}(\Omega)\times W^{1,p}(\Omega)\ :\
w,z\geq 0\ {\rm on}\ \Omega\ ,\ \frac{w}{z},\frac{z}{w}\in
L^{\infty}(\Omega)\right\\}.$

Furthermore, $I(w,z)=0$ if and only if $w=Cz$ on $\Omega$, for some constant
$C>0$.

###### Proof.

Since $w,z\in\mathcal{D}_{\Omega}$, it is easy to see that
$\frac{w^{p}}{z^{p-1}},\frac{z^{p}}{w^{p-1}}\in W^{1,p}(\Omega)$. We can
therefore expand the integrand in (3.7) and rearrange to deduce that

$I(w,z)=\int_{\Omega}\big{[}\mathcal{L}(w,z)+\mathcal{L}(z,w)\big{]}\mathrm{d}\mu_{f},$

with $\mathcal{L}$ as in (3.5). The first part of previous proposition then
gives the desired inequality. ∎

Now, we investigate property $Q_{V}\geq 0$ and its consequences. By its very
definition, $Q_{V}\geq 0$ on an open set $\Omega\subset M$ is equivalent to
the non-negativity of the fundamental tone

(3.8)
$\lambda_{V}(\Omega)=\inf_{0\not\equiv\varphi\in\mathrm{Lip}_{c}(\Omega)}\frac{pQ_{V}(\varphi)}{\|\varphi\|^{p}_{L^{p}(\Omega,\mathrm{d}\mu_{f})}}.$

If $\Omega$ is a relatively compact domain with smooth boundary, then it is
well-known that the infimum (3.8) is attained by a first eigenfunction
$\phi\not\equiv 0$ solving Euler-Lagrange equation

$\left\\{\begin{array}[]{l}\displaystyle
Q_{V}^{\prime}(\phi)=\lambda_{V}(\Omega)|\phi|^{p-2}\phi\qquad\text{on
}\,\Omega,\\\\[5.69046pt] \displaystyle\phi=0\qquad\text{on
}\,\partial\Omega,\end{array}\right.$

and $\phi>0$ on $\Omega$ up to changing its sign 222Briefly, $|\phi|$ still
minimizes the Rayleigh quotient in (3.8), thus it satisfies the Euler-Lagrange
equation $Q_{V}^{\prime}(|\phi|)=\lambda_{V}(\Omega)|\phi|^{p-1}$, hence
$|\phi|>0$ on $\Omega$ by Harnack inequality in Theorem 3.1, $(3)$..
Furthermore, by Harnack inequality, if $\Omega\subset\Omega^{\prime}$ are two
relatively compact open sets and $\Omega^{\prime}\backslash\Omega$ has non-
empty interior, then $\lambda_{V}(\Omega)>\lambda_{V}(\Omega^{\prime})$.

The next comparison result will be used throughout the paper, and improves on
Theorem 5 of [33].

######  Proposition 3.3.

Let $M^{m},p,f$ be as above and, for $A\in L^{\infty}_{\mathrm{loc}}(M)$,
define $Q_{A},Q_{A}^{\prime}$ as in (2.7), (2.8) with $V(x)=A(x)$. Consider a
relatively compact, open set $\Omega\Subset M$ with smooth boundary, and let
$u_{1},u_{2}\in C^{1,\mu}(\overline{\Omega})$, for some $\mu\in(0,1)$.
Furthermore, suppose that, for some non-negative $B\in L^{\infty}(\Omega)$ and
a nonlinearity $F(t)$ satisfying (2.6),

(3.9)
$\left\\{\begin{array}[]{l}\Delta_{p,f}u_{1}+A|u_{1}|^{p-2}u_{1}-BF(u_{1})\geq
0,\\\\[5.69046pt] \Delta_{p,f}u_{2}+A|u_{2}|^{p-2}u_{2}-BF(u_{2})\leq
0,\\\\[5.69046pt] u_{1}\leq u_{2}\ \text{ on }\partial\Omega,\qquad u_{1}\geq
0,\,u_{2}>0\ \text{ on }\Omega.\end{array}\right.$

Then, either

  * $i)$

$u_{1}\leq u_{2}$ on $\Omega$, or

  * $ii)$

$B(x)\equiv 0$ on $\Omega$, $u_{2}$ satisfies $Q_{A}^{\prime}(u_{2})=0$,
$u_{2}\equiv 0$ on $\partial\Omega$, and $\lambda_{A}(\Omega)=0$.

###### Proof.

We let $\displaystyle\xi={u_{2}}_{|\partial\Omega}\in
C^{1,\mu}(\partial\Omega)$ and let

$V=A(x)-B(x)\frac{F(u_{2})}{u_{2}^{p-1}}.$

Note that $V\in L^{\infty}(\Omega)$ by (2.6). For $x\in\partial\Omega$ such
that $u_{2}(x)=0$, we let $\nu$ be the inward unit normal to $\partial\Omega$
at $x$. Then, applying Theorem 3.1 $(4)$ we deduce that

$\langle\nabla u_{2}(x),\nu\rangle(x)>0,$

by continuity, there thus exists a constant $C>0$ such that

(3.10) $u_{1}\leq Cu_{2}\qquad\text{on }T(\partial\Omega),$

for some tubular neighbourhood $T(\partial\Omega)$ of $\partial\Omega$. Using
assumption $u_{2}>0$ on $\Omega$ we can suppose that (3.10) is true on all of
$\overline{\Omega}$ with $C>1$. Because of (3.9) and since $B(x)\geq 0$, $C>1$
and, by (2.6), $F(t)/t^{p-1}$ is increasing on $\mathbb{R}^{+}$, $Cu_{2}$ is
still a supersolution:

$\begin{array}[]{l}\displaystyle\Delta_{p,f}(Cu_{2})+A(Cu_{2})^{p-1}-BF(Cu_{2})=C^{p-1}\big{[}\Delta_{p,f}u_{2}+Au_{2}^{p-1}\big{]}-BF(Cu_{2})\\\\[5.69046pt]
\displaystyle\leq C^{p-1}BF(u_{2})-BF(Cu_{2})\leq
B(Cu_{2})^{p-1}\left[\frac{F(u_{2})}{u_{2}^{p-1}}-\frac{F(Cu_{2})}{(Cu_{2})^{p-1}}\right]\leq
0.\end{array}$

Using $u_{1}\geq 0$ as a subsolution, by (3.10) and applying the method of
sub- and supersolutions, see Theorem 4.14, page 272, in [25], we find a
solution $v$ of

(3.11)
$\left\\{\begin{array}[]{l}\Delta_{p,f}v+A|v|^{p-2}v-BF(v)=0,\\\\[5.69046pt]
v=u_{2}\qquad\text{on }\partial\Omega,\end{array}\right.$

satisfying

(3.12) $u_{1}\leq v\leq Cu_{2}\qquad{\rm on}\ \Omega$

By the $C^{1,\mu}$-regularity of Theorem 3.1, $v\in
C^{1,\alpha}(\overline{\Omega})$ for some $\alpha\in(0,1)$. If we show that
$v\leq u_{2}$ then (3.12) implies $u_{1}\leq u_{2}$ on $\Omega$ which is the
conclusion $i)$ of the Proposition.

Suppose that this is not the case, that is, assume that the open set
$U=\\{v>u_{2}\\}$ is non-empty. We are going to prove that $ii)$ holds. Since
$v\geq u_{1}\geq 0$ and $v$ is positive on $U$, then $v>0$ on $\Omega$ as a
consequence of the Harnack inequality in Theorem 3.1 (use
$V=A(x)-B(x)F(v)/v^{p-1}$, which by (2.6) is bounded on $\Omega$).
Alternatively, one can use the version of the strong maximum principle in
Theorem 5.4.1 in [70]. Now, again by the Hopf Lemma of Theorem 3.1,
$\langle\nabla v(x),\nu(x)\rangle>0$, $\nu$ the inward unit normal to
$\partial\Omega$ at $x$, at each point $x\in\partial\Omega$ where
$v(x)=u_{2}(x)=0$. Hence, the ratio $u_{2}/v$ is well defined at $x$ along the
half line determined by $\nu$. This shows that $u_{2}/v$ and similarly
$v/u_{2}$ are in $L^{\infty}(\Omega)$. Applying Proposition 3.2 on $U$ we
deduce $I(u_{2},v)\geq 0$, and $I(u_{2},v)=0$ if and only if $u_{2}$ and $v$
are proportional on $U$. However, the positivity of the test function
$(v^{p}-u_{2}^{p})/u_{2}^{p-1}>0$ on $U$ implies, by (3.9) and (3.11), that

$0\leq\displaystyle
I(u_{2},v)\leq\displaystyle-\int_{U}B\big{(}u_{2}^{p}-v^{p}\big{)}\left(\frac{F(u_{2})}{u_{2}^{p-1}}-\frac{F(v)}{v^{p-1}}\right)\mathrm{d}\mu_{f}.$

Being $F(t)/t^{p-1}$ strictly increasing on $\mathbb{R}^{+}$ and $B\geq 0$, we
deduce

(3.13)
$B\big{(}u_{2}^{p}-v^{p}\big{)}\left(\frac{F(u_{2})}{u_{2}^{p-1}}-\frac{F(v)}{v^{p-1}}\right)\geq
0$

on $U$, whence $I(u_{2},v)=0$. We therefore conclude that $u_{2}=cv$ on $U$,
for some constant $c$ which, because of the definition of $U$, satisfies
$c>1$. Using that $v=u_{2}$ on $\partial U$, we necessarily have $v=u_{2}=0$
on $\partial U$, hence $U\equiv\Omega$. Substituting $u_{2}=cv$ on $\Omega$
into (3.13) we deduce

$B\left(c^{p}-1\right)v^{p-1}\left(\frac{F(cv)}{(cv)^{p-1}}-\frac{F(v)}{v^{p-1}}\right)\equiv
0.$

Since $v>0$ on $\Omega$ and $F(t)/t^{p-1}$ is strictly increasing, $B\equiv 0$
and, from (3.11), $v>0$ solves

$\left\\{\begin{array}[]{l}\displaystyle\Delta_{p,f}v+A(x)|v|^{p-2}v=0\qquad\text{on
}\,\Omega\\\\[5.69046pt] \displaystyle v=0\qquad\text{on
}\,\partial\Omega.\end{array}\right.$

Consequently, $0$ admits a positive eigenfunction of $Q_{A}^{\prime}$. By a
result in [6], $\lambda_{A}(\Omega)=0$, showing the validity of $ii)$. ∎

######  Remark 3.2.

_We underline that, in the above proposition, the non-negativity of $Q_{A}$ is
not required. However, if $B\equiv 0$, $u_{2}$ turns out to be a positive
solution of $Q_{A}(u)\geq 0$, and using Proposition 3.4 below we automatically
have $\lambda_{A}(\Omega)\geq 0$. _

In what follows we shall frequently use the next formula: for
$I\subset\mathbb{R}$ and $\alpha\in C^{2}(I)$ with $\alpha^{\prime}>0$ on $I$,
and for $u\in C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ with $u(M)\subset I$ we
have, weakly on $M$,

(3.14)
$\Delta_{p,f}\alpha(u)=\alpha^{\prime}(u)\left|\alpha^{\prime}(u)\right|^{p-2}\Delta_{p,f}u+(p-1)\alpha^{\prime\prime}(u)\left|\alpha^{\prime}(u)\right|^{p-2}|\nabla
u|^{p}.$

A second ingredient is the following existence result that goes under the name
of the Allegretto-Piepenbrink theorem, see [5, 4, 33]. We include a proof of
the next slightly more general version, for the sake of completeness.

######  Proposition 3.4.

Let $(M,\langle\,,\,\rangle)$ be a non-compact Riemannian manifold, $f\in
C^{\infty}(M)$, $p\in(1,+\infty)$ and, for $V\in
L^{\infty}_{\mathrm{loc}}(M)$, set $Q^{\prime}_{V},Q_{V}$ as in (2.8) and
(2.7). Then, the following statements are equivalent:

  * $i)$

There exists $w\in C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$, $w>0$ weak
solution of

(3.15) $Q^{\prime}_{V}(w)\geq 0\qquad\text{on }\,M;$

  * $ii)$

There exists $u\in C^{1,\mu}_{\mathrm{loc}}(M)$, $u>0$ weak solution of

(3.16) $Q^{\prime}_{V}(u)=0\qquad\text{on }\,M;$

  * $iii)$

$Q_{V}\geq 0$ on $M$.

