# A non-smooth Brezis-Oswald uniqueness result

Sunra Mosconi  Department of Mathematics and Computer Science
University of Catania
Viale A. Doria 6, I-95125 Catania, Italy<EMAIL_ADDRESS>
###### Abstract.

We classify the non-negative critical points in $W^{1,p}_{0}(\Omega)$ of

$J(v)=\int_{\Omega}H(Dv)-F(x,v)\,dx$

where $H$ is convex and positively $p$-homogeneous, while
$t\mapsto\partial_{t}F(x,t)/t^{p-1}$ is non-increasing. Since $H$ may not be
differentiable and $F$ has a one-sided growth condition, $J$ is only l. s. c.
on $W^{1,p}_{0}(\Omega)$. We employ a weak notion of critical point for non-
smooth functionals, derive sufficient regularity of the latter without an
Euler-Lagrange equation available and focus on the uniqueness part of the
results in [6], through a non-smooth Picone inequality.

###### Key words and phrases:

Non-smooth Critical Point Theory, Regularity Theory, Uniqueness of solutions
to nonlinear PDE, Differential Inclusions, Picone Inequality

###### 2010 Mathematics Subject Classification:

35J62, 35B65, 35A02, 35P30

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
    2. 1.2 Related results and motivations
    3. 1.3 Outline of the proof
    4. 1.4 Structure of the paper
  2. 2 Preliminaries
    1. 2.1 Notations
    2. 2.2 Functional inequalities
    3. 2.3 Convex $p$-homogeneous functions
    4. 2.4 Energy critical points
  3. 3 Regularity
  4. 4 Dirichlet eigenvalues
  5. 5 Main result

##  1\. Introduction

The celebrated Brezis-Oswald theorem [6] states, among other things, that the
problem

$\begin{cases}-\Delta u=f(x,u)&\text{in $\Omega$}\\\ u=0&\text{on
$\partial\Omega$}\\\ u\geq 0&\text{in $\Omega$}\end{cases}$ (1.1)

in a connected smooth bounded domain $\Omega\subseteq\mathbb{R}^{N}$ has a
unique solution whenever $f(\cdot,t)\in L^{\infty}(\Omega)$ for all $t\geq 0$
and $t\mapsto f(x,t)/t$ is strictly decreasing in $t>0$, for a. e.
$x\in\Omega$. This result has later been generalised to the quasi-linear
setting in [13], to the nonlocal one (see [18]) and even to mixed local-
nonlocal problems in [1], producing countless applications in nonlinear
analysis.

The general principle underlying Brezis-Oswald theorem appears to be the
following. If $L$ denotes a (pseudo)-differential elliptic operator enjoying a
strong minimum principle and which is homogeneous of degree $q$, then
uniqueness of the solution

$\begin{cases}L(u)=f(x,u)&\text{in $\Omega$}\\\ u=0&\text{on
$\partial\Omega$}\\\ u\geq 0&\text{in $\Omega$}\end{cases}$ (1.2)

should follow under the assumption that $t\mapsto f(x,t)/t^{q}$ is strictly
decreasing. These kind of results, therefore, are called in the literature
”Brezis-Oswald uniqueness theorems”.

A number of additional assumptions, however, are generally required for such
statements to hold, and we discuss here a few of them.

  1. (1)

Regularity of the domain. Often, the domain $\Omega$ is assumed to have smooth
(say, $C^{1,\alpha}$) boundary, so that a Hopf-type maximum principle holds
true for the corresponding solutions. This usually requires rather fine
regularity theory for solutions of the corresponding problems, and is usually
avoided requiring additional technical assumptions on the reaction (see e. g.
[1]).

  2. (2)

Growth conditions on $f$. The monotonicity assumption on $t\mapsto
f(x,t)/t^{q}$ only grants an upper bound on the growth of $f$. Even if the
operator $L$ is variational, namely $L=dE$ for a suitable differentiable
convex functional $E:X\to\mathbb{R}$ on a Banach space $X\hookrightarrow
L^{p}(\Omega)$, this poses delicate issues. Indeed, it is then natural to
consider solutions of (1.2) as critical points of the functional

$J(v)=E(v)-\int_{\Omega}F(x,u)\,dx,\qquad F(x,t)=\int_{0}^{t}f(x,s)\,ds$

but without assuming any further lower growth control on $f$, $J$ is,
typically, only lower semicontinuous. This problem is often side-stepped by
either requiring a-priori boundedness in the notion of solution of (1.2) (so
that actually $f$ can be truncated), or by directly requiring a two sided
growth condition on $f$.

  3. (3)

Monotonicity assumption. It is clear that uniqueness fails even for (1.1)
without strict monotonicity, as seen by considering the case
$f(x,t)=\lambda_{1}(\Omega)\,t$, where $\lambda_{1}(\Omega)$ is the first
eigenfunction of the Laplacian with Dirichlet boundary conditions. In this
case $t\mapsto f(x,t)/t$ is only non-increasing and the corresponding problem
(1.1) has infinitely many positive solutions, namely all first eigenfunctions
of the Dirichlet Laplacian. However, at least for autonomous reactions
independent of $x$, this is essentially the only obstruction to uniqueness.
Indeed, in [2, Proposition 3.8] it is proved that, for $L=-\Delta_{p}$ and
$t\mapsto f(t)/t^{p-1}$ only non-increasing in $t>0$, either there is a unique
solution to the corresponding problem (1.2), or all solutions are first
eigenfunctions for the Dirichlet $p$-Laplacian in $\Omega$. In particular,
uniqueness holds true if $t\mapsto f(t)/t^{p-1}$ is non-increasing and
strictly decreasing only on an arbitrarily small interval $]0,\delta[$,
$\delta>0$.

  4. (4)

Regularity of $L$. As anticipated in point (2) above, the typical framework
for a variational Brezis-Oswald uniqueness result is $L(u)=dE(u)$, for
$E(u)=\|u\|_{X}^{p}$ and $\|\ \|_{X}$ is the norm of a Banach space
$X\hookrightarrow L^{p}(\Omega)$, smooth away from the origin.
Differentiability of the norm is typically used to have a single valued
operator $L$ in equation (1.2). Considering non-differentiable and only
locally Lipschitz energies $E$ yields a differential inclusion formally of the
form

$f(x,u)\in\partial E(u)$

for critical points of the relevant functional, where here $\partial$ denotes
the multivalued convex subdifferential. In this more general setting,
uniqueness results are clearly stronger statements.

###  1.1. Main results

In this manuscript, we plan to address the previous themes, aiming to obtain a
general version of the uniqueness part of Brezis-Oswald theorem. In
particular, we seek for a classification of the critical points of functionals
of the form

$J(v)=\int_{\Omega}H(Dv)-F(x,v)\,dx,\qquad v\in W^{1,p}_{0}(\Omega)$ (1.3)

under fairly general conditions on $\Omega$, $H$, and
$f(x,t)=\partial_{t}F(x,t)$. Our main result reads as follows (for a more
complete statement, see Theorem 5.3).

######  Theorem 1.1.

Let $\Omega\subseteq\mathbb{R}^{N}$ be open, connected and of finite measure.
For some $p>1$ suppose that

  * i)

$H:\mathbb{R}^{N}\to[0,\infty[$ is strictly convex and positively
$p$-homogeneous

  * ii)

$f:\Omega\times\mathbb{R}\to\mathbb{R}$ is Caratheodory and fulfills

$\sup_{x\in\Omega,|s|\leq t}|f(x,s)|<\infty\qquad\forall t>0$

  * iii)

The map $t\mapsto f(x,t)/t^{p-1}$ is non-increasing on $]0,\infty[$.

For $J$ as in (1.3) with

$F(x,t)=\int_{0}^{t}f(x,s)\,ds,$

let $C_{+}(J)$ be the set of its non-negative critical points $v\in
W^{1,p}_{0}(\Omega)\setminus\\{0\\}$. Then

  1. (1)

All elements in $C_{+}(J)$ are multiple of each other and each of them
minimise $J$ among non-negative functions in $W^{1,p}_{0}(\Omega)$

  2. (2)

If $t\mapsto f(x,t)/t^{p-1}$ is strictly decreasing on $t>0$, $C_{+}(J)$
contains at most one element

  3. (3)

If $f$ does not depend on $x$ and $C_{+}(J)$ contains more than one element,
each function in $C_{+}(J)$ minimises the Rayleigh quotient

$\frac{\displaystyle{\int_{\Omega}H(Dv)\,dx}}{\displaystyle{\int_{\Omega}v^{p}\,dx}}$
(1.4)

among all $v\in W^{1,p}_{0}(\Omega)\setminus\\{0\\}$, $v\geq 0$.

######  Remark 1.2.

Let us make some comments on the features of the previous theorem.

  * •

No regularity assumptions are made on the boundary of the open set $\Omega$.
The finite measure assumption could be further weakened requiring additional
conditions on the behaviour of $f(x,t)$ near $0$, but we chose not to analyse
this case in order to have a cleaner statement. Notice that in this setting,
there is no hope to have a Hopf maximum principle available.

  * •

The stated assumption on $H$ enforces the two sided bound

$c\,|z|^{p}\leq H(z)\leq d\,|z|^{p},\qquad\forall z\in\mathbb{R}^{N}$

for some $0<c\leq d<\infty$. However, since $H$ is not supposed to be $C^{1}$,
the first term of $J$ in (1.3) is only locally Lipschitz, with multi-valued
differential. Notice also that $H$ is not supposed to be even. The strict
convexity of $H$ is also not needed in assertion (2), as well as in the second
statement of assertion (1).

  * •

Conditions ii) and iii) on the reaction $f$ are equivalent to

$f(\cdot,t)\in L^{\infty}(\Omega),\qquad f(x,t)\leq C\,(1+t^{p-1}),\qquad
t\mapsto\frac{f(x,t)}{t^{p-1}}\ \text{non-increasing}$

for $t>0$, which are more commonly found in Brezis-Oswald type results.

  * •

Under the sole assumptions ii) and iii), even for smooth $H$, $J$ is only l.
s. c., and the notion of critical point has to be dealt with care. We employ
here the notion of energy critical point used in [10], meaning those $u\in
W^{1,p}_{0}(\Omega)$ which, roughly speaking, minimise the corresponding semi-
linearised convex functional

$v\mapsto\int_{\Omega}H(Dv)-f(x,u)\,v\,dx.$

For the sake of this introductory discussion, let us note that the usually
encountered notions of critical point, when available, yield an energy
critical point. For more details, we refer to Definition 2.4 and Remark 2.5.

  * •

Notice that we don’t give any condition ensuring that $C_{+}(J)$ is actually
not empty. One of the main features of the original Brezis-Oswald result (and
of the related literature on the subject) was actually to provide necessary
and sufficient condition for $J$ to have non-negative, non-trivial critical
points, under a strict monotonicity assumption on $t\mapsto f(x,t)/t^{p-1}$.
We won’t investigate this theme here.

  * •

Under assumption iii) alone, the structure of $C_{+}(J)$ is described in an
optimal way by conclusion (1) of the theorem. Indeed, when $H(z)=|z|^{2}$,
given a first positive eigenfunction $u$ of the Dirichlet Laplacian in the
unit ball $B_{1}$ and an interval $[a,b]\subseteq[0,+\infty[$, any reaction
$f$ obeying $f(x,t)=\lambda_{1}(B_{1})\,t$ if $a\,u(x)\leq t\leq b\,u(x)$ and
$f(x,t)/t$ strictly decreasing otherwise has $C_{+}(J)=\\{t\,u:t\in[a,b]\\}$.

  * •

A similar statement holds true for non-positive critical points of $J$, with
obvious modifications to ii) and iii). Notice, however, that since we are not
assuming that $H$ is even, the infimum of the ratios (1.4) may change under
the corresponding constrain $v\leq 0$. This feature of possibly non-even,
positively $p$-homogeneous energies $H$ and their corresponding first
Dirichlet eigenvalues is discussed in section 4.

In order to prove Theorem 1.1, we need to develop basic regularity theory for
energy critical points. The main step is contained in the following a-priori
bound, which is of independent interest. In its statement, we let as usual
$p^{*}=N\,p/(N-p)$ if $p<N$, $p^{*}=\infty$ if $p>N$ and $p^{*}$ be any finite
number if $p=N$.

######  Theorem 1.3.

Let $\Omega\subseteq\mathbb{R}^{N}$ have finite measure and, for given $p\geq
1$, suppose that

  * i)

$H:\mathbb{R}^{N}\to[0,\infty[$ is convex and fulfills $H(z)\geq c\,|z|^{p}$
for some $c>0$ and all $z\in\mathbb{R}^{N}$

  * ii)

$f:\Omega\times\mathbb{R}\to\mathbb{R}$ is Caratheodory and fulfills

$\sup_{x\in\Omega,|s|\leq t}|f(x,s)|<\infty\qquad\forall t>0$

  * iii)

There exist $\mu,\theta\geq 0$ and $1\leq q\leq p^{*}$ such that

${\rm sign}\,(t)f(x,t)\leq\mu\,(|t|+\theta)^{q}\qquad\text{for
a.\,e.\,$x\in\Omega$ and all $t\in\mathbb{R}$}.$ (1.5)

Then, any critical point $u\in W^{1,p}_{0}(\Omega)$ of $J$ is bounded. If
additionally $q<p^{*}$, the estimate

$\|u\|_{L^{\infty}(\Omega)}\leq\max\left\\{\theta,C\,\left((\mu/c)^{\frac{N}{p}}\int_{\Omega}|u|^{q}\,dx\right)^{\frac{p}{p\,q+(p-q)\,N}}\right\\}$
(1.6)

holds true for some $C=C(N,p,q)$ which, for any $N$, is positive and
continuous in its other arguments.

######  Remark 1.4.

Regarding this second result, the following should be noted.

  * •

Not only $H$ is only assumed to be locally Lipschitz, but no growth control
from above is required in order to obtain boundedness of the critical points
of $J$.

  * •

Assumption iii) is understood to be empty in the case $q=\infty$, which forces
$p>N$ thanks to $q\leq p^{*}$. Clearly, in this case boundedness is trivial
thanks to Morrey embedding. Anyway, (1.6) only holds for $q<p^{*}$ and
therefore for finite $q$’s.

