# Concave solutions to Finsler $p$-Laplace type equations

Sunra Mosconi  ,  Giuseppe Riey  and  Marco Squassina  Department of
Mathematics and Computer Science
University of Catania
Viale A. Doria 6, I-95125 Catania, Italy<EMAIL_ADDRESS>Dipartimento
di Matematica e Informatica,
Università della Calabria
Ponte Pietro Bucci 31B, I-87036 Arcavacata di Rende, Cosenza, Italy
<EMAIL_ADDRESS>College of Science
Princess Nourah Bint Abdul Rahman University
Saudi Arabia, Riyadh, PO Box 84428<EMAIL_ADDRESS>Dipartimento di
Matematica e Fisica
Università Cattolica del Sacro Cuore
Via della Garzetta 48, I-25133 Brescia, Italy<EMAIL_ADDRESS>
###### Abstract.

We prove concavity properties for solutions to anisotropic quasi-linear
equations, extending previous results known in the Euclidean case. We focus
the attention on nonsmooth anisotropies and in particular we also allow the
functions describing the anisotropies to be not even.

###### Key words and phrases:

Nonsmooth analysis, anisotropic problems, convexity of solutions

2010 Mathematics Subject Classification: 35J92, 35B33, 35B06

The second author is supported by the “Gruppo Nazionale per l’Analisi
Matematica, la Probabilità e le loro Applicazioni (GNAMPA)” of the Istituto
Nazionale di Alta Matematica (INdAM) and by PRIN project 2017JPCAPN (Italy):
Qualitative and quantitative aspects of nonlinear PDEs.
The third author is supported by Princess Nourah bint Abdulrahman University
Researchers Supporting Project number (PNURSP-HC2023/3), Princess Nourah bint
Abdulrahman University, Saudi Arabia

###### Contents

  1. 1 Introduction
  2. 2 Preliminary results
    1. 2.1 Main assumptions
    2. 2.2 Concavity function
  3. 3 Critical point theory
    1. 3.1 Energy critical points
    2. 3.2 Existence and uniqueness
  4. 4 Regularised problems
    1. 4.1 Approximation scheme
    2. 4.2 Hopf Lemma
  5. 5 Proof of the main result
  6. A Ellipticity estimates

##  1\. Introduction

A natural question in the framework of nonlinear elliptic PDEs is whether a
solution inherits some qualitative properties from it domain of definition.
Starting from [26] extensive research has been developed in order to deduce
symmetry of solutions from the symmetry of the domain, via the so called
Alexandroff-Serrin moving plane method. But when the symmetry of the domain is
dropped, one may wonder if the solutions still exhibit some convexity
properties just from the convexity of their domain. As it turns out, classical
concavity of, say, positive solutions with zero Dirichlet boundary conditions
is often rather demanding: it can be achieved for the torsion problem

(1) $\begin{cases}-\Delta u=1&\text{in $\Omega$}\\\ u=0&\text{on
$\partial\Omega$},\end{cases}$

for example, only for convex $\Omega$ which are suitably small perturbations
of ellipsoids [30] and the first eigenfunctions of the Dirichlet Laplacian are
actually never concave, in any convex domain [31, Remark 3.4]. One may instead
look for quasi-concavity of positive solutions in convex domains, meaning that
all their super-level sets are convex. This is usually accomplished by
requiring that for a suitable strictly increasing functions $\varphi$ the
composition $\varphi(u)$ is concave, a property called $\varphi$-concavity of
$u$. Indeed, in the seminal paper [35] it is shown that the solution of the
torsion problem (1) for $\Omega$ convex is such that $\sqrt{u}$ is concave and
in [13] the authors show that the logarithm of a first positive eigenfunctions
of the Dirichlet Laplacian is always concave if the domain is convex. More
generally, for the positive solution of

$\begin{cases}-\Delta u=u^{\beta}&\text{in $\Omega$}\\\ u=0&\text{on
$\partial\Omega$},\end{cases}$

in a convex $\Omega$, the power $u^{(1-\beta)/2}$ is concave for any
$\beta\in[0,1]$, the $0$-th power being formally identified with $\log u$, see
[32, 33].

A general class of reactions $f$ ensuring quasi-concavity of the positive
solutions of

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

(and more generally of quasi-linear problems involving the $p$-Laplace
operator) has been singled out in [11, 12], providing a precise connection on
how the quasi-concavity of the solution is affected by the nonlinear term $f$
through the above mentioned $\varphi$. Indeed, the class of continuous
increasing $\varphi$ can be partially ordered in a natural way according to
their concavity, and one of the goals of [11] was, given $f$ obeying suitable
conditions, to determine a ”minimally concave” $\varphi$ ensuring
$\varphi$-concavity of the solutions of the corresponding quasilinear problem.
Note that, for general positive reactions $f$, quasi-concavity of positive
solutions of (2) can fail for some smooth convex $\Omega$, as shown in [29].