  * $iv)$

For each relatively compact domain $\Omega\Subset M$ with $C^{1,\alpha}$
boundary for some $\alpha\in(0,1)$, and for each $\xi\in
C^{1,\alpha}(\partial\Omega)$, $\xi\geq 0$, there exists a unique solution
$\varphi\in C^{1,\mu}(\overline{\Omega})$ of

(3.17) $\left\\{\begin{array}[]{ll}Q_{V}^{\prime}(\varphi)=0&\quad\text{on
}\,\Omega,\\\\[5.69046pt] \varphi=\xi&\quad\text{on
}\,\partial\Omega\end{array}\right.$

satisfying $\varphi\geq 0$ on $\Omega$. Moreover, if $\xi\not\equiv 0$, then
$\varphi>0$ on $\Omega$.

###### Proof.

The scheme of proof is $ii)\Rightarrow i)\Rightarrow iii)\Rightarrow
iv)\Rightarrow ii);$ Note that the first implication is trivial.

$i)\Rightarrow iii)$. It follows immediately from Proposition 3.1 and the non-
negativity of $\mathcal{L}$.

$iii)\Rightarrow iv)$. By assumption $\lambda_{V}(M)\geq 0$; it follows that,
for $\Omega$ as in $iv)$, by the monotonicity property for eigenvalues
$\lambda_{V}(\Omega)>0$. Hence, the variational problem associated to (3.17)
is coercive and sequentially weakly lower-semicontinuous (see also Theorem 7.1
in Appendix). Therefore (3.17) admits a weak solution $\varphi\in
W^{1,p}(\Omega)$. By the $C^{1,\mu}$-regularity of Theorem 3.1 we have that
$\varphi\in C^{1,\mu}(\overline{\Omega})$ for some $\mu\in(0,1)$. Moreover, by
the local Harnack inequality of item $(3)$, $\varphi>0$ on $\overline{\Omega}$
whenever $\phi\geq 0$ on $\Omega$, unless $\varphi\equiv 0$.

By contradiction suppose that $\varphi$ is somewhere negative in $\Omega$.
Since $\xi\geq 0$, on $\partial\Omega$,
$\varphi_{-}=-\min\\{\varphi,0\\}\in\mathrm{Lip}_{0}(\Omega)$ and it is thus
an admissible test function for (3.17) on $\Omega$. We have

$0=\displaystyle
Q_{V}^{\prime}(\varphi)[-\varphi_{-}]\doteq-\int_{\Omega}\left\\{|\nabla\varphi|^{p-2}\langle\nabla\varphi,\nabla\varphi_{-}\rangle-V|\varphi|^{p-2}\varphi\varphi_{-}\right\\}\mathrm{d}\mu_{f}\equiv
pQ_{V}(\varphi_{-})$

and therefore $\lambda_{V}(\Omega)\leq 0$, a contradiction.

$iv)\Rightarrow ii)$. Choose an exhaustion $\\{\Omega_{j}\\}$ of $M$. Let
$u_{j}>0$, $u_{j}\in C^{1,\mu_{j}}(\overline{\Omega}_{j})\subset
W^{1,p}(\Omega_{j})$ be a solution of

(3.18) $\left\\{\begin{array}[]{l}Q_{V}^{\prime}(u_{j})=0\qquad\text{on
}\Omega_{j}\\\\[5.69046pt] u_{j}=1\qquad\text{on
}\partial\Omega_{j}.\end{array}\right.$

Fix $x_{0}\in\Omega_{1}$ and rescale $u_{j}$ in such a way that
$u_{j}(x_{0})=1$ for every $j$. By Theorem 3.1 $3)$, $\\{u_{j}\\}$ is
uniformly locally bounded in $\Omega$, thus by Theorem 3.1 $2)$ $\\{u_{j}\\}$
is uniformly locally bounded in $C^{1,\mu}(\Omega)$. It follows that
$\\{u_{j}\\}$ has a subsequence converging weakly and pointwise to a weak
solution $u\in C^{1,\mu}_{\mathrm{loc}}(M)$ of

$\left\\{\begin{array}[]{l}Q_{V}^{\prime}(u)=0\qquad\text{on }\
M\\\\[5.69046pt] u(x_{0})=1.\end{array}\right.$

Since $u\geq 0$ and $u\not\equiv 0$, again by $3)$ of Theorem 3.1 we deduce
$u>0$ on $M$. This shows the validity of $ii)$. ∎

Next, we need a gluing result which we will call the pasting lemma. Although
for $V\equiv 0$ this is somehow standard (a simple proof can be given by
adapting Lemma 2.4 in [65]), the presence of a nonzero $V$ makes things more
delicate. First, we introduce some definitions. We recall that, given an open
subset $\Omega\subset M$ possibly with non-compact closure, the space
$W^{1,p}_{\mathrm{loc}}(\overline{\Omega})$ is the set of all functions $u$ on
$\Omega$ such that, for every relatively compact open set $U\Subset M$ with
$U\cap\Omega\not=\emptyset$, $u\in W^{1,p}(\Omega\cap U)$. A function $u$ in
this space is thus well behaved on relatively compact portions of
$\partial\Omega$, while no global control is assumed on the $W^{1,p}$ norm of
$u$.

######  Lemma 3.1 (The pasting lemma).

Let $(M,\langle\,,\,\rangle)$ be a Riemannian manifold, $f\in C^{\infty}(M)$,
$p\in(1,\infty)$, $V\in L^{\infty}_{\mathrm{loc}}(M)$. Let $\Omega_{1}$,
$\Omega_{2}$ be open sets such that $\Omega_{1}\subset\Omega_{2}$. For
$j=1,2$, let $u_{j}\in C^{0}(\overline{\Omega}_{j})\cap
W^{1,p}_{\mathrm{loc}}(\overline{\Omega}_{j})$ be a positive supersolution of
$Q_{V}^{\prime}$ on $\Omega_{j}$, that is, $Q_{V}^{\prime}(u_{j})\geq 0$ on
$\Omega_{j}$. If

(3.19) $u_{2}\leq u_{1}\qquad\text{on }\partial\Omega_{1}\cap\Omega_{2},$

then the positive function

(3.20)
$u\doteq\left\\{\begin{array}[]{ll}\displaystyle\min\\{u_{1},u_{2}\\}&{\rm
on}\ \overline{\Omega}_{1}\\\\[11.38092pt] \displaystyle u_{2}&{\rm on}\
\Omega_{2}\backslash\Omega_{1}\end{array}\right.$

is in $W^{1,p}_{\mathrm{loc}}(\overline{\Omega}_{2})$ and it satisfies
$Q^{\prime}_{V}(u)\geq 0$ on $\Omega_{2}$.

When $\Omega_{1}\equiv\Omega_{2}$, a general result of V.K. Le, [46],
guarantees that $\min\\{u_{1},u_{2}\\}$ is a supersolution. The pasting lemma
can then be deduced by an approximation argument, along the lines described in
[2], and we leave the details to the interested reader. In the Appendix below,
we give a quite different proof by using the obstacle problem for $Q_{V}$ and
the minimizing properties of its solutions, that might have an independent
interest.

##  4\. Criticality theory for $Q_{V}$, capacity and Hardy weights

The criticality theory for $Q_{V}$ reveals an interesting scenario, and
extends in a nontrivial way the parabolicity theory for the standard Laplacian
and for the $p$-Laplacian (developed, among others, in [34, 64, 79]). Although
a thorough description goes beyond the scope of this paper, nevertheless the
validity of the pasting lemma gives us the opportunity to complement known
results (especially those in [67, 68]) by relating them to a capacity theory
for $Q_{V}$, see Theorem 4.1 below. We underline that, although the
$Q_{V}$-capacity theory is investigated by following the same lines as those
for the standard $p$-Laplacian, as in the previous results the presence of a
nontrivial $V$ makes things subtler.

Let $Q_{V}\geq 0$ on $M$, and fix a positive _supersolution_ $g\in
C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ of $Q^{\prime}_{V}$, that is, a
solution of

(4.1) $Q_{V}^{\prime}(g)\geq 0.$

For each $K\Subset\Omega\Subset M$, $K$ compact, $\Omega$ open, let

$\mathcal{D}(K,\Omega,g)=\Big{\\{}\varphi\in C^{0}(\overline{\Omega})\cap
W^{1,p}_{0}(\Omega)\ :\ \varphi\geq g\ \text{ in a neighbourhood of
}K\Big{\\}}$

and define the $Q_{V}$-capacity

$\displaystyle\mathrm{cap}_{Q_{V}}(K,\Omega,g)\doteq\displaystyle\inf_{\varphi\in\mathcal{D}(K,\Omega,g)}Q_{V}(\varphi)$

Clearly, $\mathrm{cap}_{Q_{V}}(K,\Omega,g)$ grows if we decrease $\Omega$, as
well as if we increase $K$. If $V\equiv 0$, it is customary to choose $g\equiv
1$ as solution of $\Delta_{p,f}g=0$, and we recover the classical definition
of capacity. We however underline that, for the next arguments to work, it is
essential that the fixed $g$ solves $Q_{V}^{\prime}(g)\geq 0$ on $M$, for
otherwise the basic properties needed in the next results could not hold.

######  Proposition 4.1.

Let $K$ be the closure of an open domain and suppose that
$\partial\Omega,\,\partial K$ are of class $C^{1,\alpha}$ for some
$\alpha\in(0,1)$. Then

(4.2) $\mathrm{cap}_{Q_{V}}(K,\Omega,g)=Q_{V}(u)$

where $u$ is the unique positive solution $u\in C^{0}(\overline{\Omega})\cap
W^{1,p}_{0}(\Omega)\cap C^{1,\mu}(\overline{\Omega}\backslash K)$ of

(4.3) $\left\\{\begin{array}[]{l}Q_{V}^{\prime}(u)=0\qquad\text{on
}\Omega\backslash K,\\\\[5.69046pt] u=g\quad\text{on }K,\qquad
u=0\quad\text{on }\partial\Omega.\end{array}\right.$

We call such a solution $u$ the $Q_{V}$-capacitor of $(K,\Omega,g)$.

######  Remark 4.1.

_The existence, uniqueness and positivity of $u$ is granted by $iv)$ in
Proposition 3.4. As for regularity, the interior estimate $u\in
C^{1,\mu}_{\mathrm{loc}}(\overline{\Omega}\backslash K)$ follows by [78, 27],
and the boundary continuity at $\partial K\cup\partial\Omega$ by Theorem 5.4,
page 235 in [52]. The fact that $u\in W^{1,p}(\Omega)$ follows by standard
theory of Sobolev functions333In fact, $u-g\in W^{1,p}(\Omega\backslash K)$
has zero trace on $\partial K$, thus it is the $W^{1,p}$-limit (and also, up
to extracting a subsequence, the pointwise limit) of some sequence
$\\{\varphi_{j}\\}\subset C^{\infty}(\overline{\Omega\backslash K})$ where
$\varphi_{j}\equiv 0$ in a neighbourhood of $\partial K$. Extending
$\varphi_{j}$ to be zero on $K$ we have that $g+\varphi_{j}$ is Cauchy in
$W^{1,p}$ and pointwise convergent to $u$, thus $g+\varphi_{j}\rightarrow u\in
W^{1,p}(\Omega)$.. _

###### Proof.

Let $\varphi\in\mathcal{D}(K,\Omega,g)$. First, we claim that
$\hat{\varphi}=\min\\{\varphi,g\\}\in\mathcal{D}(K,\Omega,g)$ solves
$Q_{V}(\hat{\varphi})\leq Q_{V}(\varphi)$, whence we can assume, without loss
of generality, that $\varphi\leq g$ on $\Omega$ (and hence $\varphi=g$ on a
neighbourhood of $K$).