  * •

Assumption ii) and iii) are required in order for the functional $J$ in (1.3)
to be well defined (possibly assuming the value $+\infty$) on
$W^{1,p}_{0}(\Omega)$. In any case, under these assumptions $J$ turns out to
be only l. s. c. on $W^{1,p}_{0}(\Omega)$ with the strong topology, and again
we understand by critical point an energy one (see subsection 2.4).

  * •

The form (1.6) of the sup bound is optimal, as discussed in Remark 3.4.
Moreover, it is invariant by scaling and scalar multiples, meaning the
following. A function $u$ is a critical point for $J$ on $W^{1,p}_{0}(\Omega)$
if and only if for any $\lambda,r>0$ $u_{\lambda,r}(x)=\lambda\,u(x/r)$ is a
critical point for a suitable functional $J_{\lambda,r}$ on
$W^{1,p}_{0}(r\,\Omega)$ (where $r\,\Omega=\\{r\,x:x\in\Omega\\}$). More
precisely, the functional $J_{\lambda,r}$ is of the form (1.3) for a reaction
$\hat{f}$ obeying (1.5) with $\hat{\theta}=\lambda\,\theta$,
$\hat{\mu}=\mu/\lambda^{q}$ and and $\hat{H}$ obeying
$\hat{H}(z)\geq\hat{c}\,|z|^{p}$ with $\hat{c}=c\,\lambda/r$. The
corresponding estimates (1.6) turn out to be independent of $\lambda$ and $r$.

  * •

Regarding the continuity statement for the constant $C$ in the last part of
the theorem, it is to be understood in its domain $p\geq 1$, $q<p^{*}$. As is
to be expected, the constant $C$ blows up as $q\uparrow p^{*}$.

###  1.2. Related results and motivations

Applications and extensions of the Brezis-Oswald theorem are countless and it
would be impossible to list all the contributions on the subject. The
principal part of the operator considered in the various generalisation can be
classical quasi-linear operators, as in [13], nonlocal operators as in [18]
and even mixed local-nonlocal ones as in [1]. In all these cases the operator
is $(p-1)$-homogeneous and the corresponding monotonicity assumption on the
reaction pertains $t\mapsto f(x,t)/t^{p-1}$. Non-homogeneous operators of
Orlicz type are considered in [7]. The case when $t\mapsto f(x,t)/t^{p-1}$ is
non-increasing but not strictly decreasing is naturally related to weighted
nonlinear eigenvalue problems, where $f(x,t)=a(x)\,t^{p-1}$. Corresponding
eigenfunctions are considered in [4, 25, 26] when the energy density $H$ is
$p$-homogeneous and $C^{1}$, while the case where $H$ may be non-
differentiable is studied in [21]. Various applications to nonlinear parabolic
problems, encompassing the non-differentiable setting, are described in [12].

A complete Brezis-Oswald theorem usually consists of two statements. The first
one is devoted to derive necessary and sufficient condition for the
solvability of (1.2), under the aforementioned monotonicity assumption of
$t\mapsto f(x,t)/t^{p-1}$, as is done in [6, 13, 18]. The second one regards
uniqueness of the solution, and is clearly the focus of this research.

As briefly outlined in the first section, our main intent with this
investigation was to clarify how the various technical assumptions which
usually appear in Brezis-Oswald type result could be weakened to have a
cleaner picture, especially regarding its uniqueness part. It is clear, for
instance, that the regularity of the domain or of $H$ is not necessary for the
uniqueness statement to hold. Our interest in this part of Brezis-Oswald
theorem, however, lies mainly in its applications to the qualitative study of
solutions to PDE’s.

Indeed, uniqueness is the basic step needed to effectively approximate the
particular PDE problem under scrutiny of the form (1.2) with solutions of
regularised problems

$L_{n}(u)=f_{n}(x,u),$ (1.7)

each having a smooth solution $u_{n}$ allowing computations can be more
effectively performed. The sequence $(u_{n})$ then shown to enjoy uniform
bounds and therefore to converge up to subsequences to a solution of the
limiting, original problem (1.2). Then, uniqueness of solutions of (1.2) is
the key point ensuring that the approximating solutions actually converge to
the original one, which then (hopefully) inherits some properties of the
$u_{n}$’s. This scheme of proof is nowadays standard in both regularity theory
and qualitative properties of solution to PDE’s.

As a typical example, let us consider positive solutions of the quasi-linear
problem

$-\Delta_{p}u=f(u)$

with Dirichlet boundary conditions and a qualitative property of these
solutions known as quasi-concavity, meaning that the super-level sets of $u$
are convex. It is quite fortunate that the monotonicity assumption on
$t\mapsto f(t)/t^{p-1}$ is strictly connected to the general conditions found
in [2] ensuring quasi-concavity of the aforementioned solutions. Actually, the
results in [2, 3] are proved via the smoothing technique outlined above under
the assumption that the domain is convex with $C^{1,\alpha}$ boundary, but
Theorem 1.1 allow to extend the results to general convex domains.

As a motivation for Theorem 1.3, let us also stress that, in order for the
previously described approximation scheme to work, uniform bounds are also
essential to ensure compactness. This is especially true when, in the family
of problems (1.7), both the operators $L_{n}$ and the reactions $f_{n}$ may
have a varying growth control, only qualitatively bounded. In this regard,
Theorem 1.3 is inspired by [5], where the stability of the available a-priori
bounds had to be treated with extreme care for the employed contradiction
argument to carry on. In particular, (1.6) should be compared with [5,
Proposition 2.4], corresponding to the case $\theta=0$ in (1.5). In [5],
stability with respect to $p$ wasn’t an issue, and the a-priori bound was
derived via a Moser iteration technique involving the $p$-Sobolev inequality.
Since the constant in the latter blows up as $p\uparrow N$, the obtained
bounds in [5] are not stable in this limit. To be honest, we initially hoped
for this apparent instability to be substantial, suggesting the possibility of
interesting phenomena exhibited by solutions of, say, the $p$-Laplace
equations as $p\uparrow N$. Unfortunately, a different and, we think, more
streamlined proof of the $L^{\infty}$ bound in the spirit of De Giorgi method,
shows that this is not the case.

Let us finally mention that our proof of Theorem 1.1, detailed in the next
section, relies on a non-smooth version of the Picone inequality which relies
only on the positive $p$-homogeneity of $H$. This should be compared to [19]
where, however, differentiable convex $H$ are considered. A natural
alternative route to the Brezis-Oswald uniqueness result, as in [12, 13, 26],
is to consider the hidden convexity properties of the functional $J$ in (1.3).
Indeed, the core assumption that $t\mapsto f(x,t)/t^{p-1}$ be non-increasing
is implied by the weaker hypothesis that $t\mapsto F(x,t^{1/p})$ is concave.
Notice that this latter condition does not require $F$ to be differentiable in
its second variable, nor any bound on its derivative is needed to define the
functional $J$. Concavity of $t\mapsto F(x,t^{1/p})$ implies hidden convexity
of

$J_{F}(v)=-\int_{\Omega}F(x,v)\,dx,$

on the cone of non-negative $v\in W^{1,p}_{0}(\Omega)$, which means convexity
of $J_{F}$ along the curves

$t\mapsto\left(t\,u^{p}+(1-t)\,v^{p}\right)^{1/p}\qquad\forall u,v\in
W^{1,p}_{0}(\Omega),\quad u,v\geq 0.$

On the other hand, by [4], our non-smooth Picone inequality is equivalent to
the hidden convexity of $J_{H}$. Hence the whole $J$ turns out to be hidden
convex on the cone of nonnegative functions in $W^{1,p}_{0}(\Omega)$ and it is
only natural to expect few critical points from such a functional. We chose
not to adopt this approach since our notion of energy critical point, while
very weak, requires at least that the reaction $f(x,t)=\partial_{t}F(x,t)$ is
well defined and Caratheodory, but we point out that the interplay between
hidden convexity and the notion of critical point for l. s. c. functionals in
the sense of [9] has been initiated in [11] in the framework of nonlinear
eigenvalue problems.

###  1.3. Outline of the proof

The proof of Theorem 1.1 is split in two main parts: regularity and
uniqueness. The first obstacle to regularity is the lack of a proper Euler-
Lagrange equation to work with, in particular when dealing with critical
points of the l. s. c. functional $J$. Here we take advantage of the
variational characterisation of critical points $u$ as minima of the semi-
linearised functional

$J_{u}(v)=\int_{\Omega}H(Dv)-f(x,u)\,v\,dx.$

This allows to test the minimality of $J_{u}(u)$ against $J_{u}(u_{k})$, where
$u_{k}:=\max\\{k,u\\}$ for $k\geq 0$, and $J_{u}(u)\leq J_{u}(u_{k})$ reads

$\int_{\Omega}H(D(u-k)_{+})\,dx\leq\int_{\Omega}f(x,u)\,(u-k)_{+}\,dx.$

This inequality and the one-sided growth control on $f$ suffice to derive a
”fast decay” relation of the $L^{q}$ norms of $(u-k)_{+}$ much in the spirit
of De Giorgi regularity proof. However, in order to ensure stability of the
estimates in $p$, we avoid the natural Sobolev inequality and instead use an
interpolation inequality á la Gagliardo-Nirenberg with a more tamed constant
(see Proposition 2.2). Once the $L^{\infty}$ bound of Theorem 1.3 is proved,
the reaction is bounded and Giaquinta-Giusti hole-filling method [15] allows
to prove that the solutions belong to suitable De Giorgi classes, thus
providing continuity and a strong minimum principle for non-negative critical
points. In particular, an Euler-Lagrange equation (or rather, inclusion) is
satisfied by critical points, namely

$\int_{\Omega}(\xi,D\varphi)=\int_{\Omega}f(x,u)\,\varphi\,dx\qquad\forall\varphi\in
W^{1,p}_{0}(\Omega)$ (1.8)

where $\xi:\Omega\to\mathbb{R}^{N}$ is a measurable choice of the
multifunction $x\mapsto\partial H(Du(x))$ and $(\ ,\ )$ denotes the Euclidean
scalar product.

With these regularity tools at our disposal we can prove the uniqueness part.
First we derive a non-smooth version of Picone inequality in Lemma 5.1. If $H$
positively $p$-homogeneous and convex, the point-wise inequality

$p\,H(z)\geq(\xi,z),\qquad\forall\xi\in\partial H(z)$ (1.9)

holds true. The non-smooth Picone inequality follows from the latter and reads
as follows: if $u$ and $v$ are positive functions in $W^{1,1}_{\rm
loc}(\Omega)$, then a. e. in $\Omega$ it holds

$p\,H(Dv)\geq\left(\xi,D\frac{v^{p}}{u^{p-1}}\right),$

for any measurable $\xi:\Omega\to\mathbb{R}^{N}$ such that $\xi(x)\in\partial
H(Du(x))$ a. e.. Now, if $H$ is differentiable its convex subdifferential is
single-valued, i. e. $\partial H=\\{DH\\}$ and actually (1.9) is an equality,
so the smooth Picone inequality becomes

$\left(DH(Dv),Dv\right)\geq\left(DH(Du),D\frac{v^{p}}{u^{p-1}}\right).$

If both $u$ and $v$ are positive critical points, (1.8) gives (cheating a
little bit, since we are testing it for $u$ with $\varphi=v^{p}/u^{p-1}$)

$\int_{\Omega}f(x,v)\,v\,dx\geq\int_{\Omega}f(x,u)\,\frac{v^{p}}{u^{p-1}}\,dx.$

Exchanging the role of $u$ and $v$ and summing the corresponding inequalities,
one arrives at

$\int_{\Omega}\left(\frac{f(x,u)}{u^{p-1}}-\frac{f(x,v)}{v^{p-1}}\right)(u-v)\,dx\geq
0$ (1.10)

which yields uniqueness when $t\mapsto f(x,t)/t^{p-1}$ is strictly decreasing.

The point, in the non-differentiable case, is that in general the inequality
in (1.9) can be strict. However, it turns out that the measurable selection
$\xi$ of $\partial H(Du)$ arising from (1.8) actually attains equality in
(1.9) almost everywhere. This property is related to our variational notion of
critical point, and allows to carry on the proof as us usual. Finally, the
characterisation of the set of critical points described in Theorem 1.1 under
the non-increasing assumption on $t\mapsto f(x,t)/t^{p-1}$ is based on the
analysis of the equality case in the non-smooth Picone inequality, derived
from the identity

$\frac{f(x,u)}{u^{p-1}}=\frac{f(x,v)}{v^{p-1}}$

which must hold a. e. if $t\mapsto f(x,t)/t^{p-1}$ is non-increasing, thanks
to (1.10).

###  1.4. Structure of the paper

Section 2 contains preliminary material: we collect the notations used in the
paper, provide some basic functional inequalities, derive basic properties of
convex, positively $p$ homogeneous functions and describe the notion of energy
critical points and its basic features. Section 3 contains the proof of
Theorem 1.3 and some of its consequences, such as continuity and minimum
principles for energy critical points. In Section 4 we define and study the
first Dirichlet eigenvalues associated to possibly non-even convex energies,
aiming at the proof of point (3) of Theorem 1.1. In the final section we prove
the non-smooth Picone inequality and Theorem 1.1.

##  2\. Preliminaries

###  2.1. Notations

Throughout the paper we will make use of the following notations.

Given a number $a\in\mathbb{R}$, we set $a_{+}=\max\\{a,0\\}$ and
$a_{-}=-\min\\{0,a\\}$ and similarly for real valued functions. For every
$v\in\mathbb{R}^{N}$ we denote by $|v|$ its Euclidean norm while $(v,w)$
stands for the scalar product, $v,w\in\mathbb{R}^{N}$. The symbol $B$ denotes
a generic ball with unspecified center and radius while $B_{r}$ denotes a ball
of radius $r$ centred at the origin. $S_{1}$ stands for the unit sphere
$\partial B_{1}$.