In [1] the optimality of the assumptions of [11] was discussed and the results
were then extended to cover positive solutions to the quasi-linear problem

$-{\rm div}(\alpha(u)Du)+\frac{1}{2}\alpha^{\prime}(u)|Du|^{2}=f(u)$

(coupled with zero boundary conditions), related to the so called modified
nonlinear Schrödinger equation, under suitable joint hypothesis on $\alpha$
and $f$.

Another direction recently investigated in the literature is the quasi-
concavity of the solutions if the nonlinearity is perturbed [15] or if the
equation is nonautonomous in the diffusion or on the source term, like in the
second order semilinear problem

$-{\rm Tr}\,\left(A(x)\,D^{2}u\right)=a(x)\,u^{\beta},$

see [2]. Finally, strict quasi-concavity of positive solutions to (2) has also
been investigated, but turns out to be a much more delicate theme. We refer to
[12] for the relevant literature and open questions in the quasilinear case.

In this paper, given a bounded convex $\Omega\subseteq\mathbb{R}^{N}$, we
investigate the quasi-concavity of positive solutions of

(3) $\left\\{\begin{array}[]{ll}-{\rm div}\,\left(DH(Du)\right)=f(u)&\hbox{ in
}\Omega\\\ u=0&\hbox{ on }\partial\Omega\,,\end{array}\right.$

where $H$ is a continuous convex $p$-homogenous function for some fixed $p>1$
vanishing only at the origin and $f$ is continuous and fulfils suitable
concavity conditions detailed below. We are particularly interested in the
case where $H$ is not even and possibly non-smooth, in which case (3) requires
a suitable variational formulation due to the possible lack of
differentiability of $H$.

When $H$ is even, the operator appearing on the right of (3) is also known as
the _Finsler $p$-Laplacian_, since the corresponding kinetic energy

$u\mapsto\int_{\Omega}H(Du)\,dx$

has density which can be expressed as

$H(Du)=\Phi(Du)^{p}$

for a suitable Finsler norm $\Phi$. Clearly, when the norm is the standard
Euclidean one, we are reduced to the usual $p$-Laplacian.

This kind of energies can be used to model several anisotropic phenomena,
related for instance to continuum mechanics, image processing and biology [3,
16, 25, 28, 37]. In materials science and chemistry a central role is played
by non-smooth Finsler norms in order to describe the behavior of crystalline
microstructures [8, 9, 10, 41, 42]. Non-smooth Finsler norms are also used in
control theory to describe the cost functional in some optimization problems
[22]. Moreover, in differential geometry it is possible to consider non-even
energy densities $H$, as for instance those related to Randers metrics [4,
19], which have applications also in relativity [38].

Problem (3) and the qualitative properties of its solutions has been
thoughtfully investigated in recent years under the assumption that $H$ is
smooth and its corresponding Finsler norm has strongly convex unit ball, see
for instance [6, 17, 18, 20, 21]. Even under these more stringent assumptions,
however, the quasi-concavity of solutions to (3) was not generally known.

Main result. In order to state our main result, given a continuous $f\in
C^{0}(\mathbb{R}_{+})$, set

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

and let $\varphi$ be

(4) $\varphi(t)=\int_{1}^{t}F(s)^{-\frac{1}{p}}ds\,.$

which is well defined on $]0,M_{f}]\cap\mathbb{R}$, for

(5) $M_{f}=\inf\\{t>0:f(t)\leq 0\\}$

(in most instances we will actually have $M_{f}=+\infty$). Note that in
general $\varphi$ may be unbounded near $0$ and also possibly near some
$\bar{t}>M_{f}$. In order to deal with possibly non-smooth convex $H$, we
restrict to special variational solutions of (3), namely minimisers of the
corresponding energy

(6) $\left\\{w\in W^{1,p}_{0}(\Omega):w\geq 0\right\\}\ni w\mapsto
J(w)=\int_{\Omega}\frac{1}{p}\,H(Dw)-F(w)\,dx.$

This is not restrictive, since if $H$ is differentiable and $f$ fulfils the
additional requirements specified in the statement below, any positive
solution of (3) turns out to minimise (6).