Consider the open set $U=\\{\varphi>g\\}\Subset\Omega$. We test
$Q_{V}^{\prime}(g)\geq 0$ with the non-negative function
$(\varphi^{p}-g^{p})_{+}/g^{p-1}\in W^{1,p}_{0}(\Omega)$ and use the non-
negativity of the Lagrangian in (3.5) to deduce

(4.4) $\begin{array}[]{lcl}0&\leq&\displaystyle\displaystyle\int_{U}|\nabla
g|^{p-2}\langle\nabla
g,\nabla\left(\frac{\varphi^{p}}{g^{p-1}}\right)\rangle\mathrm{d}\mu_{f}-\int_{U}V\varphi^{p}\mathrm{d}\mu_{f}-pQ_{V}\big{(}g_{|U}\big{)}\\\\[11.38092pt]
&=&\displaystyle p\int_{U}\left(\frac{\varphi}{g}\right)^{p-1}|\nabla
g|^{p-2}\langle\nabla
g,\nabla\varphi\rangle\mathrm{d}\mu_{f}-(p-1)\int_{U}\left(\frac{\varphi}{g}\right)^{p}|\nabla
g|^{p}\mathrm{d}\mu_{f}\\\\[11.38092pt]
&&\displaystyle-\int_{U}V\varphi^{p}\mathrm{d}\mu_{f}-pQ_{V}\big{(}g_{|U}\big{)}\\\\[11.38092pt]
&=&\displaystyle\int_{U}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}-\int_{U}\mathcal{L}(\varphi,g)\mathrm{d}\mu_{f}-\int_{U}V\varphi^{p}\mathrm{d}\mu_{f}-pQ_{V}\big{(}g_{|U}\big{)}\\\\[11.38092pt]
&\leq&\displaystyle
pQ_{V}\big{(}\varphi_{|U}\big{)}-pQ_{V}\big{(}g_{|U}\big{)},\end{array}$

hence $Q_{V}(\hat{\varphi}_{|U})=Q_{V}(g_{|U})\leq Q_{V}(\varphi_{|U})$. Since
$\varphi\equiv\hat{\varphi}$ on $\Omega\backslash U$, the claim follows. Let
now $\varphi\in\mathcal{D}(K,\Omega,g)$ be such that $\varphi=g$ on $K$. By
density, we can assume that $\varphi\in\mathrm{Lip}_{0}(\Omega)$. We therefore
have $u-\varphi\in\mathrm{Lip}_{0}(\Omega\backslash K)$. Again by density, we
can further assume that $\varphi=0$ in a neighbourhood of $\partial\Omega$.
Thus, testing $Q_{V}^{\prime}(u)=0$ with
$(\varphi^{p}-u^{p})/u^{p-1}\in\mathrm{Lip}_{0}(\Omega\backslash K)$ and
proceeding as above we obtain

(4.5)
$\begin{array}[]{lcl}0&=&\displaystyle\displaystyle\int_{\Omega\backslash
K}|\nabla u|^{p-2}\langle\nabla
u,\nabla\left(\frac{\varphi^{p}}{u^{p-1}}\right)\rangle\mathrm{d}\mu_{f}-\int_{\Omega\backslash
K}V\varphi^{p}\mathrm{d}\mu_{f}-pQ_{V}\big{(}u_{|\Omega\backslash
K}\big{)}\\\\[11.38092pt] &=&\displaystyle p\int_{\Omega\backslash
K}\left(\frac{\varphi}{u}\right)^{p-1}|\nabla u|^{p-2}\langle\nabla
u,\nabla\varphi\rangle\mathrm{d}\mu_{f}-(p-1)\int_{\Omega\backslash
K}\left(\frac{\varphi}{u}\right)^{p}|\nabla
u|^{p}\mathrm{d}\mu_{f}\\\\[11.38092pt] &&\displaystyle-\int_{\Omega\backslash
K}V\varphi^{p}\mathrm{d}\mu_{f}-pQ_{V}\big{(}u_{|\Omega\backslash
K}\big{)}\\\\[11.38092pt] &=&\displaystyle\int_{\Omega\backslash
K}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}-\int_{\Omega\backslash
K}\mathcal{L}(\varphi,u)\mathrm{d}\mu_{f}-\int_{\Omega\backslash
K}V\varphi^{p}\mathrm{d}\mu_{f}-pQ_{V}\big{(}u_{|\Omega\backslash
K}\big{)}\\\\[11.38092pt] &\leq&\displaystyle
pQ_{V}\big{(}\varphi_{|\Omega\backslash
K}\big{)}-pQ_{V}\big{(}u_{|\Omega\backslash K}\big{)}.\end{array}$

As $u=\varphi=g$ on $K$, we conclude $Q_{V}(u)\leq Q_{V}(\varphi)$ and whence
$Q_{V}(u)\leq\mathrm{cap}(K,\Omega,g)$. Since $u$ lies in the $W^{1,p}$
closure of $\mathcal{D}(K,\Omega,g)$, equality (4.2) follows. ∎

######  Remark 4.2.

_By the pasting Lemma 3.1, note that $u$ solving (4.3) is a supersolution on
the whole $\Omega$, that is, $Q_{V}^{\prime}(u)\geq 0$ on $\Omega$. _

######  Proposition 4.2.

In the assumptions of the previous theorem, suppose that $Q_{V}^{\prime}(g)=0$
on a neighbourhood of $K$, and that $\partial K$ is smooth. Then,

(4.6) $Q_{V}(u)=\frac{1}{p}\int_{\partial K}g\left[|\nabla
g|^{p-2}\frac{\partial g}{\partial\nu}-|\nabla u|^{p-2}\frac{\partial
u}{\partial\nu}\right]\mathrm{d}\sigma_{f},$

where $\nu$ is the unit normal to $\partial K$ pointing outward of $K$.

###### Proof.

Let $T\approx\partial K\times(-\varepsilon_{0},\varepsilon_{0})\Subset\Omega$
be a tubular neighbourhood of $\partial K$ where Fermi coordinates are
defined, and let $\rho(x)$ be the smooth signed distance from $\partial K$,
that is, $\rho(x)=\mathrm{dist}(x,\partial K)$ if $x\not\in K$, and
$\rho(x)=-\mathrm{dist}(x,\partial K)$ if $x\in K$. Let
$h\in\mathrm{Lip}(\mathbb{R}^{+}_{0})$ be such that $h(t)=0$ if $t\leq 0$,
$h(t)=t$ for $t\in[0,1]$ and $h(t)=1$ for $t\geq 1$ and, for small
$\varepsilon\in(0,\varepsilon_{0})$, set $h_{\varepsilon}(t)\doteq
h(t/\varepsilon)$. Applying (4.3) on $\Omega\backslash K$ to the test function
$h_{\varepsilon}(\rho)u\in\mathrm{Lip}_{0}(\Omega\backslash K)$, using the
coarea’s formula and letting $\varepsilon\rightarrow 0$, since $g$ is $C^{1}$
we deduce

(4.7) $\displaystyle 0=\lim_{\varepsilon\rightarrow
0}Q_{V}^{\prime}(u)\big{[}h_{\varepsilon}(\rho)u\big{]}=p\displaystyle
Q_{V}\big{(}u_{\Omega\backslash K}\big{)}+\int_{\partial K}u|\nabla
u|^{p-2}\frac{\partial u}{\partial\nu}\mathrm{d}\sigma_{f}.$

In a similar way, applying $Q_{V}^{\prime}(g)=0$ on $K$ to the non-negative
test function $h_{\varepsilon}(-\rho)g\in\mathrm{Lip}_{0}(K)$ and letting
$\varepsilon\rightarrow 0$ we deduce that

(4.8) $\displaystyle\displaystyle 0=\lim_{\varepsilon\rightarrow
0}Q_{V}^{\prime}(u)\big{[}h_{\varepsilon}(-\rho)g\big{]}=\displaystyle
pQ_{V}\big{(}u_{K}\big{)}-\int_{\partial K}g|\nabla g|^{p-2}\frac{\partial
g}{\partial\nu}\mathrm{d}\sigma_{f}.$

Subtracting the two identities and using $u=g$ on $\partial K$ yields (4.6). ∎

Next, we consider the $Q_{V}$-capacity of $K$ in the whole $M$:

(4.9) $\begin{array}[]{c}\displaystyle\mathcal{D}(K,g)=\Big{\\{}\varphi\in
C^{0}_{c}(M)\cap W^{1,p}_{\mathrm{loc}}(M)\ :\ \varphi\geq g\ \text{ in a
neighbourhood of }K\Big{\\}}\\\\[11.38092pt]
\displaystyle\mathrm{cap}_{Q_{V}}(K,g)\doteq\displaystyle\inf_{\varphi\in\mathcal{D}(K,g)}Q_{V}(\varphi).\end{array}$

Let $\\{\Omega_{j}\\}$ be an exhaustion of $M$ with $K\Subset\Omega_{1}$.
Then, from the definitions it readily follows that

$\mathrm{cap}_{Q_{V}}(K,g)=\inf_{j}\mathrm{cap}_{Q_{V}}(K,\Omega_{j},g)=\lim_{j\rightarrow+\infty}\mathrm{cap}_{Q_{V}}(K,\Omega_{j},g).$

If $K$ is the closure of a open set and $\partial K$ is of class
$C^{1,\alpha}$ for some $\alpha\in(0,1)$, let $u_{j}$ be the $Q_{V}$-capacitor
of $(K,\Omega_{j},g)$. By Proposition 4.1,

(4.10) $\mathrm{cap}_{Q_{V}}(K,g)=\lim_{j\rightarrow+\infty}Q_{V}(u_{j}).$

Proposition 3.3 implies that $0\leq u_{j}\leq u_{j+1}\leq g$ for each $j$,
whence, by Dini theorem and elliptic estimates, $u_{j}$ converges locally
uniformly on $M$, in $W^{1,p}_{\mathrm{loc}}(M)$ and in the $C^{1}$ topology
on $M\backslash K$ to a weak solution $u\in C^{0}(M)\cap
W^{1,p}_{\mathrm{loc}}(M)\cap C^{1,\mu}_{\mathrm{loc}}(M\backslash K)$ of

(4.11) $\left\\{\begin{array}[]{l}Q_{V}^{\prime}(u)=0\qquad\text{on
}M\backslash K,\\\\[5.69046pt] u=g\quad\text{on }K,\qquad 0<u\leq
g\quad\text{on }M\backslash K.\end{array}\right.$

The pasting Lemma 3.1 guarantees that $Q_{V}^{\prime}(u)\geq 0$ on the whole
$M$. We call such a $u$ the $Q_{V}$-capacitor of $(K,g)$.

######  Proposition 4.3.

In the assumptions of Proposition 4.2, if $Q_{V}^{\prime}(g)=0$ on a
neighbourhood of $K$,

(4.12) $\mathrm{cap}_{Q_{V}}(K,g)=\frac{1}{p}\int_{\partial K}g\left[|\nabla
g|^{p-2}\frac{\partial g}{\partial\nu}-|\nabla u|^{p-2}\frac{\partial
u}{\partial\nu}\right]\mathrm{d}\sigma_{f},$

where $\nu$ is the unit normal to $\partial K$ pointing outward of $K$.

###### Proof.

By Proposition 4.1, $\mathrm{cap}_{Q_{V}}(K,g)=\lim_{j}Q_{V}(u_{j})$, where
$u_{j}$ is the $Q_{V}$-capacitor of $(K,\Omega_{j},g)$. Now, since
$u_{j}\rightarrow u$ in $C^{1}(\partial K)$, it is enough to pass to the limit
in 4.6. ∎

Next Theorem, the core of this section, relates the subcriticality of $Q_{V}$
and the $Q_{V}$-capacity with other basic properties, which we will define
below. It is due to Y. Pinchover and K. Tintarev (see [67]), and it is known
in the literature as the ground state alternative. The authors state it for
$f$ constant and $M=\mathbb{R}^{m}$. Our contribution here is to include the
$Q_{V}$-capacity properties to the above picture. However, since at some point
of [67] the authors use inequalities for which we found no counterpart in a
manifold setting, we prefer to provide a full proof which sometimes uses
arguments that differ from those in [67, 68], though keeping the same
guidelines.

######  Definition 4.1.

For $V\in L^{\infty}_{\mathrm{loc}}(M)$, define $Q_{V}$ as in (2.7) and let
$\Omega\subseteq M$ be an open set.