For a Lebesgue measurable $E\subseteq\mathbb{R}^{N}$, $|E|$ denotes its
measure and we set $\omega_{N}=|B_{1}|$. We will omit the domain of
integration when it is the whole $\mathbb{R}^{N}$, if this cause no confusion.
For a measurable $E\subseteq\mathbb{R}^{N}$, ${\mathcal{L}}(E;\mathbb{R}^{M})$
will denote the set of Lebesgue measurable functions from $E$ to
$\mathbb{R}^{M}$. Given an open set $\Omega\subseteq\mathbb{R}^{N}$, $f$ will
denote a Caratheodory function $f:\Omega\times\mathbb{R}\to\mathbb{R}$ and

$F(x,t)=\int_{0}^{t}f(x,s)\,ds\qquad\forall t\in\mathbb{R}.$ (2.1)

Given $u\in{\mathcal{L}}(\Omega;\mathbb{R})$, we’ll denote by $f(\cdot,u)$ the
measurable function $x\mapsto f(x,u(x))$ and similarly for $F(\cdot,u)$. The
Lebesgue spaces are denoted as usual by $L^{p}(E)$ for $p\geq 1$, with norm
$\|\cdot\|_{L^{p}(E)}$, while $\sup_{E}u$ and $\inf_{E}u$ stand for the
essential supremum and infimum respectively. The Lebesgue spaces of vector
valued functions $u:E\to\mathbb{R}^{M}$, will be denoted by
$\left(L^{p}(E)\right)^{M}$ with norms $\||u|\|_{L^{p}(E)}$, where $|u|$ is
the standard Euclidean norm of $\mathbb{R}^{M}$. Furthermore, we set for the
sake of notational simplicity $\|f\|_{p}:=\|f\|_{L^{p}(\mathbb{R}^{N})}$.

Given $p\in[1,\infty]$, $p^{\prime}$ denotes its conjugate, namely
$p^{\prime}=p/(p-1)$ if $p\in\,]1,\infty[$, $1^{\prime}=\infty$ and
$\infty^{\prime}=1$. Given an integer $N\geq 2$ and $p\in[1,\infty]$, we
denote by

$p^{*}=\begin{cases}\dfrac{N\,p}{N-p}&\text{if $p<N$}\\\ \text{any positive
number}&\text{if $p=N$}\\\ \infty&\text{if $p>N$}.\end{cases}$

For an open $\Omega\subseteq\mathbb{R}^{N}$ with finite measure, the Sobolev
space $W^{1,p}_{0}(\Omega)$ is the closure of $C^{\infty}_{c}(\Omega)$ w. r.
t. the seminorm $\|Du\|_{p}$. In the finite measure case,
$W^{1,p}_{0}(\Omega)$ is a Banach space, continuously embedded in
$L^{q}(\Omega)$ for all $q\in[1,p^{*}]$. Its dual will be denoted by
$W^{-1,p^{\prime}}(\Omega)$, with duality pairing denoted by
$\langle\Lambda,\varphi\rangle$ for $\Lambda\in W^{-1,p^{\prime}}(\Omega)$ and
$\varphi\in W^{1,p}_{0}(\Omega)$. If $p<N$, the homogeneous Sobolev space
$D^{1,p}(\mathbb{R}^{N})$ is the space of functions in $L^{1}_{\rm
loc}(\mathbb{R}^{N})$ with distributional derivative in
$L^{p}(\mathbb{R}^{N})$ and which vanish at infinity, meaning that

$|\\{|u|>k\\}|<\infty\qquad\text{ for all $k>0$}.$

###  2.2. Functional inequalities

Functions in $D^{1,p}(\mathbb{R}^{N})$ for $1\leq p<N$ enjoy the Sobolev
inequality

$\|u\|_{p^{*}}\leq C(N,p)\,\|Du\|_{p}.$ (2.2)

We are particularly interested in the $p=1$ case, for which
$C(N,1)=N^{-1}\,\omega_{N}^{-1/N}$ for all $N\geq 2$.

The following result is probably well known, but we couldn’t find a precise
references and thus provide its proof.

######  Proposition 2.1.

For $1\leq p,q<\infty$, let $X_{p,q}(\mathbb{R}^{N})$ be the vector space of
functions in $L^{q}(\mathbb{R}^{N})$ with distributional derivative in
$L^{p}(\mathbb{R}^{N})$, equipped with the norm $\|u\|=\|Du\|_{p}+\|u\|_{q}$.
Then $C^{\infty}_{c}(\mathbb{R}^{N})$ is dense in $X_{p,q}(\mathbb{R}^{N})$.

###### Proof.

The set $C^{\infty}(\mathbb{R}^{N})\cap X_{p,q}$ is dense in $X_{p,q}$ by
known results (see [22, Theorem 1, Section 1.1.5]), so it suffices to show
that any $u\in C^{\infty}(\mathbb{R}^{N})\cap X_{p,q}$ can be approximated by
functions in $C^{\infty}_{c}(\mathbb{R}^{N})$. Fix a cutoff $\eta\in
C^{\infty}_{c}(B_{2};[0,1])$ such that $\eta\equiv 1$ in $B_{1}$ and, for any
$u\in C^{\infty}(\mathbb{R}^{N})\cap X_{p,q}$, define for $k\in\mathbb{N}$

$\eta_{k}(x)=\eta(x/k),\qquad u_{k}=u\,\eta_{k},\qquad A_{k}=B_{2\,k}\setminus
B_{k},$

so that $u_{k}\in C^{\infty}_{c}(\mathbb{R}^{N})$, ${\rm
supp}\,(D\eta_{k})\subseteq A_{k}$ and
$\|D\eta_{k}\|_{\infty}=\|D\eta\|_{\infty}/k$. Clearly
$\|u-u_{k}\|_{q}\leq\|u\|_{L^{q}(\mathbb{R}^{N}\setminus B_{k})}\to 0$ as
$k\to\infty$, so we only have to estimate $\|D(u-u_{k})\|_{p}$. It holds

$\begin{split}\int|Du-Du_{k}|^{p}\,dx&\leq
C_{p}\int|Du|^{p}\,(1-\eta_{k})^{p}\,dx+C_{p}\,\int|u|^{p}\,|D\eta|^{p}\,dx\\\
&\leq C_{p}\,\int_{\mathbb{R}^{N}\setminus
B_{k}}|Du|^{p}\,dx+\frac{C_{p}\,\|D\eta\|_{\infty}^{p}}{k^{p}}\,\int_{A_{k}}|u|^{p}\,dx.\end{split}$
(2.3)

and the first term on the right vanishes for $k\to\infty$ since $|Du|\in
L^{p}(\mathbb{R}^{N})$, so we are left to estimate the second term.

The case $p<N$ is standard: by Chebichev inequality any $u\in
L^{q}(\mathbb{R}^{N})$ vanishes at infinity, hence $X_{p,q}\subseteq
D^{1,p}(\mathbb{R}^{N})\subseteq L^{p^{*}}(\mathbb{R}^{N})$. Therefore, by
Hölder inequality

$\int_{A_{k}}|u|^{p}\,dx\leq\left(\int_{A_{k}}|u|^{p^{*}}\,dx\right)^{\frac{p}{p^{*}}}\,|A_{k}|^{1-\frac{p}{p^{*}}}=C\,k^{p}\,\left(\int_{A_{k}}|u|^{p^{*}}\,dx\right)^{\frac{p}{p^{*}}},$

and the last factor on the right vanishes as $k\to\infty$ due to $u\in
L^{p^{*}}(\mathbb{R}^{N})$, hence also the last term in (2.3) does so. If
$p\geq N$, we use the inequality

$\int_{B_{2}\setminus B_{1}}|v|^{p}\,dx\leq C\,\int_{B_{2}\setminus
B_{1}}|Dv|^{p}\,dx+C\,\left(\int_{B_{2}\setminus
B_{1}}|v|^{q}\,dx\right)^{p/q}$

which is trivial by Hölder when $q\geq p$ and holds true thanks to Gagliardo-
Nirenberg inequality when $q<p$. Applying it to $v(x)=u(k\,x)$ and changing
variables, we get

$\frac{1}{k^{p}}\,\int_{A_{k}}|u|^{p}\,dx\leq
C\,\int_{A_{k}}|Du|^{p}\,dx+\frac{C\,k^{N\,\left(1-\frac{p}{q}\right)}}{k^{p}}\left(\int_{A_{k}}|u|^{q}\,dx\right)^{p/q}.$

As $k\to\infty$, the first term on the right vanishes again, as well as the
second, since

$N\left(1-\frac{p}{q}\right)<p\qquad\forall p\geq N,q<\infty.$

∎

The next proposition is a particular case of the general Gagliardo-Nirenberg
inequality. In order to get stable $L^{\infty}$ bounds in Section 3, however,
we have to pay special attention to the dependance from the parameters and
thus give a direct proof depending only on (2.2).

######  Proposition 2.2 (Gagliardo-Nirenberg).

Let $N\geq 2$, $u\in W^{1,1}_{\rm loc}(\mathbb{R}^{N})$ and $1\leq
p,q<\infty$. Then for any $u\in W^{1,1}_{\rm loc}(\mathbb{R}^{N})$ it holds

$\left(\int|u|^{\frac{N\,(q\,p+p-q)}{(N-1)\,p}}\,dx\right)^{\frac{N-1}{N}}\leq
G_{p,q,N}\left(\int|Du|^{p}\,dx\right)^{1/p}\left(\int|u|^{q}\,dx\right)^{1/p^{\prime}},$
(2.4)

where

$G_{p,q,N}=\frac{1}{N\,\omega_{N}^{1/N}}\,\frac{q\,p+p-q}{p}.$

###### Proof.

We can suppose that the right-hand side is finite, which amounts to $u\in
X_{p,q}$. By the previous Proposition we can also suppose that $u\in
C^{\infty}_{c}(\mathbb{R}^{N})$. For $p=1$ (2.4) reduces to (2.2), so we can
suppose that $p>1$. Let

$r=\frac{q}{p^{\prime}}+1=\frac{q\,p+p-q}{p}$

and apply (2.2) to $v=|u|^{r}$ and then Hölder inequality, to get

$\begin{split}\left(\int|u|^{\frac{q\,p+p-q}{p}1^{*}}\,dx\right)^{1/1^{*}}&\leq\frac{1}{N\,\omega_{N}^{1/N}}\,\int|D|u|^{r}|\,dx=\frac{1}{N\,\omega_{N}^{1/N}}\,r\,\int|u|^{r-1}\,|Du|\,dx\\\
&\leq\frac{1}{N\,\omega_{N}^{1/N}}\,r\,\left(\int|Du|^{p}\,dx\right)^{1/p}\left(\int|u|^{q}\,dx\right)^{1/p^{\prime}}.\end{split}$

∎

###  2.3. Convex $p$-homogeneous functions

Let $X$ be a Banach space in duality with $X^{*}$ via the duality pairing
$\langle\cdot,\cdot\rangle$. Given a continuous convex function
$C:X\to\mathbb{R}$ (we will alway restrict to this setting) its
subdifferential at $x\in X$ is

$\partial C(x)=\left\\{x^{*}\in X^{*}:C(x+v)-C(x)\geq\langle x^{*},v\rangle\
\forall v\in X\right\\}.$

When $X=\mathbb{R}^{N}$ we’ll identify $X$ with $X^{*}$ via the scalar product
as a duality pairing, hence in this case $\partial
C(x)\subseteq\mathbb{R}^{N}$.

Throughout the paper, $H:\mathbb{R}^{N}\to[0,\infty[$ will denote a convex
function. Since its domain is $\mathbb{R}^{N}$, any such function is locally
Lipschitz. The function $H$ is positively $p$-homogeneous for some $p\geq 1$
if

$H(\lambda\,z)=\lambda^{p}\,H(z)\qquad\text{ for all $\lambda>0$ and
$z\in\mathbb{R}^{N}$}.$

Notice that we do not assume regularity of $H$, nor that $H$ is even. In the
positively $p$-homogeneous case the whole $H$ is completely determined by its
associated ball, i. e. the convex closed set

$B_{H}=\\{z\in\mathbb{R}^{N}:H(z)\leq 1\\},$

which in general is not symmetric around the origin but always contains $0$ in
its interior. Vice-versa, given any convex $K$ containing $0$ in its interior,
the function

$H_{K}(z)=\left(M_{K}(z)\right)^{p}$

where $M_{K}$ is the Minkowski functional of $K$

$M_{K}(z)=\inf\\{t>0:z/t\in K\\},$

is a convex, positively $p$-homogeneous function with associated ball $K$. The
set of symmetries of $H$ is the set of symmetries of its associated ball,
namely

${\rm Sym}\,(H)=\\{T\in O(N):T(B_{H})=B_{H}\\}$

where $O(N)$ is the orthogonal group in dimension $N$. By construction, for
any $T\in{\rm Sym}\,(H)$ it holds $H\circ T=H$.

######  Proposition 2.3.

Let $H:\mathbb{R}^{N}\to[0,\infty[$ be convex and positively $p$-homogeneous
for some $p\geq 1$. Then for any $w\in\mathbb{R}^{N}$ and $\xi\in\partial
H(w)$, it holds

$p\,H(z)\geq(\xi,z)\qquad\forall\xi\in\partial H(z).$ (2.5)

###### Proof.

Let $\xi\in\partial H(z)$ for some $z\in\mathbb{R}^{N}$. By $p$-homogeneity
and convexity

$t^{p}\,H(z)=H(t\,z)\geq H(z)+(\xi,t\,z-z)$

for all $t\geq 1$, while the left hand side coincides with the right hand side
when $t=1$. Therefore

$\left.\frac{d}{dt}\big{[}t^{p}\,H(z)-H(z)-(\xi,t\,z-z)\big{]}\right|_{t=1}\geq
0,$

which gives (2.5).