######  Theorem 1.1.

Suppose that $H\in C^{0}(\mathbb{R}^{N})$ is convex, positively
$p$-homogeneous and vanishes only at the origin, while $f\in C^{\alpha}_{\rm
loc}(]0,\infty[,\mathbb{R})\cap C^{0}([0,\infty[,\mathbb{R})$ fulfils
$M_{f}>0$. Let $u\in W^{1,p}_{0}(\Omega)$ be a non-negative, nontrivial
minimiser for (6).

Assume that

  1. (1)

$F^{\frac{1}{p}}$ is concave and $F/f$ is convex on $]0,M_{f}[$

and one of the following conditions

  1. ($2_{H}$)

$H$ is strictly convex

  2. ($2_{F}$)

$t\mapsto F(t^{1/p})$ is strictly concave.

Then $u\leq M_{f}$ and the function $v=\varphi(u)$ is concave in $\Omega$,
where $\varphi$ is given in (4).

Comments on the statement.

  * •

We are not assuming any regularity or evenness hypothesis on $H$ in Theorem
1.1. This lack of regularity and symmetry on $H$ forces us to build suitable
tools such as comparison principles and Hopf type Lemma without relying on PDE
arguments and are therefore, as far as we know, new.

  * •

Due to the assumed $p$-positive homogeneity of $H$, its strict convexity is
equivalent to the strict convexity of $\\{H\leq 1\\}$. Note that in general
the strong convexity of $\\{H\leq 1\\}$ (meaning that the principal curvature
of its boundary are positively bounded from below) is required to get
classical $C^{1,\alpha}$ regularity of the corresponding solution.

  * •

By elementary means, one can show that if $t\mapsto F^{1/p}(t)$ is concave, so
is $t\mapsto F(t^{1/p})$, which in turn means that $t\mapsto f(t)/t^{p-1}$ is
non-increasing, a common notion in Brezis-Oswald uniqueness type results, cfr.
[14, 24].

  * •

If $H\in C^{1}(\mathbb{R}^{N})$ and $t\mapsto F(t^{1/p})$ is concave it will
turn out that any solution of (3) is actually a non-negative minimiser for
$J$. This fact can be proved (see (26)) in the more general framework of
possibly non-differentiable $H$, by using the notion of energy critical point
developed in Section 3 and inspired by [23].

  * •

Either of condition ($2_{H}$) and ($2_{F}$) above, coupled with the concavity
of $t\mapsto F(t^{1/p})$ implies that the non-negative minimisers of $J$ are
essentially unique. More precisely, either $t\mapsto F(t^{1/p})$ is linear and
$u$ is a first Dirichlet positive eigenfunction minimising the corresponding
Rayleigh quotient (see (7) below), or $u$ is the unique non-negative minimiser
of $J$ on $W^{1,p}_{0}(\Omega)$. This has been proved in [36], see Proposition
3.4 for a precise statement.

  * •

Existence of non-negative minimisers (or, equivalently, of non-negative
critical points) for $J$ can be characterised in terms of the asymptotic
behaviour at $0$ and $+\infty$ of $t\mapsto f(t)/t^{p-1}$, assuming it is non-
increasing. This is the content of a Brezis-Oswald type result proved in
Proposition 3.3 below, which complement the results of [36].

  * •

Assumption ($1$) can be weakened to

    1. (1’)

$t\mapsto f(t)/t^{p-1}$ is non increasing and $t\mapsto e^{(p-1)t}/f(e^{t})$
is convex

allowing to establish $\log$-concavity of $u$. This follows coupling the
arguments in [11] with tools developed in this manuscript. Note that ($1$)
implies ($1^{\prime}$), but in this case if $\varphi$-concavity is strictly
stronger that $\log$-concavity.

Applications.

  * •

A natural choice for the energy density is $H(z)=|z|_{r}^{p}$ for given
$r\in[1,\infty]$ and $p\in\ ]1,\infty[$, where $|z|_{r}$ denotes the
$\ell^{r}$ norm on $\mathbb{R}^{N}$. Note that with this choice $\\{H\leq
1\\}$ is never strongly convex unless $r=2$, but $H$ is strictly convex for
any $r\in\ ]1,\infty[$, so that assumption ($1$) on the reaction suffices to
get quasi-concavity of the corresponding solutions of (3). If $r=1$ or
$\infty$, $H$ fails to be strictly convex and assumption ($2_{F}$) is needed
to obtain quasi-concavity of the minimisers.
In the non-even setting we can choose any convex, bounded, open
$K\subseteq\mathbb{R}^{N}$ containing $0$ (but not necessarily symmetric) and
consider its Minkowski functional

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

defining the energy density as $H=\Phi^{p}$. The resulting $H$ fulfils
($2_{H}$) as long as $K$ is strictly convex.