  * iii)

$Q_{V}$ has a _weighted spectral gap_ on $\Omega$ if there exists $W\in
C^{0}(\Omega)$, $W>0$ on $\Omega$ such that

(4.13) $\displaystyle\int_{\Omega}W(x)|\varphi|^{p}\mathrm{d}\mu_{f}\leq
Q_{V}(\varphi)\qquad\forall\ \varphi\in\mathrm{Lip}_{c}(\Omega).$

  * iv)

A sequence $\\{\eta_{j}\\}\in L^{\infty}_{c}(\Omega)\cap W^{1,p}(\Omega)$ is
said to be a _null sequence_ if $\eta_{j}\geq 0$ a.e. for each $j$,
$Q_{V}(\eta_{j})\rightarrow 0$ as $j\rightarrow+\infty$ and there exists a
relatively compact open set $B\Subset M$ and $C>1$ such that
$C^{-1}\leq\|\eta_{j}\|_{L^{p}(B)}\leq C$ for each $j$.

  * v)

A function $0\leq\eta\in W^{1,p}_{\mathrm{loc}}(\Omega)$, $\eta\geq 0$,
$\eta\not\equiv 0$ is a _ground state_ for $Q_{V}$ on $\Omega$ if it is the
$L^{p}_{\mathrm{loc}}(\Omega)$ limit of a null sequence.

######  Theorem 4.1.

Let $(M,\langle\,,\,\rangle)$ be connected and non-compact, and for $V\in
L^{\infty}_{\mathrm{loc}}(M)$ consider an operator $Q_{V}\geq 0$ on $M$. Then,
either $Q_{V}$ has a weighted spectral gap or a ground state on $M$, and the
two possibilities mutually exclude. Moreover, the following properties are
equivalent:

  * $(i)_{\mathrm{S}}$

$Q_{V}$ has a weighted spectral gap.

  * $(ii)_{\mathrm{S}}$

$Q_{V}$ is subcritical on $M$.

  * $(iii)_{\mathrm{S}}$

There exist two positive solutions $u_{1},u_{2}\in C^{0}(M)\cap
W^{1,p}_{\mathrm{loc}}(M)$ of $Q^{\prime}_{V}(u)\geq 0$ which are not
proportional.

  * $(iv)_{\mathrm{S}}$

For some (any) $K\Subset M$ compact with non-empty interior, and for some
(any) $0<g\in C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ solving
$Q_{V}^{\prime}(g)\geq 0$, $\mathrm{cap}(K,g)>0$.

When $Q_{V}$ has a ground state $\eta$, $\eta$ solves
$Q_{V}^{\prime}(\eta)=0$, and in particular $\eta\in
C^{1,\mu}_{\mathrm{loc}}(M)$, $\eta>0$ on $M$. Furthermore, the next
properties are equivalent:

  * $(i)_{\mathrm{GS}}$

$Q_{V}$ has a ground state.

  * $(ii)_{\mathrm{GS}}$

All positive solutions $g\in C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ of
$Q_{V}^{\prime}(g)\geq 0$ are proportional; in particular, each positive
supersolution is indeed a solution (hence, a ground state).

  * $(iii)_{\mathrm{GS}}$

For some (any) $K\Subset M$ compact with non-empty interior, and for some
(any) $0<g\in C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ solving
$Q_{V}^{\prime}(g)\geq 0$, $\mathrm{cap}(K,g)=0$.

###### Proof.

Hereafter, $L^{p}$ and $W^{1,p}$ spaces will be considered with respect to the
measure $\mathrm{d}\mu_{f}$. We begin with the following fact.
Claim 1:  Fix an open set $U\Subset M$. If $\\{\phi_{j}\\}\subset
L^{\infty}_{c}(M)\cap W^{1,p}(M)$ is such that
$\|\phi_{j}\|_{L^{p}(U)}+Q_{V}(\phi_{j})\leq C$ for some $C>0$ independent of
$j$, then $\\{\phi_{j}\\}$ is locally bounded in $W^{1,p}(M)$.
Proof of Claim 1.  Up to replacing $\phi_{j}$ with $|\phi_{j}|$, we can assume
that $\phi_{j}\geq 0$ a.e. on $M$. Using that $Q_{V}$ is non-negative on $M$,
choose a positive solution $g\in C^{1,\mu}_{\mathrm{loc}}(M)$ of
$Q_{V}^{\prime}(g)=0$ and consider the Lagrangian representation of
$Q_{V}(\phi_{j})$:

$Q_{V}(\phi_{j})=\int_{M}\mathcal{L}(\phi_{j},g)\mathrm{d}\mu_{f}.$

Fix $\Omega\Subset M$ containing $\overline{U}$. Since $Q_{V}(\phi_{j})\leq C$
and $\mathcal{L}(\phi_{j},g)\geq 0$ on $M$, it holds

(4.14)
$0\leq\limsup_{j\rightarrow+\infty}\int_{\Omega}\mathcal{L}(\phi_{j},g)\mathrm{d}\mu_{f}\leq
C.$

By using Cauchy-Schwarz and Young inequalities on the third addendum of the
expression of $\mathcal{L}(\phi_{j},g)$ we deduce that, for each
$\varepsilon>0$,

(4.15)
$\mathcal{L}(\phi_{j},g)\geq(1-\varepsilon^{p})|\nabla\phi_{j}|^{p}+(p-1)\left(1-\varepsilon^{-\frac{p}{p-1}}\right)\left(\frac{\phi_{j}}{g}\right)^{p}|\nabla
g|^{p}.$

In our assumptions, $|\nabla\log g|\in L^{\infty}(\Omega)$. Setting
$\varepsilon=1/2$ in (4.15), integrating and using (4.14) we get

(4.16)
$\limsup_{j\rightarrow+\infty}\left[\left(1-2^{-p}\right)\int_{\Omega}|\nabla\phi_{j}|^{p}\mathrm{d}\mu_{f}+(p-1)\left(1-2^{\frac{p}{p-1}}\right)\|\nabla\log
g\|_{L^{\infty}(\Omega)}\int_{\Omega}|\phi_{j}|^{p}\mathrm{d}\mu_{f}\right]\leq
C.$

From this inequality we argue the existence of constants $C_{1},C_{2}>0$
independent of $j$ such that

(4.17) $\|\nabla\phi_{j}\|_{L^{p}(\Omega)}\leq
C_{1}\|\phi_{j}\|_{L^{p}(\Omega)}+C_{2}.$

Suppose now, by contradiction, that $\\{\phi_{j}\\}$ is not bounded in
$W^{1,p}(\Omega)$. By (4.17), $\|\phi_{j}\|_{L^{p}(\Omega)}$ diverge. Set
$\bar{\phi}_{j}=\phi_{j}/\|\phi_{j}\|_{L^{p}(\Omega)}$. Then, by (4.17)
$\\{\bar{\phi}_{j}\\}$ is bounded in $W^{1,p}(\Omega)$, thus it has a
subsequence (still called $\\{\bar{\phi}_{j}\\}$) converging weakly in
$W^{1,p}$, strongly in $L^{p}$ and pointwise almost everywhere to some non-
negative function $\Psi$ satisfying
$\|\Psi\|_{L^{p}(\Omega)}=\lim_{j}\|\bar{\phi}_{j}\|_{L^{p}(\Omega)}=1$. Since
$|\nabla\log g|\in L^{\infty}(\Omega)$ we straightforwardly have that

(4.18) $\begin{array}[]{rll}(i)&\bar{\phi}_{j}^{p}|\nabla\log
g|^{p}\rightarrow\Psi^{p}|\nabla\log g|^{p}&\quad\text{in
}L^{1}(\Omega),\\\\[5.69046pt] (ii)&\bar{\phi}_{j}^{p-1}|\nabla\log
g|^{p-1}\rightarrow\Psi^{p-1}|\nabla\log g|^{p-1}&\quad\text{in
}L^{\frac{p}{p-1}}(\Omega).\end{array}$

We just prove $(ii)$, the other being a consequence of the proof. By the
elementary inequalities

(4.19) $(x-y)^{\beta}\leq x^{\beta}-y^{\beta}\leq
Cx^{\beta-1}(x-y)\qquad\text{for }y\in[0,x]\text{ and }\beta>1,$

for some $C$ depending on $\beta$, with a final application of Hölder
inequality we deduce that

(4.20)
$\begin{array}[]{l}\displaystyle\|(\bar{\phi}_{j}^{p-1}-\Psi^{p-1})|\nabla\log
g|^{p-1}\|_{L^{p/(p-1)}(\Omega)}^{p/(p-1)}\leq\|\nabla\log
g\|_{L^{\infty}(\Omega)}^{p}\int_{\Omega}\big{|}\bar{\phi}_{j}^{p-1}-\Psi^{p-1}\big{|}^{\frac{p}{p-1}}\mathrm{d}\mu_{f}\\\\[8.5359pt]
\displaystyle\leq\|\nabla\log
g\|_{L^{\infty}(\Omega)}^{p}\int_{\Omega}\big{|}\bar{\phi}_{j}^{p}-\Psi^{p}\big{|}\mathrm{d}\mu_{f}\leq
C\|\nabla\log
g\|^{p}_{L^{\infty}(\Omega)}\int_{\Omega}\big{(}\max\\{\bar{\phi}_{j},\Psi\\}\big{)}^{p-1}\big{|}\bar{\phi}_{j}-\Psi|\mathrm{d}\mu_{f},\\\\[8.5359pt]
\displaystyle\leq C\|\nabla\log
g\|^{p}_{L^{\infty}(\Omega)}\left\|\bar{\phi}_{j}+\Psi\right\|_{L^{p}(\Omega)}^{(p-1)/p}\|\bar{\phi}_{j}-\Psi\|_{L^{p}(\Omega)},\end{array}$

and this latter goes to zero $\bar{\phi}_{j}\rightarrow\Psi$ in
$L^{p}(\Omega)$. Now, coupling $(ii)$ with the weak convergence of
$\bar{\phi}_{j}$ to $\Psi$ in $W^{1,p}(\Omega)$, we get

$\int_{\Omega}\left(\frac{\bar{\phi}_{j}}{g}\right)^{p-1}|\nabla
g|^{p-2}\langle\nabla
g,\nabla\bar{\phi}_{j}\rangle\mathrm{d}\mu_{f}\longrightarrow\int_{\Omega}\left(\frac{\Psi}{g}\right)^{p-1}|\nabla
g|^{p-2}\langle\nabla g,\nabla\Psi\rangle\mathrm{d}\mu_{f},$

so that, combining $(i),(ii)$ and the weak lower semicontinuity of
$\|\cdot\|_{W^{1,p}(\Omega)}$,

$0\leq\int_{\Omega}\mathcal{L}(\Psi,g)\mathrm{d}\mu_{f}\leq\liminf_{j\rightarrow+\infty}\int_{\Omega}\mathcal{L}(\bar{\phi}_{j},g)\mathrm{d}\mu_{f}=\|\phi_{j}\|^{-p}_{L^{p}(\Omega)}\int_{\Omega}\mathcal{L}(\phi_{j},g)\mathrm{d}\mu_{f}\rightarrow
0$

as $j\rightarrow+\infty$. Hence $\mathcal{L}(\Psi,g)=0$ on $\Omega$, so by
Proposition 3.1 $\Psi=cg$ for some constant $c\geq 0$. However, since
$\|\phi_{j}\|_{L^{p}(U)}\leq C$ for each $j$,

$0\leq\|\Psi\|_{L^{p}(U)}=\lim_{j\rightarrow+\infty}\|\bar{\phi}_{j}\|_{L^{p}(U)}=\lim_{j\rightarrow+\infty}\frac{\|\phi_{j}\|_{L^{p}(U)}}{\|\phi_{j}\|_{L^{p}(\Omega)}}\leq\limsup_{j\rightarrow+\infty}\frac{C}{\|\phi_{j}\|_{L^{p}(\Omega)}}=0,$

hence $c=0$ and $\Psi\equiv 0$ on $\Omega$, contradicting the fact that
$\|\Psi\|_{L^{p}(\Omega)}=1$. This concludes the proof of the claim.
Claim 2:  Either $Q_{V}$ has a weighted spectral gap or a ground state, but
not both.
Proof of Claim 2. For a relatively compact open set $U$, define the
$L^{p}$-capacity $c$ as follows:

$\mathcal{D}(U)=\Big{\\{}\varphi\in L^{\infty}_{c}(M)\cap
W^{1,p}(M):\|\varphi\|_{L^{p}(U)}=1\Big{\\}},\qquad
c(U)=\inf_{\varphi\in\mathcal{D}(U)}Q_{V}(\varphi),$

where, as usual, the $L^{p}$-norm is computed with respect to
$\mathrm{d}\mu_{f}$. Then, two mutually exclusive cases may occur: ether
$c(U)>0$ for each $U$, or $c(U)=0$ for some $U$. In the first case, it is easy
to see that $Q_{V}$ has a weighted spectral gap. Indeed, let $\\{U_{j}\\}$ be
a locally finite covering of $M$ via relatively compact, open sets, set
$c_{j}=c(U_{j})>0$ and let $\\{t_{j}\\}$ be a sequence of positive numbers
such that $\sum_{j}t_{j}=1$. For each $\varphi\in\mathrm{Lip}_{c}(M)$, by the
definition of $c_{j}$ we get $Q_{V}(\varphi)\geq
c_{j}\|\varphi\|_{L^{p}(U_{j})}^{p}$, and summing up we get

$Q_{V}(\varphi)=\left[\sum_{j=1}^{+\infty}t_{j}\right]Q_{V}(\varphi)\geq\sum_{j=1}^{+\infty}t_{j}c_{j}\int_{U_{j}}|\varphi|^{p}\mathrm{d}\mu_{f}=\int_{M}\hat{W}|\varphi|^{p}\mathrm{d}\mu_{f},$

where

$\hat{W}(x)=\sum_{j=1}^{+\infty}t_{j}c_{j}1_{U_{j}}(x)>0\qquad\text{on }M.$

We can thus choose a weighted spectral gap $W$ by taking a positive,
continuous function $W$ not exceeding $\hat{W}$.
Now, suppose that $c(U)=0$ for some $U$. We show that there exists a ground
state. Indeed, by the definition of $c(U)$ there exists $\\{\eta_{j}\\}\subset
L^{\infty}_{c}(M)\cap W^{1,p}(M)$ such that $\|\eta_{j}\|_{L^{p}(U)}=1$ and
$Q_{V}(\eta_{j})\rightarrow 0$. Up to replacing $\eta_{j}$ with $|\eta_{j}|$,
we can suppose that $\eta_{j}\geq 0$ a.e. on $M$. By Claim 1, $\eta_{j}$ is
locally bounded in $W^{1,p}$, and a Cantor type argument on an increasing
exhaustion of $M$ ensures the existence of a subsequence, still called
$\\{\eta_{j}\\}$, converging weakly in $W^{1,p}_{\mathrm{loc}}(M)$ and
strongly in $L^{p}_{\mathrm{loc}}(M)$ to some function $\eta\in
W^{1,p}_{\mathrm{loc}}(M)$. By definition, $\eta$ is a ground state.
We now show our desired equivalences. To establish those involving
$(iv)_{\mathrm{S}}$ and $(iii)_{\mathrm{GS}}$, where the “some/all"
alternative appears, then we will always assume the weakest alternative and
prove the strongest one.
$\mathbf{(i)_{\mathrm{GS}}\Rightarrow(ii)_{\mathrm{GS}}}$. Let $\eta\geq 0$ be
a ground state, and let $\\{\eta_{j}\\}\subset L^{\infty}_{c}(M)\cap
W^{1,p}(M)$ be a null sequence converging in $L^{p}_{\mathrm{loc}}$ to $\eta$.
Then, by Claim 1 $\\{\eta_{j}\\}$ is locally bounded in $W^{1,p}$, thus up to
passing to a subsequence we can assume that also $\eta_{j}\rightarrow\eta$
weakly in $W^{1,p}_{\mathrm{loc}}$. Consider a positive solution $g\in
C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ of $Q_{V}^{\prime}(g)\geq 0$. Fix
$\Omega\Subset M$. Up to multiply $g$ by a large positive constant, we can
suppose that $\\{x\in\Omega:\eta(x)<g(x)\\}$ has positive measure. Let
$\bar{\eta}_{j}=\min\\{\eta_{j},g\\}$, and note that $\\{\bar{\eta}_{j}\\}$ is
still a null sequence, converging weakly in $W^{1,p}_{\mathrm{loc}}$ to
$\bar{\eta}=\min\\{\eta,g\\}$. Consider the Lagrangian representation

(4.21)
$Q_{V}(\bar{\eta}_{j})\geq\int_{M}\mathcal{L}(\bar{\eta}_{j},g)\mathrm{d}\mu_{f}$

guaranteed by Proposition 3.1. We claim that

(4.22)
$\frac{\bar{\eta}_{j}^{p}}{g^{p-1}}\rightarrow\frac{\bar{\eta}^{p}}{g^{p-1}}\qquad\text{weakly
in }W^{1,p}(\Omega).$

To see this, we follows arguments analogous to those yielding (4.18). Choose a
constant $c_{\Omega}>0$ large enough to satisfy $g\geq c_{\Omega}^{-1}$ on
$\Omega$ and $\bar{\eta}_{j}\leq c_{\Omega}$. By a direct computation,
$\|\bar{\eta}_{j}^{p}/g^{p-1}\|_{W^{1,p}(\Omega)}$ is uniformly bounded so
that, by density, it is enough to check the weak convergence with test
function $\varphi\in\mathrm{Lip}_{0}(\Omega)$. From

$\left|\int_{\Omega}\frac{\bar{\eta}_{j}^{p}-\bar{\eta}^{p}}{g^{p-1}}\varphi\mathrm{d}\mu_{f}\right|\leq
c_{\Omega}^{p-1}\|\varphi\|_{L^{\infty}(\Omega)}\int_{\Omega}\big{|}\bar{\eta}_{j}^{p}-\bar{\eta}^{p}\big{|}\mathrm{d}\mu_{f},$

applying the inequalities in (4.20) from the second line to the end we deduce
that $\bar{\eta}_{j}^{p}/g^{p-1}\rightarrow\bar{\eta}^{p}/g^{p-1}$ weakly in
$L^{p}(\Omega)$. Regarding the gradient part,

$\left|\int_{\Omega}\langle\nabla\left(\frac{\bar{\eta}_{j}^{p}-\bar{\eta}^{p}}{g^{p-1}}\right),\nabla\varphi\rangle\mathrm{d}\mu_{f}\right|\leq\mathrm{(I)}+\mathrm{(II)},$

where

$\begin{array}[]{rcl}\mathrm{(I)}&=&\displaystyle(p-1)\int_{\Omega}\big{|}\bar{\eta}_{j}^{p}-\bar{\eta}^{p}\big{|}|\nabla
g|\frac{|\nabla\varphi|}{g^{p}}\\\\[8.5359pt] \mathrm{(II)}&=&\displaystyle
p\left|\int_{\Omega}\langle\left(\frac{\bar{\eta}_{j}}{g}\right)^{p-1}\nabla\varphi,\nabla\bar{\eta}_{j}\rangle\mathrm{d}\mu_{f}-\int_{\Omega}\langle\left(\frac{\bar{\eta}}{g}\right)^{p-1}\nabla\varphi,\nabla\bar{\eta}\rangle\mathrm{d}\mu_{f}\right|.\end{array}$

As for (I), by Hölder and both the inequalities in (4.19) we deduce

$\begin{array}[]{rcl}\frac{1}{p-1}\mathrm{(I)}&\leq&\displaystyle
c_{\Omega}^{p}\|\nabla\varphi\|_{L^{\infty}(\Omega)}\|\nabla
g\|_{L^{p}(\Omega)}\left(\int_{\Omega}\big{|}\bar{\eta}_{j}^{p}-\bar{\eta}^{p}\big{|}^{\frac{p}{p-1}}\mathrm{d}\mu_{f}\right)^{\frac{p-1}{p}}\\\\[8.5359pt]
&\leq&\displaystyle
c_{\Omega}^{p}\|\nabla\varphi\|_{L^{\infty}(\Omega)}\|\nabla
g\|_{L^{p}(\Omega)}\left(\int_{\Omega}\left|\bar{\eta}_{j}^{\frac{p^{2}}{p-1}}-\bar{\eta}^{\frac{p^{2}}{p-1}}\right|\mathrm{d}\mu_{f}\right)^{\frac{p-1}{p}}\\\\[8.5359pt]
&\leq&\displaystyle
c_{\Omega}^{p}\|\nabla\varphi\|_{L^{\infty}(\Omega)}\|\nabla
g\|_{L^{p}(\Omega)}C^{\frac{p-1}{p}}\left(\int_{\Omega}\max\\{\bar{\eta}_{j},\bar{\eta}\\}^{\frac{p^{2}}{p-1}-1}\left|\bar{\eta}_{j}-\bar{\eta}\right|\mathrm{d}\mu_{f}\right)^{\frac{p-1}{p}}\\\\[8.5359pt]
&\leq&\displaystyle
c_{\Omega}^{p+\frac{p}{p-1}-\frac{p-1}{p}}\|\nabla\varphi\|_{L^{\infty}(\Omega)}\|\nabla
g\|_{L^{p}(\Omega)}C^{\frac{p-1}{p}}\left(\int_{\Omega}\left|\bar{\eta}_{j}-\bar{\eta}\right|\mathrm{d}\mu_{f}\right)^{\frac{p-1}{p}}.\end{array}$

Again by Hölder inequality, the last integral goes to zero since
$\bar{\eta}_{j}\rightarrow\bar{\eta}$ in $L^{p}(\Omega)$, which shows that
(I)$\rightarrow 0$ as $j\rightarrow+\infty$. Finally, we consider (II). To
show that (II)$\rightarrow 0$, using the weak convergence if $\bar{\eta}_{j}$
to $\bar{\eta}$ in $W^{1,p}(\Omega)$ and standard estimates it is enough to
prove that

$\left(\frac{\bar{\eta}_{j}}{g}\right)^{p-1}\nabla\varphi\rightarrow\left(\frac{\bar{\eta}}{g}\right)^{p-1}\nabla\varphi\qquad\text{strongly
in }L^{\frac{p}{p-1}}(\Omega).$

This follows from

$\int_{\Omega}|\nabla\varphi|^{\frac{p}{p-1}}\left|\left(\frac{\bar{\eta}_{j}}{g}\right)^{p-1}-\left(\frac{\bar{\eta}}{g}\right)^{p-1}\right|^{\frac{p}{p-1}}\mathrm{d}\mu_{f}\leq
c_{\Omega}^{p}\|\nabla\varphi\|_{L^{\infty}(\Omega)}^{\frac{p}{p-1}}\int_{\Omega}\left|\bar{\eta}_{j}^{p-1}-\bar{\eta}^{p-1}\right|^{\frac{p}{p-1}}\mathrm{d}\mu_{f}$

and inequalities analogous to those in the second and third lines of (4.20).
This concludes the proof of (4.22).
Now, (4.22) implies that

$\int_{\Omega}|\nabla g|^{p-2}\langle\nabla
g,\nabla\left(\frac{\bar{\eta}_{j}^{p}}{g^{p-1}}\right)\rangle\mathrm{d}\mu_{f}\longrightarrow\int_{\Omega}|\nabla
g|^{p-2}\langle\nabla
g,\nabla\left(\frac{\bar{\eta}^{p}}{g^{p-1}}\right)\rangle\mathrm{d}\mu_{f},$

so that integrating on $\Omega$ the Lagrangian identity

$\mathcal{L}(\bar{\eta}_{j},g)=|\nabla\bar{\eta}_{j}|^{p}-|\nabla
g|^{p-2}\langle\nabla
g,\nabla\left(\frac{\bar{\eta}_{j}^{p}}{g^{p-1}}\right)\rangle$

and using the weak lower semicontinuity of the $W^{1,p}$ norm we deduce

(4.23)
$0\leq\int_{\Omega}\mathcal{L}(\bar{\eta},g)\mathrm{d}\mu_{f}\leq\liminf_{j\rightarrow+\infty}\int_{\Omega}\mathcal{L}(\bar{\eta}_{j},g)\mathrm{d}\mu_{f}.$