∎

###  2.4. Energy critical points

In this section we consider functionals of the form (1.3) with more general
structural conditions than those required in Theorem 1.1. More precisely, we
assume that

$\begin{split}\text{ $H:\mathbb{R}^{N}\to[0,+\infty[$ is convex and for some
$p\geq 1$ and $0\leq c<d\leq\infty$ obeys}\\\ c\,|z|^{p}\leq H(z)\leq
d\,|z|^{p}\qquad\forall
z\in\mathbb{R}^{N},\qquad\qquad\qquad\qquad\,\end{split}$ (2.6)
$\text{$f:\Omega\times\mathbb{R}\to\mathbb{R}$ is Caratheodory
and}\quad\sup_{|s|\leq
t}\|f(\cdot,s)\|_{L^{\infty}(\Omega)}<\infty\quad\forall t>0,$ (2.7)
$\text{there exist $\theta,\mu\geq 0$, $1\leq q\leq p^{*}$ s.\,t. }{\rm
sign}\,(t)\,f(x,t)\leq\mu\,(|t|+\theta)^{q-1}\quad\forall t\in\mathbb{R}.$
(2.8)

Regarding $F$ as per (2.1), it follows from (2.7) that when $p>N$ then
$F(\cdot,v)\in L^{\infty}(\Omega)$ for any $v\in W^{1,p}_{0}(\Omega)$ by
Morrey embedding, while, when $p\leq N$, from (2.8) we readily obtain the one-
sided estimate

$\sup_{\Omega}F(\cdot,t)\leq\frac{\mu}{q}\,\left(|t|+\theta\right)^{q}$

for a finite $q$. Therefore, as long as $\Omega$ has finite measure (which
will be assumed henceforth),

$\left(H(Dv)-F(\cdot,v)\right)_{-}\in L^{1}(\Omega)\qquad\forall v\in
W^{1,p}_{0}(\Omega).$ (2.9)

and the functional $J$ in (1.3) is well defined on $W^{1,p}_{0}(\Omega)$,
eventually assuming the value $\infty$ if $F(\cdot,v)\notin L^{1}(\Omega)$.
Fatou’s lemma ensures that $J$ is l. s. c. on $W^{1,p}_{0}(\Omega)$ with the
strong topology (regardless of $p\geq 1$), hence the non-smooth critical point
theory of [9] can be applied. Notice also that if (2.8) is strengthened to
hold for both $f$ and $-f$, then $J$ turns out to be locally Lipschitz, since
its second term is actually $C^{1}$ on $W^{1,p}_{0}(\Omega)$, allowing to
employ Clarke critical point theory [8]. We will not pursue these approaches
here for sake of brevity and will use the following notion of critical point,
taken from [10].

######  Definition 2.4 (Energy critical point).

Suppose $\Omega\subseteq\mathbb{R}^{N}$ has finite measure. Under assumptions
(2.6), (2.7) and, if $p\leq N$, (2.8), we say that $u\in W^{1,p}_{0}(\Omega)$
is an energy critical point for $J$, if $J(u)<\infty$ and $u$ minimises the
convex functional

$J_{u}(v)=\int_{\Omega}H(Dv)-f(x,u)\,v\,dx$ (2.10)

defined on the space

$V_{u}=\left\\{v\in W^{1,p}_{0}(\Omega):f(\cdot,u)\,v\in
L^{1}(\Omega)\right\\}.$

A few remarks on this notion of critical point are in order.

######  Remark 2.5.

  * •

Notice that an energy critical point $u$ is assumed to lie in $V_{u}$, i. e.
$f(\cdot,u)\,u\in L^{1}(\Omega)$. The role of $p$ in the previous definition
is connected to the validity of (2.9) (granted, as noted, by (2.8)), so that
$J$ is at least well defined. It should be remarked that, at this level of
generality, it has nothing to do with (2.6), where one could also have $c=0$
and $d=\infty$. Regardless, from $H\geq 0$ and the assumption $J(u)<\infty$,
we see that any energy critical point $u$ fulfills $H(Du)\in L^{1}(\Omega)$.
In particular, if $c$ in (2.6) is finite and positive and $p\leq N$, then
$u\in L^{q}(\Omega)$ by the finite measure assumption on $\Omega$ and Sobolev
embedding. Let us also remark that if $u\in W^{1,p}_{0}(\Omega)$ is assumed,
or proved, to be bounded, assumption (2.8) is irrelevant as $J$ is always well
defined as long as $\Omega$ has finite measure and in this case
$V_{u}=W^{1,p}_{0}(\Omega)$.

  * •

We stress here that the notion of energy critical point is a very general one
as it does not involve any topological structure on $W^{1,p}_{0}(\Omega)$. As
such, it is useless if one wants to develop an effective critical point
theory, where topological considerations are paramount. Under conditions (2.6)
with $c>0$ and $p>1$, as well as (2.7) and (2.8) if $p\leq N$, a critical
point for $J$ in the sense of [9] (with the strong topology on
$L^{p^{*}}(\Omega)$) is an energy critical point (see [10, Theorem 5]).
Similarly, if $d<\infty$ and $p>1$ in (2.6) and (2.8) holds for both $f$ and
$-f$, then any critical point in the sense of Clarke for $J$ is an energy
critical point.

  * •

In general, (2.7) and the finite measure assumption on $\Omega$ ensure that
$f(x,u)\in L^{1}(\Omega)$, hence $V_{u}$ contains $C^{\infty}_{c}(\Omega)$ and
is dense in $W^{1,p}_{0}(\Omega)$. In the case $d<\infty$ of (2.6), being

$J_{H}(v):=\int_{\Omega}H(Dv)\,dx$ (2.11)

convex and finite on $W^{1,p}_{0}(\Omega)$, it is locally Lipschitz, hence for
any energy critical point $u$ it holds

$\int_{\Omega}f(x,u)\,(v-u)\,dx\leq J_{H}(v)-J_{H}(u)\leq
C_{r}\,\|D(v-u)\|_{p}$

for all $v\in V_{u}$ with $\|D(v-u)\|_{p}\leq r$. Therefore the linear
functional

$V_{u}\ni v\mapsto\int_{\Omega}f(x,u)\,v\,dx$

has a unique extension $T$ to $W^{1,p}_{0}(\Omega)$ as an element in
$W^{-1,p}(\Omega)$. Notice that while $\langle T,v\rangle$ is finite for all
$v\in W^{1,p}_{0}(\Omega)$, this does not imply that $f(\cdot,u)\,v\in
L^{1}(\Omega)$, unless a one-sided bound on $f(x,u)\,v$ holds true, enabling
to apply Brezis-Browder theorem.

######  Proposition 2.6.

Let $\Omega\subseteq\mathbb{R}^{N}$ be open with finite measure and assume
(2.6) with $d<\infty$, (2.7) and, if $p\leq N$, (2.8). If $u\in
W^{1,p}_{0}(\Omega)$ is an energy critical point for $J$, there exists $\xi\in
L^{p^{\prime}}(\Omega;\mathbb{R}^{N})$ such that

$\begin{cases}\text{$\xi(x)\in\partial H(Du(x))$}&\text{ for
a.\,e.\,$x\in\Omega$}\\\\[10.0pt]
\displaystyle{\int_{\Omega}\left(\xi,D\varphi\right)-f(x,u)\,\varphi\,dx=0}&\
\forall\varphi\in V_{u}.\end{cases}$ (2.12)

###### Proof.

Let us compute $\partial J_{H}$ (see (2.11)), following [23]. The operator

$W^{1,p}_{0}(\Omega)\ni v\mapsto A(v)=Dv\in\left(L^{p}(\Omega)\right)^{N}$

is linear and continuous with transpose

$\big{(}L^{p^{\prime}}(\Omega)\big{)}^{N}\ni g\mapsto A^{*}(g)={\rm div}\,g\in
W^{-1,p^{\prime}}(\Omega)$

where, as usual, ${\rm div}\,g\in W^{-1,p^{\prime}}(\Omega)$ acts as

$\langle{\rm
div}\,g,\varphi\rangle=\int_{\Omega}\left(g,D\varphi\right)\,dx\qquad\varphi\in
W^{1,p}_{0}(\Omega).$ (2.13)

The convex functional

$\left(L^{p}(\Omega)\right)^{N}\ni g\mapsto I_{H}(g)=\int_{\Omega}H(g)\,dx$

is locally Lipschitz thanks to $d<\infty$ in (2.6) and has convex
subdifferential (see [24, Theorem 21])

$\partial I_{H}(g)=\left\\{\xi\in
L^{p^{\prime}}(\Omega;\mathbb{R}^{N}):\text{$\xi(x)\in\partial H(g(x))$ for
a.\,e.\,$x\in\Omega$}\right\\}.$

Clearly $J_{H}=I_{H}\circ A$ and by [24, Theorem 19] it holds

$\partial\left(I_{H}\circ A\right)(u)=A^{*}\left(\partial I_{H}(A(u))\right).$

It follows that at any $v\in W^{1,p}_{0}(\Omega)$

$\partial J_{H}(v)=\left\\{{\rm div}\,\xi:\xi\in
L^{p^{\prime}}(\Omega;\mathbb{R}^{N})\text{ and $\xi(x)\in\partial H(Dv(x))$
a.\,e.\,in $\Omega$}\right\\},$

where an element of the set on the right acts as in (2.13). To conclude, let
$T\in W^{-1,p^{\prime}}(\Omega)$ be the unique extension of

$v\mapsto\int_{\Omega}f(\cdot,u)\,v\,dx,\qquad v\in V_{u}$

granted by Remark 2.5. Then $u$ also minimises $v\mapsto J_{H}(v)-\langle
T,v\rangle$ on $W^{1,p}_{0}(\Omega)$, by the continuity of $J_{H}-T$ and the
density of $V_{u}$ in $W^{1,p}_{0}(\Omega)$. Therefore

$0\in\partial(J_{H}-T)(u)=\partial J_{H}(u)-T$

which implies (2.12) when bracketed with $v\in V_{u}$.

∎

The following easy consequence will be useful later.

######  Corollary 2.7.

Suppose that the assumptions of the previous Proposition hold true and
additionally that $H$ is positively $p$-homogeneous. Then any energy critical
point $u$ for $J$ fulfills

$(\xi,Du)=p\,H(Du)\qquad\text{a.\,e.\,in $\Omega$}$ (2.14)

for any $\xi\in L^{p^{\prime}}(\Omega;\mathbb{R}^{N})$ obeying (2.12). In
particular,

$p\,\int_{\Omega}H(Du)\,dx=\int_{\Omega}f(x,u)\,u\,dx.$ (2.15)

###### Proof.

Since $u$ minimises the convex functional $J_{u}$ defined in (2.10), it does
so on the half-line through $u$, namely on $\\{t\,u:t>0\\}\subseteq V_{u}$.
The function

$h(t):=J_{u}(t\,u)=t^{p}\,\int_{\Omega}H(Du)\,dx-t\,\int_{\Omega}f(x,u)\,u\,dx$

is smooth and convex on $]0,\infty[$, therefore the previous statement forces
$h^{\prime}(1)=0$, which is (2.15). Let now $\xi\in
L^{p^{\prime}}(\Omega;\mathbb{R}^{N})$ fulfil (2.12), so that testing with
$\varphi=u\in V_{u}$, we get

$\int_{\Omega}(\xi,Du)\,dx=\int_{\Omega}f(x,u)\,u\,dx.$

By using (2.15), we deduce

$\int_{\Omega}p\,H(Du)-(\xi,Du)\,dx=0,$

but by (2.5) the integrand is non-negative, therefore it must vanish a. e. in
$\Omega$, proving (2.14). ∎

##  3\. Regularity

The following lemma enucleates the De Giorgi method for proving $L^{\infty}$
bounds.

######  Lemma 3.1.

Let $\Omega\subseteq\mathbb{R}^{N}$, $q\geq 1$ and $h_{0}>0$. Let $u$ be such
that $(u-h_{0})_{+}\in L^{q}(\Omega)$ and $u$ satisfies the inequalities

$\int_{\Omega}(u-k)_{+}^{q}\,dx\leq\frac{b}{(k-h)^{\beta}}\left(\frac{k}{k-h}\right)^{\delta}\left(\int_{\Omega}(u-h)_{+}^{q}\,dx\right)^{1+\gamma}$
(3.1)

for all $h_{0}\leq h<k$ and parameters $b,\beta,\delta\geq 0$ and $\gamma>0$.
Then $u$ is bounded from above and, if furthermore $\beta$ is positive, it
holds

$\sup_{\Omega}u\leq\max\left\\{2\,h_{0},\,C\,\left(b^{1/\gamma}\int_{\Omega}(u-h_{0})_{+}^{q}\,dx\right)^{\gamma/\beta}\right\\}$
(3.2)

for $C=C(b,\gamma,\beta,\delta)>0$ smoothly depending on the data.

###### Proof.

Let $M\geq 2\,h_{0}$ to be determined and set $k_{n}=M\,(1-2^{-1-n})$ for
$n\geq 0$. Notice that $k_{n}\uparrow M$, $k_{0}=M/2\geq h_{0}$ and

$\frac{b}{(k_{n+1}-k_{n})^{\beta}}\left(\frac{k_{n+1}}{k_{n+1}-k_{n}}\right)^{\delta}\leq\frac{b}{M^{\beta}}\,2^{(n+2)\,(\beta+\delta)}.$

We can then apply (3.1) for $k=k_{n+1}$ and $h=k_{n}$ to get

$\int_{\Omega}(u-k_{n+1})_{+}^{q}\,dx\leq\frac{A\,b\,A^{n}}{M^{\beta}}\left(\int_{\Omega}(u-k_{n})_{+}^{q}\,dx\right)^{1+\gamma}.$

for $A=4^{\beta+\delta}$. By [17, Proposition 7.1], if

$\int_{\Omega}(u-k_{0})_{+}^{q}\,dx\leq\frac{M^{\beta/\gamma}}{(A\,b)^{1/\gamma}\,A^{1/\gamma^{2}}}$
(3.3)

then

$\lim_{n}\int_{\Omega}(u-k_{n})_{+}^{q}\,dx=0,$

which forces $u\leq M$ in $\Omega$. Condition (3.3) is satisfied for
sufficiently large $M$, since $(u-h_{0})_{+}\in L^{q}(\Omega)$ and hence, for
$M\uparrow\infty$,

$\int_{\Omega}(u-k_{0})_{+}^{q}\,dx=\int_{\Omega}(u-M/2)_{+}^{q}\,dx\to 0.$

Therefore the upper boundedness of $u$ is proved. In the case $\beta>0$,
observe that (3.3) is fulfilled (and hence $u\leq M$) for all $M$ such that

$M\geq 2\,h_{0}\qquad\text{and}\qquad M\geq
A^{\frac{\gamma+1}{\gamma\,\beta}}\,\left(b^{1/\gamma}\int_{\Omega}(u-h_{0})_{+}^{q}\,dx\right)^{\gamma/\beta},$

proving the bound (3.2). ∎

######  Theorem 3.2 ($L^{\infty}$ bounds).

Let $\Omega\subseteq\mathbb{R}^{N}$ be open with finite measure and suppose
(2.6) holds with $c>0$, as well as (2.7) and, if $p\leq N$, (2.8). Then, any
energy critical point $u\in W^{1,p}_{0}(\Omega)$ of $J$ is bounded. Moreover,
if (2.8) holds true (regardless of $p$) for some $q<p^{*}$, then

$\|u\|_{L^{\infty}(\Omega)}\leq\max\left\\{\theta,C\,\left((\mu/c)^{\frac{N}{p}}\int_{\Omega}|u|^{q}\,dx\right)^{\frac{p}{p\,q+(p-q)\,N}}\right\\}$
(3.4)

for some $C=C(N,p,q)$ independent of $d$, $\theta$, $\mu$ and $\Omega$.
Moreover, for any fixed $N$,
$C(N,\cdot,\cdot):\,[1,\infty[\times[1,p^{*}[\,\to\mathbb{R}_{+}$ is positive
and continuous.