  * •

Given $p>1$, the typically used reaction is $F(t)=c\,t^{q}$ for $1\leq q<p$,
$c>0$, which then fulfils (1) and ($2_{F}$) above. In this case, given any
convex, positively $p$-homogeneous $H$ vanishing only at the origin and a
convex bounded $\Omega\subseteq\mathbb{R}^{N}$, a non-negative minimiser $u$
of $J$ has the property that $u^{(p-q)/q}$ is concave. In this case, since
($2_{F}$) holds true, $H$ may fail to be strictly convex allowing for example
to establish the power-concavity of the minimisers of the non-homogeneous
Rayleigh quotient

$\inf\left\\{\dfrac{\displaystyle{\int_{\Omega}|Du|_{\infty}^{2}\,dx}}{\displaystyle{\left(\int_{\Omega}|u|^{q}\,dx\right)^{2/q}}}:u\in
W^{1,2}_{0}(\Omega)\setminus\\{0\\}\right\\}$

for any $q\in[1,2[$ and convex, bounded $\Omega$.
Other explicit examples where ($2_{F}$) holds true, thus allowing such a
generality for $H$, can be found in [11, Section 2].

  * •

Another application of our main result is when $u$ is a first Dirichlet
positive eigenfunction of a convex bounded domain $\Omega$, thus minimising
the homogeneous Rayleigh quotient

(7)
$\lambda_{1,H}^{+}(\Omega)=\inf\left\\{\dfrac{\displaystyle{\int_{\Omega}H(Du)\,dx}}{\displaystyle{\int_{\Omega}u^{p}}}:u\in
W^{1,p}_{0}(\Omega)\setminus\\{0\\},u\geq 0\right\\}.$

Then $F(t)=\lambda_{1,H}^{+}(\Omega)\,t^{p}/p$ and $u$ is $\log$-concave as
long as $H$ is strictly convex. Note that $F$ fulfils (1) in this case, but
not ($2_{F}$), and that the requirement $u\geq 0$ in (7) is needed since $H$
may fail to be even.

A sample consequence is the $\log$-concavity of the first positive Dirichlet
eigenfunction of the pseudo $p$-Laplacian studied in [5], solving

$-\tilde{\Delta}_{p}u=-\sum_{i=1}^{N}|\partial_{i}u|^{p-2}\partial_{i}u=\tilde{\lambda}_{1,p}(\Omega)\,u^{p-1}$

for $p\in\ ]1,\infty[$. The energy-density of the corresponding kinetic energy
is $H(z)=|z|_{p}^{p}$ which, as already noted, is strictly convex but
$\\{H\leq 1\\}$ is not strongly convex.

Sketch of proof. The proof of Theorem 1.1 relies on a two-steps approximation
tailored on both $H$ and $F$. First we smooth out $H$, by keeping its positive
$p$-homogeneity and ensuring a form of strong $p$-ellipticity that the
original $H$ may not have. The uniqueness (up to scalar multiples when
$t\mapsto F(t^{1/p})$ is linear) of the minimiser proved in Section 3 is key
to grant the convergence of the minimisers corresponding to the smoothed
functional to the original one. Thus we are reduced to prove
$\varphi$-concavity of a minimiser $u$ when $H$ is smooth and $p$-elliptic.
Standard regularity theory ensures in this setting up to
$C^{1,\alpha}(\overline{\Omega})$ regularity, but since $H$ is not assumed to
be even, nor can be its regularisations. An appropriate and apparently new
anisotropic version of the Hopf Lemma is proved in subsection 4.2 and turns
out to be essential for the second regularisation procedure, since we can
infer from it uniform $C^{2}$ bounds in an inner thin strip arbitrarily near
the boundary of $\Omega$. A family of different approximating problem is then
built, whose corresponding minimisers are globally $C^{2}$. The form of this
approximation (see (44)) has be to chosen carefully, in order to ensure that
classical results of Kennington and Korevaar can be applied under conditions
involving solely the reaction $f$. Then the strategy of Sakaguchi [40] can be
employed, namely to consider separately the concavity function related the
corresponding solutions far from $\partial\Omega$ and on the boundary of the
aforementioned strip, where uniform $C^{2}$ bounds are available. By passing
to the limit, this allows to conclude the concavity of $\varphi(u)$ on any
strongly convex subdomain sufficiently close to $\Omega$, and thus the
theorem.