Inequalities (4.21) and $\mathcal{L}(\bar{\eta}_{j},g)\geq 0$ on $M$ then
imply

$0\leq\int_{\Omega}\mathcal{L}(\bar{\eta}_{j},g)\mathrm{d}\mu_{f}\leq\int_{M}\mathcal{L}(\bar{\eta}_{j},g)\mathrm{d}\mu_{f}\leq
Q_{V}(\bar{\eta}_{j})\longrightarrow 0$

as $j\rightarrow+\infty$, so we conclude by (4.23) that
$\mathcal{L}(\bar{\eta},g)\equiv 0$ on $\Omega$. By Proposition 3.1,
$\bar{\eta}=cg$ on $\Omega$ for some $c\geq 0$. In fact, $c>0$ since
$\|\eta\|_{L^{p}(U)}=\lim_{j}\|\eta_{j}\|_{L^{p}(U)}\neq 0$, and $c<1$ since
$\\{x\in\Omega:\eta(x)<g(x)\\}$ has positive measure. Therefore,
$\bar{\eta}\equiv\eta$ on $\Omega$, showing that $g$ is a positive multiple of
$\eta$ on $\Omega$, that is, $(ii)_{\mathrm{GS}}$ holds.
Claim 3.  When $Q_{V}$ has a ground state $\eta$, $\eta\in
C^{1,\mu}_{\mathrm{loc}}(M)$, is positive and solves $Q_{V}^{\prime}(\eta)=0$.
Proof of Claim 3.  By $(i)_{\mathrm{GS}}\Rightarrow(ii)_{\mathrm{GS}}$, all
solutions $g\in C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ of
$Q_{V}^{\prime}(g)\geq 0$ are proportional. Choosing $g$ to be a positive
solution of $Q_{V}^{\prime}(g)=0$ (which exists by Proposition 3.4), we get
that $\eta>0$ solves $Q_{V}^{\prime}(\eta)=0$ and $\eta\in
C^{1,\mu}_{\mathrm{loc}}(M)$, proving the claim.
$\mathbf{(iii)_{\mathrm{GS}}\Rightarrow(i)_{\mathrm{GS}}}$. Since
$\mathrm{Int}(K)\neq\emptyset$ we select a closed smooth geodesic ball $B$
contained in $\mathrm{Int}(K)$. By the monotonicity of capacity,
$\mathrm{cap}_{Q_{V}}(B,g)=0$. Fix an exhaustion $\\{\Omega\\}_{j}$ of $M$
with $B\Subset\Omega_{1}$, let $\eta_{j}$ be the $Q_{V}$-capacitor of
$(B,\Omega_{j},g)$ extended with zero outside $\Omega_{j}$, and let $\eta$ be
the capacitor of $(B,g)$. Then, (4.10) ensures that
$Q_{V}(\eta_{j})\rightarrow\mathrm{cap}_{Q_{V}}(B,g)=0$, and since
$\eta_{j}\rightarrow\eta$ in $L^{p}_{\mathrm{loc}}(M)$ we deduce that $\eta$
is the desired ground state.
$\mathbf{(iv)_{\mathrm{S}}\Rightarrow(iii)_{\mathrm{S}}}$. Up to enlarging $K$
(capacity increases), we can assume that $K$ is the closure of a relatively
compact open set with smooth boundary. Now, consider a solution $\bar{g}\in
C^{1,\mu}_{\mathrm{loc}}(M)$ of $Q_{V}^{\prime}(\bar{g})=0$ on $M$. Up to
multiplying $\bar{g}$ by a small positive constant, we can suppose that
$\bar{g}\leq g$ on $K$. By the very definition of $Q_{V}$-capacity,

$\mathrm{cap}_{Q_{V}}(K,g)\geq\mathrm{cap}_{Q_{V}}(K,\bar{g}).$

We claim that $\mathrm{cap}_{Q_{V}}(K,\bar{g})>0$. Indeed, otherwise, by
$(iii)_{\mathrm{GS}}\Rightarrow(i)_{\mathrm{GS}}$ just proved, $Q_{V}$ has a
ground state and so all solutions $u\in C^{0}(M)\cap
W^{1,p}_{\mathrm{loc}}(M)$ of $Q_{V}^{\prime}(u)\geq 0$ are proportional. In
particular, $\bar{g}=cg$ for some constant $c>0$, which implies
$\mathrm{cap}_{Q_{V}}(K,g)=c^{-p}\mathrm{cap}_{Q_{V}}(K,\bar{g})=0$
contradicting our assumptions. Now, since $\bar{g}$ solves
$Q_{V}^{\prime}(\bar{g})=0$ and $\mathrm{cap}_{Q_{V}}(K,\bar{g})>0$, applying
formula (4.12) in Proposition 4.3 with $\bar{g}$ replacing $g$ we deduce that
necessarily the $Q_{V}$-capacitor $u$ of $(K,\bar{g})$ is different from
$\bar{g}$. As $u$ solves $Q_{V}^{\prime}(u)\geq 0$ on $M$ and $u=g$ on $K$,
$u$ and $\bar{g}$ are two non-proportional solutions of $Q_{V}^{\prime}(v)\geq
0$, proving $(iii)_{\mathrm{S}}$.
$\mathbf{(ii)_{\mathrm{GS}}\Rightarrow(iii)_{\mathrm{GS}}}$. If, by
contradiction, $\mathrm{cap}_{Q_{V}}(K,g)>0$ for some $(K,g)$, then by
$(iv)_{\mathrm{S}}\Rightarrow(iii)_{\mathrm{S}}$ above there would exists two
non-proportional solutions of $Q_{V}^{\prime}(g)\geq 0$, contradicting
$(ii)_{\mathrm{GS}}$.
Now, we have concluded the equivalence
$(i)_{\mathrm{GS}}\Leftrightarrow(ii)_{\mathrm{GS}}\Leftrightarrow(iii)_{\mathrm{GS}}$
in the “ground state" part of the theorem. Combining with Claim $2$, we
automatically have the validity of
$\mathbf{(i)_{\mathrm{S}}\Leftrightarrow(iii)_{\mathrm{S}}\Leftrightarrow(iv)_{\mathrm{S}}}$.
We are thus left to show $(i)_{\mathrm{S}}\Leftrightarrow(ii)_{\mathrm{S}}$.
Having observed that $(i)_{\mathrm{S}}\Rightarrow(ii)_{\mathrm{S}}$ is obvious
(set $w=W$), to conclude we prove that
$\mathbf{(ii)_{\mathrm{S}}\Rightarrow(i)_{\mathrm{S}}}$. Suppose by
contradiction that $(i)_{\mathrm{S}}$ is not true. Then, by Claim $2$, $Q_{V}$
has a ground state $\eta$, which is positive on $M$ and solves
$Q_{V}^{\prime}(\eta)=0$. Fix a smooth open set $U$ such that $w\not\equiv 0$
on $U$, let $\\{\Omega_{j}\\}\uparrow M$ be an exhaustion of $M$ with
$U\Subset\Omega_{1}$, and let $\eta_{j}$ be the $Q_{V}$-capacitor of
$(U,\Omega_{j},\eta)$. Then, by the equivalence
$(i)_{\mathrm{GS}}\Leftrightarrow(iii)_{\mathrm{GS}}$ (in particular, by the
proof of $(iii)_{\mathrm{GS}}\Rightarrow(i)_{\mathrm{GS}}$), $\\{\eta_{j}\\}$
is a null sequence. By the subcriticality assumption and since $\eta_{j}=\eta$
on $U$,

$\int_{U}w|\eta|^{p}\mathrm{d}\mu_{f}\leq\int_{M}w|\eta_{j}|^{p}\mathrm{d}\mu_{f}\leq
Q_{V}(\eta_{j})\rightarrow 0$

as $j\rightarrow+\infty$, contradicting the fact that $\eta>0$ on $M$ and
$w\not\equiv 0$ on $U$. ∎

######  Remark 4.3.

_We can now give a simple proof of the fact that the positivity of the Yamabe
invariant $Y(M)$ in Theorem 1.1 implies the subcriticality of the conformal
Laplacian $L_{\langle\,,\,\rangle}$. Indeed, suppose the contrary. Then, by
Theorem 4.1, there exist a ground state $\eta>0$ and a null sequence
$\\{\eta_{j}\\}$ locally $L^{2}$-converging to $\eta$. From the chain of
inequalities_

$Y(M)\left(\int_{M}|\eta_{j}|^{\frac{2m}{m-2}}\right)^{\frac{m-2}{m}}\leq\int_{M}\left[|\nabla\eta_{j}|^{2}+\frac{s(x)}{c_{m}}\eta_{j}^{2}\right]\rightarrow
0\quad\text{as }\,j\rightarrow+\infty,$

_we deduce that $\eta_{j}\rightarrow 0$ in $L^{\frac{2m}{m-2}}(M)$, hence
locally in $L^{2}(M)$, a contradiction. _

It is now immediate to prove the equivalence partly mentioned in the
Introduction (Proposition 2.1). We suggest the interested reader to consult
the very recent [23, 24] for an investigation on the optimality of the Hardy
weights given in items $5)$ and $6)$ below.

######  Proposition 4.4.

Let $M^{m}$ be a Riemannian manifold of dimension $m\geq 2$, $p\in(1,+\infty)$
and $f\in C^{\infty}(M)$. The following properties are equivalent:

  * $1)$

There exists a positive, non-constant solution $g\in C^{0}(M)\cap
W^{1,p}_{\mathrm{loc}}(M)$ of $\Delta_{p,f}g\leq 0$.

  * $2)$

For some (any) compact $K\subset M$ with non-empty interior, and for some
(any) solution $g$ of $\Delta_{p,f}g\leq 0$, $\mathrm{cap}(K,g)>0$.

  * $3)$

$Q_{0}$ is subcritical on $M$: there exists $w\in L^{1}_{\mathrm{loc}}(M)$,
$w\geq 0$, $w\not\equiv 0$ on $M$ such that

(4.24)
$\displaystyle\int_{M}w(x)|\varphi|^{p}\mathrm{d}\mu_{f}\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\qquad\forall\,\varphi\in\mathrm{Lip}_{c}(M).$

  * $4)$

$Q_{0}$ has a weighted spectral gap on $M$: there exists $W\in C^{0}(M)$,
$W>0$ on $M$ such that

(4.25)
$\displaystyle\int_{M}W(x)|\varphi|^{p}\mathrm{d}\mu_{f}\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\qquad\forall\,\varphi\in\mathrm{Lip}_{c}(M).$

  * $5)$

For each non-constant, positive weak solution $u\in W^{1,p}_{\mathrm{loc}}(M)$
of $\Delta_{p,f}u\leq 0$ the following Hardy type inequality holds:

(4.26) $\left(\frac{p-1}{p}\right)^{p}\int_{M}\frac{|\nabla
u|^{p}}{u^{p}}|\varphi|^{p}\mathrm{d}\mu_{f}\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\qquad\forall\,\varphi\in\mathrm{Lip}_{c}(M).$

  * $6)$

For each $y\in M$, $-\Delta_{p,f}$ has a positive Green kernel
$\mathcal{G}(x,y)$ and the following Hardy inequality holds:

(4.27)
$\left(\frac{p-1}{p}\right)^{p}\int_{M}\frac{|\nabla_{x}\mathcal{G}|^{p}}{\mathcal{G}^{p}}|\varphi|^{p}\mathrm{d}\mu_{f}\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\qquad\forall\,\varphi\in\mathrm{Lip}_{c}(M).$

######  Remark 4.4.

_Here, a Green kernel $\mathcal{G}(x,y)$ means a distributional solution of_

$\Delta_{p,f}\mathcal{G}(\cdot,y)=-\delta_{y},$

_where $\delta_{y}$ is the Dirac delta function at $y$. _

###### Proof.

The equivalence $1)\Leftrightarrow 2)\Leftrightarrow 3)\Leftrightarrow 4)$ is
the particular case $V\equiv 0$ of Theorem 4.1. Indeed, as any positive
constant is a solution of $\Delta_{p,f}u\leq 0$, $1)$ can be rephrased as the
existence of two non-proportional solutions of $\Delta_{p,f}u\leq 0$. Clearly
$5)\Rightarrow 3)$, thus $5)$ yields that $Q_{0}$ is subcritical on $M$;
therefore, we just need to show that $1)\Rightarrow 5)$. Towards this aim
observe that if $g\in C^{0}(M)\cap\in W^{1,p}_{\mathrm{loc}}(M)$, $g>0$ is a
positive, non-constant solution of $\Delta_{p,f}g\leq 0$, then by (3.14)
$\displaystyle z=g^{\frac{p-1}{p}}$ is a positive weak solution of

$\displaystyle\Delta_{p,f}z+\left(\frac{p-1}{p}\right)^{p}\frac{|\nabla
u|^{p}}{u^{p}}z^{p-1}\leq 0\qquad\text{on }M.$

Proposition 3.1 and the non-negativity of $\mathcal{L}$ then imply that
$Q_{V}$ is non-negative for

$V=\left(\frac{p-1}{p}\right)^{p}\frac{|\nabla u|^{p}}{u^{p}},$

which gives (4.26).