###### Proof.

By considering $H/c$ and $f/c$, $u$ is still an energy critical point of the
corresponding functional, so we can assume that $c=1$. Without loss of
generality, it suffices to prove that $u$ is bounded above (otherwise we look
at $v(x)=-u(x)$, which is a critical point of a functional with
$\check{H}(z)=H(-z)$ and $\check{f}(x,t)=-f(x,-t)$ obeying the same structural
conditions as $J$). The boundedness of $u$ being trivial from Morrey embedding
when $p>N$, we may suppose that $p\geq 1$ is arbitrary and that (2.8) holds
true. Moreover, we can assume $q<\infty$ in (2.8), since otherwise the
condition $q\leq p^{*}$ forces $p>N$ and again boundedness follows from Morrey
embedding. Finally, by the definition of energy critical point and the
discussion following Definition 2.4, we can also assume that $H(Du)$ and
$f(\cdot,u)\,u$ belong to $L^{1}(\Omega)$, while $u\in L^{q}(\Omega)$.

Given $k>0$, we compare the minimality of $u$ with $\min\\{k,u\\}$, which
belongs to $V_{u}$ since $|f(\cdot,u)\,\min\\{k,u\\}|\leq|f(\cdot,u)\,u|\in
L^{1}(\Omega)$. From

$J_{u}(u)\leq J_{u}(\min\\{k,u\\})$

we get, by the locality of the weak gradient and (2.8),

$\int_{{\mathcal{A}}_{k}}H(Du)\,dx\leq\int_{{\mathcal{A}}_{k}}f(x,u)\,(u-k)\,dx\leq\mu\,\int_{{\mathcal{A}}_{k}}(u+\theta)^{q-1}\,(u-k)\,dx$
(3.5)

where we set

${\mathcal{A}}_{k}=\\{x\in\Omega:u(x)\geq k\\}$

which has finite measure. The inequality

$\mu\,(t+\theta)^{q-1}\,(t-k)\leq\mu\,4^{q-1}(k^{q}+(t-k)^{q}),$ (3.6)

holds true for any $t\geq k\geq\theta/2$111Indeed, if $\theta/2\leq k\leq
t\leq 2\,k$ then $(t+\theta)^{q-1}\,(t-k)\leq k\,(2\,k+\theta)^{q-1}\leq
4^{q-1}\,k^{q}$, while if $t\geq 2\,k$ then $t\leq 2\,(t-k)$ and thus
$(t+\theta)^{q-1}\,(t-k)\leq 2^{q-1}\,(t-k)^{q}$.. Using (2.6), the growth
condition on $f$ and (3.6), we get

$\int_{{\mathcal{A}}_{k}}|Du|^{p}\,dx\leq\mu\,4^{q-1}\,\int_{{\mathcal{A}}_{k}}k^{q}+(u-k)^{q}\,dx$
(3.7)

which holds for any $k\geq\theta/2$.

We now aim at applying Lemma 3.1. First notice that for finite $q$, the
inequality $q\leq p^{*}$ is equivalent (even in the case $p\geq N$) to

$q\,p+(p-q)\,N\geq 0.$ (3.8)

For $N\geq 2$, (3.8) implies

$q\leq\frac{N}{N-1}\,\frac{q\,p+p-q}{p}$

so that we can chain Hölder, Gagliardo-Nirenberg inequality (2.4) and (3.7) to
get

$\begin{split}\int_{\Omega}&(u-k)_{+}^{q}\,dx\leq\left(\int_{\Omega}(u-k)_{+}^{1^{*}\frac{q\,p+p-q}{p}}\,dx\right)^{\frac{q\,p}{1^{*}\,(q\,p+p-q)}}\,|{\mathcal{A}}_{k}|^{1-\frac{q\,p}{1^{*}\,(q\,p+p-q)}}\\\
&\leq
C\,\left(\int_{\Omega}|D(u-k)_{+}|^{p}\,dx\right)^{\frac{q}{q\,p+p-q}}\left(\int_{\Omega}(u-k)_{+}^{q}\,dx\right)^{\frac{q\,p}{p^{\prime}(q\,p+p-q)}}\,|{\mathcal{A}}_{k}|^{\frac{q\,p+(p-q)\,N}{N\,(q\,p+p-q)}}\\\
&\leq
C\,\left(\mu\,\int_{{\mathcal{A}}_{k}}k^{q}+(u-k)^{q}\,dx\right)^{\frac{q}{q\,p+p-q}}\left(\int_{\Omega}(u-k)_{+}^{q}\,dx\right)^{\frac{q\,(p-1)}{q\,p+p-q}}\,|{\mathcal{A}}_{k}|^{\frac{q\,p+(p-q)\,N}{N\,(q\,p+p-q)}}\end{split}$
(3.9)

with

$C=\left(4^{q-1}\,G_{p,q,N}^{p}\right)^{\frac{q}{q\,p+p-q}}$

smoothly depending on the data.

Let now $\theta/2\leq h<k$. By Chebishev inequality

$|{\mathcal{A}}_{k}|\leq\frac{1}{(k-h)^{q}}\int_{\Omega}(u-h)_{+}^{q}\,dx,$
(3.10)

so that, since $(u-k)_{+}\leq(u-h)_{+}$,

$\int_{{\mathcal{A}}_{k}}k^{q}+(u-k)^{q}\,dx=k^{q}\,|{\mathcal{A}}_{k}|+\int_{\Omega}(u-k)_{+}^{q}\,dx\leq\left(\frac{k^{q}}{(k-h)^{q}}+1\right)\int_{\Omega}(u-h)_{+}^{q}\,dx,$

hence, since $k/(k-h)>1$,

$\int_{{\mathcal{A}}_{k}}k^{q}+(u-k)^{q}\,dx\leq
2\left(\frac{k}{k-h}\right)^{q}\int_{\Omega}(u-h)_{+}^{q}\,dx.$

Inserting the latter into (3.9) and using again (3.10) on the last factor
gives

$\int_{\Omega}(u-k)_{+}^{q}\,dx\leq\frac{C\,\mu^{\frac{q}{q\,p+p-q}}}{(k-h)^{q\,\frac{q\,p+(p-q)\,N}{N\,(q\,p+p-q)}}}\,\left(\frac{k}{k-h}\right)^{\frac{q^{2}}{q\,p+p-q}}\,\left(\int_{\Omega}(u-k)_{+}^{q}\,dx\right)^{1+\frac{q\,p}{N\,(q\,p+p-q)}}$

and Lemma 3.1 yields the stated boundedness with the choices

$b=C\,\mu^{\frac{q}{q\,p+p-q}},\qquad\beta=q\,\frac{q\,p+(p-q)\,N}{N\,(q\,p+p-q)},\qquad\gamma=\frac{q\,p}{N\,(q\,p+p-q)},\qquad\delta=\frac{q^{2}}{q\,p+p-q}$

(notice that $b,\delta,\gamma$ are positive and $\beta$ is non-negative thanks
to (3.8)). Moreover if $q<p^{*}$ then the previous choice of $\beta$ is
positive, so that (3.2) provides (3.4) by noting that
$(u-\theta/2)_{+}\leq|u|$. If $N=1$ boundedness is trivial from $u\in
W^{1,p}_{0}(\Omega)$ and $p>1$, while the estimate (3.4) is obtained as
before, this time instead of using the Gagliardo-Nirenberg inequality (2.4),
employing

$\|v\|_{\infty}^{\frac{q\,p+p-q}{p}}\leq\frac{q\,p+p-q}{p}\,\|Dv\|_{p}\,\|v\|_{q}^{q/p^{\prime}},$

which can be proved by elementary means. The details are omitted. ∎

######  Remark 3.3 (On the finite measure assumption on $\Omega$).

It is worth noting that the boundedness of $u$ stated in the previous Theorem
does not depend quantitatively on $\Omega$, not even in the case $q=p^{*}$
when $p\leq N$. Indeed, the same statement holds true for arbitrary
$\Omega\subseteq\mathbb{R}^{N}$, not necessarily of finite measure, as long as
the energy critical point $u\in W^{1,1}_{\rm loc}(\Omega)$ is assumed to
fulfill $H(Du)\in L^{1}(\Omega)$, $f(\cdot,u)\,u\in L^{1}(\Omega)$ and,
additionally, that $u$ vanishes at infinity in the measure sense. In this
case, (3.5) still holds true and the proof goes on as before since each
${\mathcal{A}}_{k}$ has finite measure for $k>0$. Moreover, assuming without
loss of generality $\theta>0$ in (2.8) provides the bound (3.4) with
$(|u|-\theta/2)_{+}$ (which is in $L^{q}(\Omega)$ by Sobolev, since its
support has finite measure) instead of $|u|$ on the right hand side.

######  Remark 3.4 (On the form of (3.4)).

It is worth analysing the optimality of the form (3.4) of the $L^{\infty}$
bound in the simple case of the Dirichlet energy, the main point being that
one cannot expect the simpler bound

$\|u\|_{L^{\infty}(\Omega)}\leq
C\,\left((\mu/c)^{\frac{N}{p}}\int_{\Omega}|u|^{q}\,dx\right)^{\frac{p}{p\,q+(p-q)\,N}}$
(3.11)

to be true when $\theta>0$ with a constant $C=C(N,p,q)$ independent of
$\theta$.

Let $\mu\in\ ]0,\lambda_{1}[$, where $\lambda_{1}$ is the first eigenvalue of
the negative Dirichlet Laplacian in the ball $B_{1}$. For any $\theta>0$, the
problem

$\begin{cases}-\Delta u=\mu\,(u+\theta)&\text{in $B_{1}$},\\\ u=0&\text{in
$\partial B_{1}$}\end{cases}$ (3.12)

has, by standard methods, a unique positive solution, which minimises a
functional obeying the assumptions of the previous Theorem with $c=1$, $q=p=2$
and $\mu,\theta$ as in (2.8). By comparison, $u$ is greater or equal than the
solution of

$\begin{cases}-\Delta v=\mu\,\theta&\text{in $B_{1}$},\\\ v=0&\text{in
$\partial B_{1}$},\end{cases}$

which is $v(x)=\mu\,\theta\,(1-|x|^{2})/(2\,N)$, therefore

$\|u\|_{L^{\infty}(B_{1})}\geq\frac{\mu\,\theta}{2\,N}.$ (3.13)

On the other hand, by testing the weak form of (4.6) with $u$ and by the
definition of $\lambda_{1}$, we get

$\mu\,\int_{B_{1}}u^{2}+\theta\,u\,dx=\int_{B_{1}}|Du|^{2}\,dx\geq\lambda_{1}\,\int_{B_{1}}u^{2}\,dx$

so that, rearranging and using Hölder inequality, we find

$(\lambda_{1}-\mu)\int_{B_{1}}u^{2}\,dx\leq\mu\,\theta\,\int_{B_{1}}u\,dx\leq\mu\,\theta\,\left(\int_{B_{1}}u^{2}\,dx\right)^{1/2}\omega_{N}^{1/2}.$

The latter gives the bound

$\left(\mu^{N/2}\,\int_{B_{1}}u^{2}\,dx\right)^{1/2}\leq\omega_{N}^{1/2}\,\frac{\mu^{1+\frac{N}{4}}\,\theta}{\lambda_{1}-\mu},$

so that, if $\mu>0$ fulfills

$C(N,2,2)\,\omega_{N}^{1/2}\,\frac{\mu^{\frac{N}{4}}}{\lambda_{1}-\mu}<\frac{1}{2\,N},$

by (3.13) we have

$\|u\|_{L^{\infty}(B_{1})}>C(N,2,2)\left(\mu^{N/2}\,\int_{B_{1}}u^{2}\,dx\right)^{1/2}$

and the bound (3.11) cannot hold.

Once boundedness is proved, assumption (2.7) ensures that $f(\cdot,u)\in
L^{\infty}(\Omega)$ and therefore $V_{u}=W^{1,p}_{0}(\Omega)$. Even if (2.12)
is just a differential inclusion, basic regularity theory is nevertheless at
hand, thanks to the theory of quasi-minima developed by Giaquinta and Giusti
in [15].

######  Corollary 3.5 (Regularity).

Suppose (2.6) holds true with $0<c<d<\infty$ and $p>1$, as well as (2.7). Let
$u$ be a bounded energy critical point for $J$ on $W^{1,p}_{0}(\Omega)$ and
$M_{0}>0$ be such that $\|u\|_{L^{\infty}(\Omega)}\leq M_{0}$. If $\Omega$ is
bounded and obeys

$L=\inf_{x\in\partial\Omega,r>0}\frac{|B_{r}(x)\setminus\Omega|}{|B_{r}(x)|}>0,$
(3.14)

then

$\|u\|_{C^{\beta}(\overline{\Omega})}\leq C$

for $C>0$ and $\beta\in(0,1)$ depending smoothly on $p$, $c$, $d$, $M_{0}$,
$f$, $L$ and $|\Omega|$, continuously for fixed $N$. If $\Omega$ has finite
measure,

$\|u\|_{C^{\beta}(\Omega^{\prime})}\leq C$

for any $\Omega^{\prime}\Subset\Omega$, with $C$ additionally depending on
$\Omega^{\prime}$ and $\Omega$.