To prove $6)\Rightarrow 1)$, take a positive Green kernel $\mathcal{G}(x,y)$
for $-\Delta_{p,f}$ at $y$. For large $c>0$, the function
$G_{c}(x)=\min\\{c,\mathcal{G}(x,y)\\}$ is a non-constant, positive weak
solution of $\Delta_{p,f}G_{c}\leq 0$ by the pasting Lemma 3.1, showing $1)$.
Vice versa, if $1)$ holds, then by the equivalence $1)\Leftrightarrow 2)$ we
deduce that each compact set has positive capacity with respect to the
“standard" supersolution $g\equiv 1$. This implies, by results in [38] and
[39], that for each $y\in M$ there exists a positive Green kernel
$\mathcal{G}(x,y)$ for $-\Delta_{p,f}$. Defining again $G_{c}$ as above,
$\Delta_{p,f}G_{c}\leq 0$ and the equivalence $1)\Leftrightarrow 5)$ ensures
that the Hardy inequality

(4.28) $\left(\frac{p-1}{p}\right)^{p}\int_{M}\frac{|\nabla
G_{c}|^{p}}{G_{c}^{p}}|\varphi|^{p}\mathrm{d}\mu_{f}\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\qquad\text{holds
for each }\varphi\in\mathrm{Lip}_{c}(M),$

and (4.27) follows by letting $c\rightarrow+\infty$ and using the monotone
convergence theorem. ∎

######  Remark 4.5.

_As a consequence of[75, 76] and Theorem 1.1 in [44] (see also [45]), if
$M=\mathbb{R}^{m}$ each positive Green kernel $\mathcal{G}$ satisfies_

(4.29)
$\frac{|\nabla_{x}\mathcal{G}|^{p}}{\mathcal{G}^{p}}(x,y)\sim\left\\{\begin{array}[]{ll}C\,\mathrm{dist}(x,y)^{-m}\log^{-m}\mathrm{dist}(x,y)&\quad\text{if
}p=m,\\\\[8.5359pt] C\,\mathrm{dist}(x,y)^{-p}&\quad\text{if
}p<m\end{array}\right.$

_as $\mathrm{dist}(x,y)\rightarrow 0$, for some explicit $C>0$. In the linear
case $p=2$, the first order expansions in [75, 76] guarantee the validity of
(4.29) on each Riemannian manifold. On the contrary, when $p\neq 2$, the
scaling arguments used in [44] are typical of the Euclidean setting and,
although we believe (4.29) to be true in general, to the best of our knowledge
there is still no proof of (4.29) in a manifold setting. _

The above proposition gives a useful, simple criterion to check the
subcriticality of some $Q_{V}$.

######  Proposition 4.5.

Let $(M,\langle\,,\,\rangle)$ be a Riemannian manifold, $f\in C^{0}(M)$ and
$p>1$. Let $V\in L^{\infty}_{\mathrm{loc}}(M)$. Suppose that $Q_{0}$ is
subcritical on $M$. If, for some Hardy weight $\hat{w}$, it holds
$V\leq\hat{w}$ and $V\not\equiv\hat{w}$, then $Q_{V}$ is subcritical.

###### Proof.

Indeed, using (4.24),

$Q_{V}(\varphi)\doteq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}-\int_{M}V|\varphi|^{p}\mathrm{d}\mu_{f}\geq\int_{M}(\hat{w}-V)|\varphi|^{p}\mathrm{d}\mu_{f};$

consequently, $w$ in (2.10) can be chosen to be $\hat{w}-V$, proving the
subcriticality of $Q_{V}$. ∎

######  Remark 4.6.

_The alternative in Theorem 4.1 is also related to the existence of a global
positive Green kernel for $Q_{V}^{\prime}$. It has been shown in [67]
(Theorems 5.4 and 5.5) that, on $\mathbb{R}^{m}$, $Q_{V}$ has a weighted
spectral gap if and only if $Q_{V}^{\prime}$ admits a global positive Green
kernel. The result depends on Lemma 5.1 therein, which has been proved via a
rescaling argument typical of the Euclidean space, and calls for a different
strategy in a manifold setting. However, in the linear case we can easily
obtain the above equivalence on general manifolds by relating $Q_{V}^{\prime}$
to a weighted Laplacian via a standard transformation. In fact, suppose that
$p=2$ and that $Q_{V}$ is non-negative. Let $g$ be a positive solution of
$Q_{V}^{\prime}(g)=0$. Then, setting $h=f-2\log g$, by a simple computation
the following formula holds weakly for $\varphi\in C^{2}(M)$:_

(4.30)
$-\Delta_{h}\left(\frac{\varphi}{g}\right)=g^{-1}Q_{V}^{\prime}(\varphi).$

_Note that, according to our notation, for smooth $\phi$_

$\Delta_{h}\phi=g^{-2}e^{f}\mathrm{div}(e^{-f}g^{2}\nabla\phi)$

_Integrating by parts, we infer_

$\frac{1}{2}\int_{M}\left|\nabla\left(\frac{\varphi}{g}\right)\right|^{2}g^{2}\mathrm{d}\mu_{f}=Q_{V}(\varphi)\qquad\forall\,\varphi\in
C^{2}_{c}(M).$

_Therefore, it is readily seen that $Q_{V}^{\prime}$ is subcritical if and
only if so is $-\Delta_{h}$ with respect to the measure
$g^{2}\mathrm{d}\mu_{f}$. Now, Proposition 4.4 guarantees that this happens if
and only if $-\Delta_{h}$ has a positive Green kernel $\mathcal{G}(x,y)$.
Coming back with the aid of (4.30), it is easy to see that
$\mathcal{G}(x,y)g(x)g(y)$ is a global positive Green kernel for
$Q_{V}^{\prime}$. _

##  5\. Hardy weights and comparison geometry

On a manifold $M$ for which $Q_{0}$ is subcritical, the criterion in
Proposition 4.5 shifts the problem of the subcriticality of $Q_{V}$ to the one
of finding explicit Hardy weights. As we will see in a moment, the
construction of weights given in $4)$ and $5)$ of Proposition 4.4 is
compatible with the usual geometric comparison theorems. Therefore, it gives a
useful way to produce simple Hardy weights on manifolds satisfying suitable
curvature assumptions. In this section, we describe some examples to
illustrate the method.

First, we underline the following simple fact. By its very definition, the set
of Hardy weights $w\in L^{1}_{\mathrm{loc}}(M)$, $w\geq 0$, $w\not\equiv 0$ is
convex in $L^{1}_{\mathrm{loc}}(M)$. More generally, given a family
$\\{w_{\alpha}\\}_{\alpha\in A}$ of Hardy weights on $M$ whose index $\alpha$
lies in a measurable space $(A,\mathscr{F})$ ($\mathscr{F}$ a
$\sigma$-algebra), and such that the map

$w\ \ :\ \ (x,\alpha)\in M\times A\longrightarrow w_{\alpha}(x)\in[0,+\infty]$

is measurable, for each measure $\lambda$ on $A$ such that $0<\lambda(A)\leq
1$ the function

(5.1) $\chi(x)=\int_{A}w(x,\alpha)\mathrm{d}\lambda(\alpha)$

is still a Hardy weight. Indeed, it is enough to apply Tonelli’s theorem: for
each $\varphi\in\mathrm{Lip}_{c}(M)$,

(5.2)
$\begin{array}[]{lcl}\displaystyle\int_{M}\chi|\varphi|^{p}\mathrm{d}\mu_{f}&=&\displaystyle\int_{A}\left[\int_{M}w_{\alpha}(x)|\varphi(x)|^{p}\mathrm{d}\mu_{f}\right]\mathrm{d}\lambda(\alpha)\\\\[11.38092pt]
&\leq&\displaystyle\int_{A}\left[\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\right]\mathrm{d}\lambda(\alpha)\leq\displaystyle\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}.\end{array}$

Clearly, this construction makes sense also if for $\lambda$-almost all
$\alpha\in A$, $w_{\alpha}$ is a Hardy weight.

######  Remark 5.1.

_By item $5)$ of Proposition 4.4, each Green kernel $\mathcal{G}$ generates a
family $w$ indexed by $A=M$:_

(5.3)
$w(x,y)\doteq\left(\frac{p-1}{p}\right)^{p}\frac{|\nabla_{x}\mathcal{G}(x,y)|^{p}}{\mathcal{G}(x,y)^{p}}\
\ \ :\ \ \ M\times M\longrightarrow[0,+\infty],$

_provided that $w$ is measurable. If $p=2$, the standard construction of a
Green kernel in [34] produces a symmetric kernel, and measurability is
obvious. However, measurability seems to be a subtle issue if $p\neq 2$, since
$\mathcal{G}(x,y)$ is constructed in [38] and [39] by fixing $y$ and finding a
solution of $\Delta_{p,f}\mathcal{G}(x,y)=-\delta_{y}$. The dependence of
$\mathcal{G}(x,y)$ from $y$ could be, a priori, wild. _

We now describe two simple measures $\lambda$ that have been considered in the
literature when $M=\mathbb{R}^{m}$, and the corresponding Hardy inequalities.

######  Example 5.1.

[Multipole Hardy weights]
_Fix a possibly infinite sequence $\\{y_{j}\\}\subset M$, $j\in
I\subset\mathbb{N}$, let $\\{t_{j}\\}_{j\in I}\subset(0,1]$ be such that
$\sum_{j}t_{j}\leq 1$ and define_

$\lambda=\sum_{j}t_{j}\delta_{y_{j}},$

_where $\delta_{y_{j}}$ is the Dirac delta function at $y_{j}$. Being
$\lambda$ discrete, measurability of $w$ follows automatically and thus, by
(5.2),_

(5.4) $\left(\frac{p-1}{p}\right)^{p}\int_{M}\left[\sum_{j\in
I}\frac{t_{j}|\nabla_{x}\mathcal{G}(x,y_{j})|^{p}}{\mathcal{G}(x,y_{j})^{p}}\right]\big{|}\varphi(x)\big{|}^{p}\mathrm{d}\mu_{f}\leq\int_{M}\big{|}\nabla\varphi(x)\big{|}^{p}\mathrm{d}\mu_{f}$

_holds for $\varphi\in\mathrm{Lip}_{c}(M)$. _

######  Example 5.2.

[Hardy weights that blow-up along a submanifold]
_If $w(x,y)$ in (5.3) is measurable, for each rectifiable subset
$\Sigma\hookrightarrow M^{m}$ of finite non-zero $k$-dimensional Hausdorff
measure $\mathcal{H}^{k}$ we can set_

$\lambda=\frac{\mathcal{H}^{k}\llcorner\Sigma}{\mathcal{H}^{k}(\Sigma)},$

_Then, we have the Hardy inequality_

(5.5)
$\left(\frac{p-1}{p}\right)^{p}\int_{M}\left[\frac{1}{\mathcal{H}^{k}(\Sigma)}\int_{\Sigma}\frac{|\nabla_{x}\mathcal{G}(x,y)|^{p}}{\mathcal{G}(x,y)^{p}}\mathrm{d}\mathcal{H}^{k}(y)\right]\big{|}\varphi(x)\big{|}^{p}\mathrm{d}x\leq\int_{M}\big{|}\nabla\varphi(x)\big{|}^{p}\mathrm{d}x$

_for each $\varphi\in\mathrm{Lip}_{c}(M)$.
Hardy weights of this type have been considered, for instance, in [31]: in
Theorem 1.1 therein, $M=\mathbb{R}^{m}=\mathbb{R}^{2}\times\mathbb{R}^{m-2}$,
$p=2$, $f\equiv 0$ and $\Sigma\subset\mathbb{R}^{m}$ is a round circle
$S_{\rho}\doteq\mathbb{S}^{1}(\rho)\times\\{0\\}\subset\mathbb{R}^{2}\times\mathbb{R}^{m-2}$
centered at the origin and of radius $\rho$. We remark that Hardy weights of
different type but still depending on the distance from a submanifold have
been investigated, among others, in [10]. The reader is also suggested to see
the references therein for deepening. _