###### Proof.

To make the statement precise, we rewrite (2.7) as

$\sup_{|s|\leq t}\|f(\cdot,s)\|_{L^{\infty}(\Omega)}\leq h(t)\qquad\forall
t>0$

for a smooth, increasing $h$, so that the constants and exponents in the
statement will depend smoothly just on $M_{0}$, through the function $h$ which
in turn depends on $f$. As $u$ is bounded, it minimises the convex functional
$J_{u}$ defined in (2.10) over the whole $W^{1,p}_{0}(\Omega)$, we can adapt
the proof of [17, Theorem 7.1] to our setting, focusing on the Dirichlet
boundary condition. We focus on the first statement, since the second one will
follow from it by choosing smooth $\Omega^{\prime}\Subset\Omega$ obeying
(3.14).

Consider the case $p\leq N$. For any ball $B_{R}$, not necessarily contained
in $\Omega$ and $0<r<R$, we fix $\eta\in C^{\infty}_{c}(B_{R})$ such that

${\rm supp}\,\eta\subseteq B_{R},\qquad 0\leq\eta\leq
1,\qquad\left.\eta\right|_{B_{r}}\equiv 1,\qquad|D\eta|\leq\frac{C(N)}{R-r},$
(3.15)

and compare the minimality of $u$ with the function $v=u-\eta\,(u-k)_{+}$ for
$k\geq 0$. Notice that $v\in W^{1,p}_{0}(\Omega)$ regardless of the relative
position of $B_{R}$ and $\Omega$, and

${\rm supp}\,(v-u)\subseteq{\mathcal{A}}(k,R):=\\{x\in\overline{\Omega}\cap
B_{R}:u(x)\geq k\\},$ $Dv=(1-\eta)\,Du-(u-k)_{+}\,D\eta\qquad\text{in\
}{\mathcal{A}}(k,R).$ (3.16)

Therefore $J_{u}(u)\leq J_{u}(v)$ entails

$\int_{{\mathcal{A}}(k,R)}H(Du)\,dx\leq\int_{{\mathcal{A}}(k,R)}H(Dv)+f(x,u)\,\eta\,(u-k)_{+}\,dx.$

By using the growth conditions on $H$, (2.7) and the boundedness of $u$, we
infer

$c\,\int_{{\mathcal{A}}(k,R)}|Du|^{p}\,dx\leq
d\,\int_{{\mathcal{A}}(k,R)}|Dv|^{p}+M_{0}\,\sup_{|s|\leq
M_{0}}\|f(\cdot,s)\|_{L^{\infty}(\Omega)}\,dx\\\ $

so that by (3.16)

$\begin{split}\int_{{\mathcal{A}}(k,r)}|Du|^{p}\,dx&\leq
C\,\int_{{\mathcal{A}}(k,R)}\left[(1-\eta)^{p}\,|Du|^{p}+|D\eta|^{p}\,(u-k)^{p}\right]\,dx+C\,|{\mathcal{A}}(k,R)|\\\
&\leq
C\,\int_{{\mathcal{A}}(k,R)\setminus{\mathcal{A}}(k,r)}|Du|^{p}\,dx+C\,\int_{{\mathcal{A}}(k,R)}\frac{(u-k)^{p}}{(R-r)^{p}}\,dx+C\,|{\mathcal{A}}(k,R)|\end{split}$

for $C=C(N,p,c,d,M_{0})>0$ and continuously depending of the data. By hole-
filling the first term on the right, i. e. adding to the inequality $C$ times
the left hand side, we get

$(C+1)\int_{{\mathcal{A}}(k,r)}|Du|^{p}\,dx\leq
C\int_{{\mathcal{A}}(k,R)}|Du|^{p}\,dx+\frac{C}{(R-r)^{p}}\,\int_{{\mathcal{A}}(k,R)}(u-k)^{p}\,dx+C\,|{\mathcal{A}}(k,R)|,$

or

$\int_{{\mathcal{A}}(k,r)}|Du|^{p}\,dx\leq(1-\delta)\left[\int_{{\mathcal{A}}(k,R)}|Du|^{p}\,dx+\frac{1}{(R-r)^{p}}\int_{{\mathcal{A}}(k,R)}(u-k)^{p}\,dx+\,|{\mathcal{A}}(k,R)|\right]$

for $\delta=1/(C+1)$, so that [17, Proposition 6.1] provides

$\int_{{\mathcal{A}}(k,r)}|Du|^{p}\,dx\leq\frac{C}{(R-r)^{p}}\int_{{\mathcal{A}}(k,R)}(u-k)^{p}\,dx+C\,|{\mathcal{A}}(k,R)|.$

A similar conclusion holds true for $-u$, which is a critical point of a
functional with the same structural conditions. Therefore $u$ belongs to the
De Giorgi class as considered in [20, p. 81 and p. 90], namely
${\mathcal{B}}_{p}(\Omega\cup\partial\Omega,M_{0},C,1,0)$, yielding the stated
Hölder regularity up to the boundary. Luckily, the Hölder exponent found in
[20, Theorem 6.1 and Proposition 7.1] is very explicit in its dependance from
the parameters, showing the stated dependance of $\beta$ and on the
$C^{\beta}(\overline{\Omega})$ norm of $u$.

If $p>N$, by comparing the minimality of $u$ with $\pm u_{\pm}$, we directly
have

$\int_{\\{\pm u\geq 0\\}}H(Du)\,dx\leq\int_{\\{\pm u\geq
0\\}}f(x,u)\,u_{\pm}\,dx\leq C$

with a constant $C$ smoothly depending on $M_{0}$. Extending $u$ as $0$
outside $\Omega$, using the growth condition (2.6) and Morrey embedding yields
$u\in C^{1-N/p}(\mathbb{R}^{N})$ with a uniform bound on the norm. ∎

######  Corollary 3.6 (Minimum principles).

Let (2.6) hold with $0<c<d<\infty$ and $p>1$, as well as (2.7), and let $u$ be
a bounded energy critical point for $J$ on $W^{1,p}_{0}(\Omega)$, where
$\Omega$ has finite measure.

  1. (1)

If $f(\cdot,t)\geq 0$ for $t\leq 0$, then $u\geq 0$ in $\Omega$.

  2. (2)

If $u\geq 0$ and there exist $A\geq 0$, $\delta>0$ such that for all
$t\in\,]0,\delta[$

$\inf_{\Omega}f(\cdot,t)\geq-A\,t^{p-1},$ (3.17)

then, in each connected component of $\Omega$, $u$ is either strictly positive
or vanishes identically.

###### Proof.

We test the minimality with respect to $J_{u}$ of $u$ against $u_{+}$, to get

$\int_{\\{u<0\\}}H(Du)\,dx\leq\int_{\Omega}f(x,u)\,(u-u_{+})\,dx=-\int_{\\{u\leq
0\\}}f(x,u)\,u_{-}\,dx$

and since $f(\cdot,u)\geq 0$ on $u\leq 0$, the term on the left must vanish,
implying that $u\geq 0$ a. e.. To prove the strict positivity of $u$ we
further test the minimality of $u$ with $v=u+\eta\,(k-u)_{+}$ for $k\geq 0$
and $\eta$ as in (3.15) with $B_{R}\subseteq\Omega$. Since

${\rm supp}\,(u-v)\subseteq\\{x\in B_{R}:u(x)\leq k\\}=:{\mathcal{B}}(k,R)$

the minimality relation $J_{u}(u)\leq J_{u}(v)$ reads

$\int_{{\mathcal{B}}(k,R)}H(Du)\,dx\leq\int_{{\mathcal{B}}(k,R)}H(Dv)+f(x,u)\,(u-v)\,dx.$

Since $u$ is bounded, we can assume (3.17) holds true for all $t\geq 0$, with
a bigger constant $A$ also depending on $\delta$ and $\|u\|_{\infty}$. Since
$u-v=-\eta\,(k-u)_{+}\leq 0$, (3.17) gives

$\begin{split}\int_{{\mathcal{B}}(k,R)}H(Du)\,dx&\leq\int_{{\mathcal{B}}(k,R)}H(Dv)+A\,u^{p-1}\,\eta\,(k-u)_{+}\,dx.\\\
&\leq\int_{{\mathcal{B}}(k,R)}H(Dv)\,dx+A\,k^{p}\,|{\mathcal{B}}(k,R)|.\end{split}$

Proceeding with the same calculations of the previous proof one arrives via
hole-filling at

$\int_{{\mathcal{B}}(k,r)}|Du|^{p}\,dx\leq\frac{C}{(R-r)^{p}}\int_{{\mathcal{B}}(k,R)}(k-u)_{+}^{p}\,dx+A\,k^{p}\,|{\mathcal{B}}(k,R)|$

for all $k\geq 0$. Therefore $u\geq 0$ belongs to the homogeneous De Giorgi
class $DG^{-}_{p}(\Omega)$ as in [14]. A weak Harnack inequality holds true in
this setting (see [14, Theorem 2]), from which the strong minimum principle
readily follows. ∎

##  4\. Dirichlet eigenvalues

For $H:\mathbb{R}^{N}\to[0,\infty[$ convex and obeying (2.6) we can define its
first Dirichlet eigenvalue as

$\lambda_{1,H}(\Omega)=\inf\left\\{\int_{\Omega}H(Dv)\,dx:v\in
W^{1,p}_{0}(\Omega),\int_{\Omega}|v|^{p}\,dx=1\right\\}.$ (4.1)

A first normalized Dirichlet eigenfunction for $\lambda_{1,H}(\Omega)$ is a
function solving the minimum problem for $\lambda_{1,H}(\Omega)$ with its
constraints. When $\Omega$ has finite measure, the embedding
$W^{1,p}_{0}(\Omega)\hookrightarrow L^{p}(\Omega)$ is compact, therefore, if
in (2.6) we assume $c>0$ and $p>1$, problem (4.1) is always solvable and its
value is positive.

The value (4.1) makes sense in general, but we focus here on the
$p$-positively homogeneous setting for $p>1$. In this case,
$\lambda_{1,H}(\Omega)$ is related to the best constant for the inequality

$\int_{\Omega}|v|^{p}\,dx\leq C_{\rm
opt}(H,\Omega)\,\int_{\Omega}H(Dv)\,dx\qquad\forall v\in W^{1,p}_{0}(\Omega)$
(4.2)

which, by positive $p$-homogeneity, satisfies $C_{\rm
opt}(H,\Omega)=1/\lambda_{1,H}(\Omega)$. Moreover, if $u$ solves problem
(4.1), any positive multiple of $u$ attains the best constant in (4.2), and
will be called a first Dirichlet eigenfunction. Notice that we are not
assuming any regularity on $H$ except the natural local Lipschitz one and,
more substantially, we do not assume that $H$ is even. In the possibly non-
even setting, it is convenient to define

$\begin{split}\lambda_{1,H}^{+}(\Omega)&=\inf\left\\{\int_{\Omega}H(Dv)\,dx:v\in
W^{1,p}_{0}(\Omega),\int_{\Omega}|v|^{p}\,dx=1,\ v\geq 0\right\\}\\\
\lambda_{1,H}^{-}(\Omega)&=\inf\left\\{\int_{\Omega}H(Dv)\,dx:v\in
W^{1,p}_{0}(\Omega),\int_{\Omega}|v|^{p}\,dx=1,\ v\leq 0\right\\},\end{split}$
(4.3)

which may in principle be different. We say that $u\geq 0$ is a first
normalized Dirichlet positive eigenfunction for $\lambda^{+}_{1,H}(\Omega)$ if
$u$ realises the minimum for $\lambda_{1,H}^{+}(\Omega)$ (with its
constraints) and that $u\leq 0$ is a first normailzed negative Dirichlet
eigenfunction if it does so for $\lambda_{1,H}^{-}(\Omega)$. Similar
considerations as in the unconstrained eigenvalue $\lambda_{1,H}(\Omega)$ are
valid for sign-constrained ones and we will drop the term normalized
accordingly.

Notice that, without any evennes assumption on $H$, it may a-priori no longer
be true that if $u$ is a first positive eigenfunction then $-u$ is a first
negative eigenfunction. Moreover, it holds

$\lambda_{1,H}^{-}(\Omega)=\lambda_{1,\check{H}}^{+}(\Omega),\qquad\check{H}(z)=H(-z),$
(4.4)

since $v$ obeys the constraints involving $\lambda_{1,H}^{-}(\Omega)$ if and
only if $-v$ does so for $\lambda_{1,H}^{+}(\Omega)$.

######  Proposition 4.1.

Suppose $H:\mathbb{R}^{N}\to[0,\infty[$ is continuous and positively
$p$-homogeneoues, for $p\geq 1$. Then, for any open
$\Omega\subseteq\mathbb{R}^{N}$

$\lambda_{1,H}(\Omega)=\min\left\\{\lambda_{1,H}^{+}(\Omega),\lambda_{1,H}^{-}(\Omega)\right\\}.$
(4.5)

###### Proof.

Clearly

$\lambda_{1,H}(\Omega)\leq\min\left\\{\lambda_{1,H}^{+}(\Omega),\lambda_{1,H}^{-}(\Omega)\right\\},$

so we prove the opposite inequality. To this end, notice that all the
eigenvalues are finite and, for arbitrary $\varepsilon>0$, choose $v$ such
that $\|v\|_{p}=1$ and

$\int_{\Omega}H(Dv)\,dx<\lambda_{1,H}(\Omega)+\varepsilon.$

If $v_{-}\equiv 0$ a. e., then $v\geq 0$ and

$\min\left\\{\lambda_{1,H}^{+}(\Omega),\lambda_{1,H}^{-}(\Omega)\right\\}\leq\lambda_{1,H}^{+}(\Omega)\leq\int_{\Omega}H(Dv)\,dx<\lambda_{1,H}(\Omega)+\varepsilon,$

so that letting $\varepsilon\downarrow 0$ we have

$\min\left\\{\lambda_{1,H}^{+}(\Omega),\lambda_{1,H}^{-}(\Omega)\right\\}\leq\lambda_{1,H}(\Omega).$
(4.6)

A similar conclusion holds if $v_{+}=0$, so we can assume that both $v_{+}$
and $v_{-}$ do not vanish identically. Thus $v_{+}/\|v_{+}\|_{p}$ and
$-v_{-}/\|v_{-}\|_{p}$ are admissible competitors for the corresponding
problems (4.3), giving, by $p$-homgeneity

$\lambda_{1,H}^{\pm}(\Omega)\,\|v_{\pm}\|_{p}^{p}\leq\int_{\Omega}H(D(\pm
v_{\pm}))\,dx.$

From $1=\|v\|_{p}=\|v_{+}\|_{p}+\|v_{-}\|_{p}$ and $H(0)=0$ we infer

$\begin{split}\min\left\\{\lambda_{1,H}^{+}(\Omega),\lambda_{1,H}^{-}(\Omega)\right\\}&=\min\left\\{\lambda_{1,H}^{+}(\Omega),\lambda_{1,H}^{-}(\Omega)\right\\}\left(\|v_{+}\|_{p}^{p}+\|v_{-}\|_{p}^{p}\right)\\\
&\leq\int_{\Omega}H(Dv_{+})\,dx+\int_{\Omega}H(D(-v_{-}))\,dx\\\
&\leq\int_{\Omega}H(Dv)\,dx\leq\lambda_{1,H}(\Omega)+\varepsilon\end{split}$

so that (4.6) follows again by letting $\varepsilon\downarrow 0$.

∎

In general, we do not expect that the two values in (4.3) coincide, although
we have not been able to produce an explicit example. If equality is always
true, however, it cannot follow from convexity, $p$-homogeneity and modularity
of the energy alone, as the following toy example shows.

######  Example 4.2.

Let $H:\mathbb{R}^{2}\to[0,\infty[$ be

$H(x,y)=a\,x_{+}^{2}+x_{-}^{2}+a\,y_{+}^{2}+y_{-}^{2}$

for some $a>1$. We look at $\mathbb{R}^{3}$ as the set of functions
$u:\Omega\to\mathbb{R}$ with $\Omega=\\{0,1,2\\}$, whose ”gradient” is
$Du:=(u(0)-u(1),u(1)-u(2))\in\mathbb{R}^{2}$. The corresponding positive and
negative Dirichlet eigenvalue problems are, respectively,

$\lambda_{1,H}^{\pm}(\Omega)=\inf\left\\{\sum H(Du):u(0)=u(2)=0,u(1)=\pm
1\right\\}.$

The function $H$ is convex, positively $2$-homogeneous and modular, meaning

$H(Du)+H(Dv)=H(D\max\\{u,v\\})+H(D\min\\{u,v\\}),$

due to [16, Remark 2.2], so the truncation arguments employed in the previous
proof still stand. However

$\lambda_{1,H}^{+}(\Omega)=1<a=\lambda_{1,H}^{-}(\Omega).$

A similar construction can clearly be generalised to finite sets $\Omega$
containing an arbitrary number of points.

A simple case where the sign-constrained eigenvalues are equal is when
$\Omega$ enjoys a symmetry opposite with $H$. In order to make a precise
statement, let

${\rm Sym}\,(\Omega)=\left\\{T\in O(N):T(\Omega)=\Omega+x_{0}\text{ for some
$x_{0}\in\mathbb{R}^{N}$}\right\\}.$

######  Proposition 4.3.

Under the assumptions of the previous proposition, suppose that

${\rm Sym}\,(\Omega)\cap-{\rm Sym}\,(H)\neq\emptyset.$ (4.7)

Then

$\lambda_{1,H}^{+}(\Omega)=\lambda_{1,H}^{-}(\Omega)=\lambda_{1,H}(\Omega).$
(4.8)

###### Proof.

Fix $T\in{\rm Sym}\,(\Omega)\cap-{\rm Sym}\,(H)$, so that it holds

$H(-T(z))=H(z)\qquad\forall z\in\mathbb{R}^{N}.$

Then, for any $v\geq 0$ in $W^{1,p}_{0}(\Omega)$ with $\|v\|_{p}=1$, the
function $w=-v\circ T$ is non positive, belongs to $W^{1,p}_{0}(\Omega-x_{0})$
for some $x_{0}\in\mathbb{R}^{N}$ and fulfills $\|w\|_{p}=1$. By translation
invariance

$\lambda_{1,H}(\Omega)=\lambda_{1,H}(\Omega-x_{0})$

and

$\int_{\Omega-x_{0}}H(Dw)\,dx=\int_{\Omega-x_{0}}H(-TDv\circ
T)\,dx=\int_{\Omega}H(Dv)\,dx.$

Therefore $\lambda_{1,H}^{+}(\Omega)\geq\lambda_{1,H}^{-}(\Omega)$. Proceeding
symmetrically from $v\leq 0$ we get the opposite inequality, while applying
(4.5) concludes the proof of (4.8). ∎

######  Remark 4.4.

By the translation invariance of the energy, the same conclusion holds true if
each connected component of $\Omega$ separately fulfills assumption (4.7). In
particular, (4.8) holds true if each connected component has at least one axis
of symmetry. Using this fact it is readily shown that (4.8) always holds true
for arbitrary open $\Omega\subseteq\mathbb{R}$. Indeed, open connected subset
of $\mathbb{R}$ are intervals, half-lines or the whole $\mathbb{R}$. But if
$\Omega$ has an unbounded connected component, a scaling argument shows that
all eigenvalues vanish, while if $\Omega$ has only bounded connected
components the previous Proposition applies, since open intervals are
symmetric with respect to their midpoint.

######  Proposition 4.5.

Suppose that $H:\mathbb{R}^{N}\to[0,+\infty[$ is convex, positively
$p$-homogeneous for some $p>1$, vanishing only at the origin and that $\Omega$
is connected with finite measure. If $u$ is a first Dirichlet eigenfunction
(sign-constrained or not), then $u\in C^{\alpha}(\Omega)$ and never vanishes
in $\Omega$.

###### Proof.

We first observe that any eigenfunction is a critical point for a functional
of the form (1.3) obeying (2.6), (2.7) and (2.8). Since $H$ vanishes only at
the origin and $H$ is continuous

$0<\inf_{|z|=1}H(z)\leq\sup_{|z|=1}H(z)<\infty,$

so that by the positive $p$-homogeneity of $H$, (2.6) holds with $0<c\leq
d<\infty$. In the unconstrained case, for the functional

$J(v)=\int_{\Omega}H(Dv)-\lambda_{1,H}(\Omega)|v|^{p}\,dx$

it holds $J\geq 0$ on $W^{1,p}_{0}(\Omega)$ by construction and $J(u)=0$,
hence $u$ minimises $J$ and is therefore an energy critical point for it. In
the sign-constrained case, by considering $\check{H}$ instead of $H$ and using
(4.4), it suffices to consider the case when $u$ is a positive eigenfunction.
The functional

$J_{+}(v)=\int_{\Omega}H(Dv)-\lambda_{1,H}^{+}(\Omega)\,v_{+}^{p}\,dx$

vanishes at $u$ and it holds $J_{+}(v)\geq J_{+}(v_{+})$. But
$J_{+}(v_{+})\geq 0$ by the definition of $\lambda^{+}_{1,H}(\Omega)$, so
again $u$ minimises $J_{+}$. Clearly in all cases (2.8) holds true with $q=p$
and Theorem 3.2 implies boundedness, and thus continuity by Corollary 3.5, of
all eigenfunctions. Moreover, Corollary 3.6 applies for positive
eigenfunction, since in this case $f(x,t)=t_{+}^{p-1}\geq 0$, yielding the
first statement for positive eigenfunctions.

For an arbitrary eigenfunction $u$, we claim that either $u_{+}$ or $-u_{-}$
are eigenfunctions for $\lambda^{\pm}_{1,H}(\Omega)$ respectively, regardless
of any assumption on $\Omega$. Indeed, it suffices to prove it for normalized
eigenfunctions, so let $u$ be a first Dirichlet eigenfunction for
$\lambda_{1,H}(\Omega)$ such that $\|u\|_{p}=1$. If $u$ is of constant sign,
the claim is trivial. Otherwise, both $u_{+}$ and $u_{-}$ are non trivial and
then $\pm u_{\pm}/\|u_{\pm}\|_{p}$ are admissible for the corresponding
problems in (4.3). By the positive $p$-homogeneity of $H$

$\lambda_{1,H}^{\pm}(\Omega)\leq\frac{1}{\|u_{\pm}\|_{p}^{p}}\,\int_{\Omega}H(D(\pm
u_{\pm}))\,dx$

which, summed up, imply

$\begin{split}\lambda_{1,H}(\Omega)&=\min\left\\{\lambda_{1,H}(\Omega)^{+},\lambda_{1,H}^{-}(\Omega)\right\\}\left(\|u_{+}\|_{p}^{p}+\|u_{-}\|_{p}^{p}\right)\\\
&\leq\int_{\Omega}H(Du_{+})+H(D(-u_{-}))\,dx=\lambda_{1,H}(\Omega),\end{split}$

so that all inequalities above are actually equalities, proving the claim. It
follows that if $u_{+}\neq 0$, then $u_{+}$ is a positive eigenfunction and
therefore never vanishes by the previous part of the proof, so $u_{-}=0$ and
$u=u_{+}>0$ in $\Omega$. A similar conclusion holds true if $u_{-}\neq 0$,
yielding $u=-u_{-}<0$ in $\Omega$ and concluding the proof.

∎

##  5\. Main result

In order to prove a Brezis-Oswald comparison principle, we’ll need the
following generalisation to the non-smooth setting of the Picone inequality.

######  Lemma 5.1 (Picone inequality).

Suppose $H$ is convex and positively $p$-homogeneous. Let $u,v\in W^{1,1}_{\rm
loc}(\Omega)\cap L^{\infty}_{\rm loc}(\Omega)$ be such that
$\inf_{\Omega}u>0$, $\inf_{\Omega}v\geq 0$ and let
$\xi\in{\mathcal{L}}(\Omega;\mathbb{R}^{N})$ be such that $\xi(x)\in\partial
H(Du(x))$ a. e.. Then it holds

$\frac{1}{p}\left(\xi,D\frac{v^{p}}{u^{p-1}}\right)\leq
H(Dv)\qquad\text{a.\,e.\,in $\Omega$. }$ (5.1)

Suppose additionally that $H$ is strictly convex, $\Omega$ is connected and
equality holds a. e. in (5.1). Then $v=k\,u$ for some $k>0$.

###### Proof.

By considering $v+\varepsilon$ instead of $v$ and then passing to the limit
for $\varepsilon\downarrow 0$, we can suppose that $\inf_{\Omega}v>0$. First
observe that by the assumptions $v^{p}/u^{p-1}\in W^{1,1}_{\rm loc}(\Omega)$
and Liebnitz rule for weak derivatives ensures

$D\frac{v^{p}}{u^{p-1}}=(1-p)\,\frac{v^{p}}{u^{p}}\,Du+p\,\frac{v^{p-1}}{u^{p-1}}\,Dv.$
(5.2)

Let $\xi\in\partial H(w)$ for some $w\in\mathbb{R}^{N}$. The following
computation are henceforth made at a point $x\in\Omega$ such that
$\xi=\xi(x)\in\partial H(Du(x))$. In the inequality

$H(z)\geq H(Du)+(\xi,z-Du)\qquad\forall z\in\mathbb{R}^{N}$

we choose $z=(u/v)\,Dv$ to get, by $p$-homogeneity,

$\left(\frac{u}{v}\right)^{p}\,H(Dv)=H\left(\frac{u}{v}\,Dv\right)\geq
H(Du)+\frac{u}{v}\,(\xi,Dv)-(\xi,Du)$

and by (2.5) at $w=Du(x)$

$\left(\frac{u}{v}\right)^{p}\,H(Dv)\geq\left(\frac{1}{p}-1\right)(\xi,Du)+\frac{u}{v}\,(\xi,Dv).$

Multiplying both sides by $(v/u)^{p}$ and using (5.2) we finally get

$H(Dv)\geq\frac{1}{p}\left(\xi,(1-p)\,\frac{v^{p}}{u^{p}}\,Du+p\,\frac{v^{p-1}}{u^{p-1}}\,Dv\right)=\frac{1}{p}\left(\xi,D\frac{v^{p}}{u^{p-1}}\right)$

and (5.1) is proved.

To prove the second statement, suppose equality holds in (5.1) a. e. in
$\Omega$ for some choice of $\xi$ obeying $\xi\in\partial H(Du)$ a. e.. Then
all the previous inequalities are equalities and in particular

$H\left(\frac{u}{v}\,Dv\right)=H(Du)+\left(\xi,\frac{u}{v}\,Dv-Du\right)$

holds for a. e. $x\in\Omega$. Let, for any such $x$, $w=Du(x)$ and
$z=(u(x)/v(x))\,Dv(x)$ and suppose that $w\neq z$. From $H(z)=H(w)+(\xi,z-w)$
and the convexity of $H$ we infer

$H\big{(}t\,z+(1-t)\,w\big{)}\leq
t\,H(z)+(1-t)\,H(w)=H(w)+\big{(}\xi,t\,(z-w)\big{)},$

while from $\xi\in\partial H(z)$ we deduce

$H(w)+\big{(}\xi,t\,(z-w)\big{)}=H(w)+\big{(}\xi,(t\,z+(1-t)\,w)-w\big{)}\leq
H\big{(}t\,z+(1-t)\,w\big{)}.$

Therefore $H$ is linear on the segment from $z$ to $w$, contradicting the
strict convexity of $H$. Therefore $z=w$, hence

$D\log u=D\log v$

a. e. in $\Omega$, which implies that $u$ and $v$ are proportional by the
connectedness of $\Omega$. ∎

######  Remark 5.2.

By inspecting the proof, it is readily seen that assumption $\inf_{\Omega}u>0$
can be weakened to

$\inf_{\Omega^{\prime}}u>0,\qquad\forall\Omega^{\prime}\Subset\Omega.$

We can now prove our main result, Theorem 1.1, which we now restate in a form
convenient for its proof.