To introduce the results below, we first recall comparison geometry, starting
with the definition of a model manifold. Briefly, fix a point
$o\in\mathbb{R}^{m}$. Given $g\in C^{2}(\mathbb{R}^{+}_{0})$ such that $g>0$
on some open interval $(0,R)\subset\mathbb{R}^{+}$, $g(0)=0$,
$g^{\prime}(0)=1$, a model $(M^{m}_{g},\mathrm{d}s_{g}^{2})$ is the manifold
$B_{R}(o)\subset\mathbb{R}^{m}$ equipped with a radially symmetric $C^{2}$
metric $\mathrm{d}s_{g}^{2}$ whose expression, in polar geodesic coordinates
$(\rho,\theta)$ centered at $o$ (where $\theta\in\mathbb{S}^{m-1}$), is given
by

$\mathrm{d}s_{g}^{2}=\mathrm{d}\rho^{2}+g(\rho)^{2}\mathrm{d}\theta^{2},$

$\mathrm{d}\theta^{2}$ being the standard metric on the unit sphere
$\mathbb{S}^{m-1}$. Clearly, $\rho$ is the distance function from $o$, and
$M_{g}$ is complete if and only if $R=+\infty$. At a point $x=(\rho,\theta)$,
the radial sectional curvature $K_{\mathrm{rad}}$ of $M_{g}$ (that is, the
sectional curvature restricted to planes containing $\nabla\rho(x)$), the
Hessian and the Laplacian of $\rho$ are given by

(5.6) $\displaystyle
K_{\mathrm{rad}}(x)=-\frac{g^{\prime\prime}(\rho)}{g(\rho)},\quad\displaystyle\nabla\mathrm{d}\rho(x)=\frac{g^{\prime}(\rho)}{g(\rho)}\Big{(}\mathrm{d}s_{g}^{2}-\mathrm{d}\rho\otimes\mathrm{d}\rho\Big{)},\quad\Delta\rho(x)=(m-1)\frac{g^{\prime}(\rho)}{g(\rho)}.$

By the first formula, in (5.6), a model can also be given by prescribing the
radial sectional curvature $-G\in C^{0}(\mathbb{R}^{+}_{0})$ and recovering
$g\in C^{2}(\mathbb{R}^{+}_{0})$ as the solution of

(5.7) $\left\\{\begin{array}[]{l}g^{\prime\prime}-Gg=0\qquad\text{on
}\mathbb{R}^{+},\\\\[5.69046pt] g(0)=0,\quad
g^{\prime}(0)=1,\end{array}\right.$

on the maximal interval where $g>0$. A sharp condition on $G$ that ensures the
positivity of $g$ on the whole $\mathbb{R}^{+}$ is given by

$G_{-}\in L^{1}(\mathbb{R}^{+}),\qquad
t\int_{t}^{+\infty}G_{-}(s)\mathrm{d}s\leq\frac{1}{4}\qquad\text{on
}\mathbb{R}^{+},$

see Proposition 1.21 in [12]. In particular, if $G\equiv\kappa^{2}$ for some
constant $\kappa\geq 0$, we will denote with $g_{\kappa}$ the solution $g$ of
(5.7):

(5.8) $g_{\kappa}(\rho)=\left\\{\begin{array}[]{ll}\rho&\ \text{if
}\kappa=0,\\\\[5.69046pt] \kappa^{-1}\sinh(\kappa\rho)&\ \text{if
}\kappa>0.\end{array}\right.$

When $\kappa=0$, $M_{g}$ is the Euclidean space $\mathbb{R}^{m}$, while if
$\kappa>0$ our model is the hyperbolic space $\mathbb{H}^{m}_{\kappa}$ of
sectional curvature $-\kappa^{2}$. Clearly, the two examples are radially
symmetric with respect to any chosen origin $o$. Hereafter, we will always
consider geodesically complete models.
Given $p\in(1,+\infty)$, $-\Delta_{p}$ is subcritical on $M_{g}$ if and only
if

(5.9) $g(\rho)^{-\frac{m-1}{p-1}}\in L^{1}(+\infty)$

(the case $p=2$ can be found, for instance, in [34], and for $p\neq 2$ the
argument of the proof goes along the same lines). In fact, under the validity
of (5.9), up to an unessential constant, the function

$G(x,o)=G\big{(}(\rho,\theta),o\big{)}=\int_{\rho}^{+\infty}g(s)^{-\frac{m-1}{p-1}}\mathrm{d}s$

is the minimal positive Green kernel for $-\Delta_{p}$ with singularity at
$o$.

A simplified version of the Hessian and Laplacian comparison theorems from
below (see [58, 62, 12]) can be stated as follows: suppose that
$(M,\langle\,,\,\rangle)$ has a pole $o$ and, denoting with
$r(x)=\mathrm{dist}(x,o)$, that the radial sectional curvature
$K_{\mathrm{rad}}$ satisfies

(5.10) $K_{\mathrm{rad}}(x)\leq-G\big{(}r(x)\big{)}$

(i.e., for each plane $\pi\leq T_{x}M$ containing $\nabla r(x)$,
$K(\pi)\leq-G\big{(}r(x)\big{)}$). Denote with $g$ the solution of (5.7), and
let $(0,R)$ be the maximal interval where $g>0$. Then, in the sense of
quadratic forms,

(5.11)
$\nabla\mathrm{d}r(x)\geq\frac{g^{\prime}(r(x))}{g(r(x))}\Big{(}\langle\,,\,\rangle-\mathrm{d}r\otimes\mathrm{d}r\Big{)}$

pointwise on $B_{R}(o)\backslash\\{o\\}$, and tracing

(5.12) $\Delta r(x)\geq(m-1)\frac{g^{\prime}(r(x))}{g(r(x))},$

whose validity holds weakly on the geodesic ball $B_{R}(o)\subset M$. As it is
apparent from (5.6), $M$ is compared with a model $M_{g}$ of radial sectional
curvature $-G$. Indeed, via Sturm comparison, for (5.12) to hold it is enough
that $g\in C^{2}(\mathbb{R}^{+}_{0})$ solves the inequality

(5.13) $\left\\{\begin{array}[]{l}g^{\prime\prime}-Gg\leq 0\qquad\text{on
}\mathbb{R}^{+},\\\\[5.69046pt] g(0)=0,\quad
g^{\prime}(0)=1,\end{array}\right.$

so that the model to which $M$ is compared has radial sectional curvature
greater than or equal to $-G$. In [13], we have collected a number of examples
of $G$ for which explicit positive solutions $g$ of (5.13) can be found. These
also include manifolds whose sectional curvature can be positive but in a
controlled way, loosely speaking manifolds whose compared model is some sort
of paraboloid.

With this preparation, we are now ready to discuss the next cases.

###  5.1. Hardy weights on manifolds with a pole

######  Theorem 5.1.

Let $(M,\langle\,,\,\rangle)$ be a Riemannian manifold with a pole $o$.
Denoting with $r(x)=\mathrm{dist}(x,o)$, assume that the radial sectional
curvature satisfies

$K_{\mathrm{rad}}\leq-G\big{(}r(x)\big{)},$

for some $G\in C^{2}(\mathbb{R}^{+}_{0})$. Suppose that $g$ solving (5.13) is
positive on $\mathbb{R}^{+}$, and, for $p\in(1,+\infty)$, assume that

(5.14) $g(r)^{-\frac{m-1}{p-1}}\in L^{1}(+\infty).$

Then, the Hardy inequality

(5.15) $\int_{M}(\chi\circ
r)|\varphi|^{p}\mathrm{d}x\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}x$

holds for each $\varphi\in\mathrm{Lip}_{c}(M)$, where

(5.16)
$\chi(t)=\left(\frac{p-1}{p}\right)^{p}\left[g(t)^{\frac{m-1}{p-1}}\int_{t}^{+\infty}g(s)^{-\frac{m-1}{p-1}}\mathrm{d}s\right]^{-p}$

###### Proof.

Consider the transplanted Green kernel of the model $M_{g}$ with pole at $o$:

(5.17) $G(x)=\int_{r(x)}^{+\infty}g(s)^{-\frac{m-1}{p-1}}.$

Then, a computation using (5.11) shows that $\Delta_{p}G\leq 0$ on
$M\backslash\\{o\\}$.

Furthermore, $G$ satisfies the asymptotic behaviour

(5.18) $G(x)\sim\left\\{\begin{array}[]{ll}C>0&\quad\text{if
}p>m,\\\\[2.84544pt] |\log r|&\quad\text{if }p=m,\\\\[2.84544pt]
\frac{p-1}{m-p}r^{-\frac{m-p}{p-1}}&\quad\text{if
}p<m,\end{array}\right.\qquad\text{as }r\rightarrow 0^{+},$

for some constant $C>0$. It thus follows that, for each $c\in(0,C)$ if $p>m$
and $c\in\mathbb{R}^{+}$ if $p\leq m$, by the pasting Lemma 3.1 the function
$G_{c}=\min\\{G,c\\}$ is a solution of $\Delta_{p}G_{c}\leq 0$, hence by (4.4)

(5.19) $\left(\frac{p-1}{p}\right)^{p}\int_{M}\frac{|\nabla
G_{c}|^{p}}{G_{c}^{p}}|\varphi|^{p}\mathrm{d}x\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}x\qquad\text{holds
for each }\varphi\in\mathrm{Lip}_{c}(M).$

Letting $c\rightarrow C^{-}$ when $p>m$, $C\rightarrow+\infty$ when $p\leq m$,
and since

$\left(\frac{p-1}{p}\right)^{p}\frac{|\nabla
G|^{p}}{G^{p}}(x)=\chi\big{(}r(x)\big{)},$

we conclude the validity of (5.15). ∎

The function $\chi\circ r$ has the following asymptotics as $r=r(x)\rightarrow
0^{+}$:

(5.20)
$\chi(r)\sim\left\\{\begin{array}[]{ll}\left(\frac{p-1}{Cp}\right)^{p}r^{-\frac{p(m-1)}{p-1}}&\quad\text{if
}p>m,\\\\[8.5359pt]
\left(\frac{m-1}{m}\right)^{m}\frac{1}{r^{m}\log^{m}r}&\quad\text{if
}p=m,\\\\[8.5359pt]
\left(\frac{m-p}{p}\right)^{p}\frac{1}{r^{p}}&\quad\text{if
}p<m,\end{array}\right.$

where, for $p>m$, the constant $C$ is the same as in (5.18). Note that, on
each case, $\chi\circ r\in L^{1}_{\mathrm{loc}}(M)$, in particular the
singularity at $o$ is integrable.

######  Remark 5.2.

_Under the same assumptions on $M$, this example can be extended to deal with
the weighted operator $\Delta_{p,f}$ provided that $f=f(r(x))$ is a radial
function. In this case, it can be seen that (5.14) must be replaced by_

$\frac{e^{f(r)}}{g(r)^{\frac{m-1}{p-1}}}\in L^{1}(+\infty),$

_and that_

$G(x)=\int_{r(x)}^{+\infty}\frac{e^{f(s)}\mathrm{d}s}{g(s)^{\frac{m-1}{p-1}}}$

_is a solution of $\Delta_{p,f}G\leq 0$ on $M\backslash\\{o\\}$ that gives
rise to a Hardy weight analogous to (5.15). _

The Hardy weight in (5.15) is explicit once we have an explicit $g$ solving
(5.13), for example those related to the families of $G$ described in the
Appendix of [13]. We refer the reader to the above mentioned paper also for a
detailed study of the corresponding Hardy weight (called the “critical curve"
therein) for $p=2$. The modifications needed to deal with general $p$ are
straightforward. Here, we just focus on the Euclidean and hyperbolic settings.

######  Example 5.3.

[Hardy weights for Euclidean and hyperbolic spaces]
_Consider the Euclidean space, where $g(r)=g_{0}(r)=r$. Condition (5.14) is
met if and only if $p<m$, and the Hardy weight (5.15) has the simple
expression_

(5.21)
$\chi\big{(}r(x)\big{)}=\left(\frac{m-p}{p}\right)^{p}\frac{1}{r(x)^{p}}.$

_Consequently, the Hardy inequality_

(5.22)
$\left(\frac{m-p}{p}\right)^{p}\int_{M}\frac{|\varphi|^{p}}{r^{p}}\mathrm{d}x\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}x\qquad\forall\,\varphi\in\mathrm{Lip}_{c}(M)$

_holds on each manifold with a pole and radial sectional curvature
$K_{\mathrm{rad}}\leq 0$. Inequality (5.22) is classical and well-known in
$\mathbb{R}^{m}$, see [10].