######  Theorem 5.3.

Let $\Omega\subseteq\mathbb{R}^{N}$ be open, connected and of finite measure.
For $p>1$ suppose that

  * i)

$H:\mathbb{R}^{N}\to[0,\infty[$ is convex, positively $p$-homogeneous and
vanishes only at the origin

  * ii)

$f$ fulfills (2.7)

  * iii)

The map $t\mapsto f(x,t)/t^{p-1}$ is non-increasing on $\mathbb{R}_{+}$ for a.
e. $x\in\Omega$.

Let $u\in W^{1,p}_{0}(\Omega)\setminus\\{0\\}$ be a non-negative energy
critical point for $J$. The following assertions hold.

  1. (1)

$u$ minimises $J$ among non-negative functions in $W^{1,p}_{0}(\Omega)$.

  2. (2)

If $t\mapsto f(x,t)/t^{p-1}$ is strictly decreasing, $u$ is the unique
positive energy critical point for $J$.

  3. (3)

If $H$ is strictly convex, any other non-negative energy critical point is a
constant multiple of $u$.

  4. (4)

If $H$ is strictly convex and $f$ doesn’t depend on $x$, either $u$ is the
unique positive energy critical point for $J$ or $u$ is a first Dirichlet
positive eigenfunction.

###### Proof.

Since $u\geq 0$ by assumption, we can assume $f(x,t)\equiv f(x,0)$ for all
$t\leq 0$. Let

$h(t)=\sup_{0\leq s\leq t}\|f(\cdot,s)\|_{\infty}.$

By the monotonicity assumption iii) and (2.7)

$f(x,t)\leq\begin{cases}h(1)\,t^{p-1}&\forall t\geq 1\\\ h(1)&\forall t\leq
1.\end{cases}$ (5.3)

so that Theorem 3.2 and Corollary 3.5 apply and $u\in C^{0}(\Omega)\cap
L^{\infty}(\Omega)$. Moreover

$f(x,t)\geq\frac{f(x,\|u\|_{\infty})}{\|u\|_{\infty}^{p-1}}\,t^{p-1}\geq-\frac{h(\|u\|_{\infty})}{\|u\|_{\infty}^{p-1}}\,t^{p-1}\qquad\forall
t\in[0,\|u\|_{\infty}]$ (5.4)

so that the connectedness of $\Omega$ and Corollary 3.6 ensure that $u>0$ in
$\Omega$.

Since $u$ is bounded, it holds $V_{u}=W^{1,p}_{0}(\Omega)$, hence its Euler-
Lagrange equation (2.12) holds true for any $\varphi\in W^{1,p}_{0}(\Omega)$.
Let $\xi\in L^{p^{\prime}}(\Omega;\mathbb{R}^{N})$ be such that

$\xi(x)\in\partial
H(Du(x))\quad\text{a.\,e.},\qquad\int_{\Omega}\left(\xi,D\varphi\right)-f(x,u)\,\varphi\,dx=0\quad\forall\varphi\in
W^{1,p}_{0}(\Omega)$

and let $v\in W^{1,p}_{0}(\Omega)\cap L^{\infty}(\Omega)$ be non-negative. It
is readily checked that for any $\varepsilon>0$ the function
$\varphi=v^{p}/(u+\varepsilon)^{p-1}$ is an admissible test function in
$W^{1,p}_{0}(\Omega)$, hence

$\int_{\Omega}\left(\xi,D\frac{v^{p}}{(u+\varepsilon)^{p-1}}\right)\,dx=\int_{\Omega}f(x,u)\,\frac{v^{p}}{(u+\varepsilon)^{p-1}}\,dx$

and by Picone inequality (5.1) (notice that $\xi\in\partial H(Du)=\partial
H(D(u+\varepsilon))$), we have

$p\,\int_{\Omega}H(Dv)\,dx\geq\int_{\Omega}\left(\xi,D\frac{v^{p}}{(u+\varepsilon)^{p-1}}\right)\,dx=\int_{\Omega}f(x,u)\,\frac{v^{p}}{(u+\varepsilon)^{p-1}}\,dx.$
(5.5)

We claim that

$\lim_{\varepsilon\downarrow
0}\int_{\Omega}\frac{f(x,u)}{(u+\varepsilon)^{p-1}}\,v^{p}\,dx=\int_{\Omega}\frac{f(x,u)}{u^{p-1}}\,v^{p}\,dx.$
(5.6)

Indeed, let

$\Omega_{+}=\big{\\{}x\in\Omega:f(x,u(x))\geq
0\big{\\}},\qquad\Omega_{-}=\big{\\{}x\in\Omega:f(x,u(x))<0\big{\\}}.$

By Beppo-Levi monotone convergence theorem, (5.6) holds true with $\Omega_{+}$
instead of $\Omega$, while on $\Omega_{-}$ it holds, by (5.4),

$0\geq\frac{f(x,u)}{(u+\varepsilon)^{p-1}}\,v^{p}\geq-\frac{h(\|u\|_{\infty})}{\|u\|_{\infty}^{p-1}}\,\frac{u^{p-1}}{(u+\varepsilon)^{p-1}}\,v^{p}\geq-\frac{h(\|u\|_{\infty})}{\|u\|_{\infty}^{p-1}}\,\|v\|_{\infty}^{p}$

so that dominated convergence yields (5.6) on $\Omega_{-}$ as well. Therefore
(5.6) is proved, and passing to the limit as $\varepsilon\downarrow 0$ in
(5.5) yields

$p\,\int_{\Omega}H(Dv)\,dx\geq\int_{\Omega}\frac{f(x,u)}{u^{p-1}}\,v^{p}\,dx.$
(5.7)

Another monotone convergence argument (separately on $\Omega_{+}$ and
$\Omega_{-}$) allows to remove the assumption $v\in L^{\infty}(\Omega)$, so
that (5.7) holds for all $v\in W^{1,p}_{0}(\Omega)$, $v\geq 0$. As a byproduct
of this argument, let us notice that the integrand on the right of (5.7) is
always in $L^{1}(\Omega)$, for any $v\in W^{1,p}_{0}(\Omega)$, $v\geq 0$.

Let us prove assertion (1). To this end, recall that $F(\cdot,v)_{+}\in
L^{1}(\Omega)$ for all non-negative $v\in W^{1,p}_{0}(\Omega)$, thanks to the
one-sided growth condition (5.3). Therefore it suffices to compare $J(u)$ with
$J(v)$ for non-negative $v\in W^{1,p}_{0}(\Omega)$ such that $F(\cdot,v)\in
L^{1}(\Omega)$. We rewrite (5.7) as

$J(v)\geq\int_{\Omega}\frac{f(x,u)}{u^{p-1}}\,\frac{v^{p}}{p}-F(x,v)\,dx$

and recall that, by (2.15),

$J(u)=\int_{\Omega}\frac{1}{p}\,f(x,u)\,u-F(x,u)\,dx,$

so we have

$J(v)-J(u)\geq\int_{\Omega}\frac{f(x,u)}{u^{p-1}}\,\left(\frac{v^{p}}{p}-\frac{u^{p}}{p}\right)+F(x,u)-F(x,v)\,dx.$

We claim that the integrand is non-negative. Indeed, for a. e. $x\in\Omega$ it
holds

$F(x,t)-F(x,s)=\int_{s}^{t}\frac{f(x,\sigma)}{\sigma^{p-1}}\,\sigma^{p-1}\,d\sigma$

so, considering separately the two cases $t\geq s\geq 0$ and $0\leq t\leq s$
and using the monotonicity of $\sigma\mapsto f(x,\sigma)/\sigma^{p-1}$, we get

$F(x,t)-F(x,s)\geq\frac{f(t)}{t^{p-1}}\,\left(\frac{t^{p}}{p}-\frac{s^{p}}{p}\right)$

for all $t,s\geq 0$. Setting $t=u(x)$ and $s=v(x)$ proves the claim, therefore
$J(v)\geq J(u)$ for all $v\geq 0$, $v\in W^{1,p}_{0}(\Omega)$, concluding the
proof of the first statement.

To prove statement (2), suppose now that $v$ is another nontrivial, non-
negative critical point for $J$, so that (5.7) holds true with $u$ and $v$
exchanged. Recalling (2.15) we get

$\int_{\Omega}f(x,v)\,v\,dx\geq\int_{\Omega}\frac{f(x,u)}{u^{p-1}}\,v^{p}\,dx,\qquad\int_{\Omega}f(x,u)\,u\,dx\geq\int_{\Omega}\frac{f(x,v)}{v^{p-1}}\,u^{p}\,dx$

which, summed up, give

$\int_{\Omega}\left(\frac{f(x,u)}{u^{p-1}}-\frac{f(x,v)}{v^{p-1}}\right)\,(u^{p}-v^{p})\,dx\geq
0.$

By assumption ii), the latter forces

$\frac{f(x,u)}{u^{p-1}}=\frac{f(x,v)}{v^{p-1}}$ (5.8)

a. e. in $\Omega$, which implies that $u=v$ if $t\mapsto f(x,t)/t^{p-1}$ is
strictly decreasing.

To prove assertion (3), recall that by Picone inequality

$p\,H(Dv)-\left(\xi,D\frac{v^{p}}{(u+\varepsilon)^{p-1}}\right)\geq 0,$

for all $\varepsilon\geq 0$. Hence, chaining Fatou lemma, the weak form of the
equation for $u$, (5.6), (5.8) and (2.15), we get

$\begin{split}\int_{\Omega}p\,H(Dv)-\left(\xi,D\frac{v^{p}}{u^{p-1}}\right)\,dx&\leq\lim_{\varepsilon\downarrow
0}\int_{\Omega}p\,H(Dv)-\left(\xi,D\frac{v^{p}}{(u+\varepsilon)^{p-1}}\right)\,dx\\\
&=\lim_{\varepsilon\downarrow
0}\int_{\Omega}p\,H(Dv)-f(x,u)\,\frac{v^{p}}{(u+\varepsilon)^{p-1}}\,dx\\\
&=\int_{\Omega}p\,H(Dv)-\frac{f(x,u)}{u^{p-1}}\,v^{p}\,dx\\\
&=\int_{\Omega}p\,H(Dv)-f(x,v)\,v\,dx=0\end{split}$

so that equality must occur a. e. in the Picone inequality. If $H$ is strictly
convex, Lemma 5.1 ensures that

$u=k\,v\,,\qquad\mbox{for some $k>0$,}$ (5.9)

giving assertion (3).

Finally, statement (4) amounts to prove that, in the case $f(x,t)=f(t)$, the
validity of (5.9) for some $k\neq 1$ forces $u$ to be a first Dirichlet
positive eigenfunction. Let $g(t):=f(t)/t^{p-1}$ and assume that $k>1$ in
(5.9). For any fixed $\delta\in\ ]0,\|u\|_{\infty}[$, by continuity
$u(\Omega)$ is connected and contains $]0,\|u\|_{\infty}-\delta]$. Hence for
any $n\in\mathbb{N}$ there exists $x_{n}\in\Omega$ such that

$u(x_{n})=\frac{\|u\|_{\infty}-\delta}{k^{n}}$

and thus (5.8) and (5.9) give

$\begin{split}g(\|u\|_{\infty}-\delta)&=g(u(x_{0}))=g(v(x_{0}))=g(u(x_{0})/k)=g((\|u\|_{\infty}-\delta)/k)=g(u(x_{1}))\\\
&=g(v(x_{1}))=g(u(x_{1})/k)=g((\|u\|_{\infty}-\delta)/k^{2})=\dots=g((\|u\|_{\infty}-\delta)/k^{n})\end{split}$

for any $n\geq 0$. Therefore $g$, being continuous and non-increasing, is
constantly equal to some $\lambda\geq 0$ on $]0,\|u\|_{\infty}-\delta]$. By
letting $\delta\downarrow 0$, the same conclusion holds on
$]0,\|u\|_{\infty}]$. Similarly, if $0<k<1$, we infer that $g\equiv\lambda\geq
0$ on

$]0,\|v\|_{\infty}]=\ ]0,\|u\|_{\infty}/k]\supseteq\ ]0,\|u\|_{\infty}].$

Notice that then $f(t)=\lambda\,t^{p-1}$ for
$t\in\left[0,\|u\|_{\infty}\right]$, hence

$F(t)=\frac{\lambda}{p}\,t^{p}\qquad\text{ for all $t\in[0,\|u\|_{\infty}]$}.$
(5.10)

By (2.15) it holds

$J(u)=\int_{\Omega}H(Du)-\frac{\lambda}{p}\,u^{p}\,dx=0$

and by the first assertion of the Theorem

$J(v)\geq J(u)\qquad\forall v\in W^{1,p}_{0}(\Omega),\ v\geq 0.$

If $0\leq v\leq\|u\|_{\infty}$, from the last two displays and (5.10) we infer

$\frac{\lambda}{p}\int_{\Omega}v^{p}\,dx\leq\int_{\Omega}H(Dv)\,dx.$

By positive $p$-homogeneity, the latter holds for any bounded $v\geq 0$,
$v\neq 0$, (simply writing it for $w=v\,\|u\|_{\infty}/\|v\|_{\infty}$, which
fulfills $0\leq w\leq\|u\|_{\infty}$) and thus for any $v\geq 0$ in
$W^{1,p}_{0}(\Omega)$ by a monotone convergence argument. It follows that
$\lambda/p\leq\lambda_{1,H}^{+}(\Omega)$ and $J(u)=0$ implies

$\lambda_{1,H}^{+}(\Omega)\,\int_{\Omega}u^{p}\,dx\leq\int_{\Omega}H(Du)\,dx=\frac{\lambda}{p}\,\int_{\Omega}u^{p}\,dx\leq\lambda_{1,H}^{+}(\Omega)\,\int_{\Omega}u^{p}\,dx$

i. e., $u$ is a first Dirichlet positive eigenfunction, as claimed.

∎

Acknowledgements
The author thanks an anonymous referee for his careful reading of the
manuscript and valuable suggestions. Prof. M. Degiovanni is warmly thanked for
outlining a mistake in a first version of the manuscript and for drawing our
attention to paper [11]. Prof. J. I. Díaz is also thanked for kindly pointing
out reference [12].

Funding informations
The author is member of Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni (GNAMPA), is partially supported by project
PIACERI - Linea 2 and 3 of the University of Catania and by GNAMPA’s project
”Equazioni alle derivate parziali di tipo ellittico o parabolico con termini
singolari”.

Conflict of interest
The author states no conflict of interest

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