As a final point of interest it may be worth mentioning some cases which,
despite natural, are not covered by Theorem 1.1. One may consider the
crystalline, $2$-homogeneous energy $H(z)=|z|_{\infty}^{2}$ and given a convex
$\Omega\subseteq\mathbb{R}^{2}$, look for a positive minimiser $u\in
W^{1,2}_{0}(\Omega)$ of the Rayleigh quotient (7). Note that $H$ is not
strictly convex and $F(t^{1/2})=\lambda_{1,H}(\Omega)\,t/2$ is not strictly
concave, thus we are not able to prove through Theorem 1.1 that $\log u$ is
concave, as one may naively guess. The reason, as should be clear from the
previously described proof of Theorem 1.1, is that we don’t known wether the
corresponding eigenvalue is simple, a fact that may well be false for some
convex $\Omega$.

Notations. In the paper $c$ and $C$ (eventually with subscripts) denote
constants which are allowed to vary from line to line; their dependance on
various parameters will be outlined only when relevant to the proof. For
$t\in\mathbb{R}$ we denote $t_{+}=\max\\{t,0\\}$ and $t_{-}=\max\\{-t,0\\}$.

For $a,b\in\mathbb{R}^{N}$ we denote by $(a,b)$ the standard Euclidean scalar
product, by $|a|$ the Euclidean norm and by $a\otimes b$ the matrix whose
entries are $(a\otimes b)_{ij}=a_{i}b_{j}$. Recall that, for
$v,w\in\mathbb{R}^{N}$, there holds:

$\big{(}a\otimes b\,v,w\big{)}=\big{(}b,v\big{)}\,\big{(}a,w\big{)}.$

For a measurable $E\subseteq\mathbb{R}^{N}$, we let $|E|$ be its
$N$-dimensional Lebesgue measure and for $p\geq 1$, the $L^{p}(E)$ norm of a
measurable $u:E\to\mathbb{R}$ will be denoted by $\|u\|_{p}$ when omitting the
domain $E$ of $u$ causes no confusion.

##  2\. Preliminary results

###  2.1. Main assumptions

Throughout the paper $\Omega$ will be an open subset of $\mathbb{R}^{N}$ with
finite measure, often assumed to be convex and bounded. Recall that a strongly
convex set is a smooth convex set such that the principal curvatures of
$\partial\Omega$ are positive. Clearly, any strongly convex $\Omega$ is
strictly convex, but the opposite may not be true.

Moreover, $H:\mathbb{R}^{N}\to[0,\infty[$ will denote a continuous convex
function, obeying at least the one-sided bound

(8) $H(z)\geq\frac{1}{C}\,|z|^{p}.$

A strengthening of the previous condition will be often assumed, namely

(9) $\frac{1}{C}\,|z|^{p}\leq H(z)\leq C\,|z|^{p}$

and In many instances, $H$ will be additionally required to be positively
$p$-homogeneous ($p>1$), meaning

$H(t\,z)=t^{p}\,H(z)\qquad\forall t>0,z\in\mathbb{R}^{N}.$

Any such $H$ clearly obeys (9).

The reaction $f$ belongs to $C^{0}(\mathbb{R})$, is even and satisfies the
one-sided growth condition

(10) $f(t)\leq C\,\big{(}t^{p-1}+1\big{)}\qquad t\geq 0$

as well as $M_{f}>0$, where $M_{f}$ is given in (5), (possibly
$M_{f}=+\infty$). Let us remark that the evenness condition on $f$ is assumed
only for convenience, since we are interested in non-negative critical points
for the corresponding functional. Given such a function, we will set

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

and

$f_{+}(t)=\max\\{0,f(t)\\},\qquad F_{+}(t)=\int_{0}^{t}f_{+}(s)\,ds.$

Note that we are making a slight abuse of notation here as
$F_{+}(t)\neq(F(t))_{+}$. We will often assume (see e. g. the following
paragraph) that $F^{1/p}$ is concave on $[0,M[$. Note that from the concavity
of $F^{1/p}$ on $[0,\infty[$ we readily infer (10) and, more importantly the
following condition

(11) $\mathbb{R}_{+}\ni t\mapsto\frac{f(t)}{t^{p-1}}\quad\text{is non-
increasing},$

which in turn is equivalent to the concavity of $t\mapsto F(t^{1/p})$ on
$\mathbb{R}_{+}$. Note that the opposite implication is not true, i. e.
$t\mapsto F(t^{1/p})$ may be concave but $t\mapsto F^{1/p}(t)$ may fail to be
concave, see [11, Remark 3.4].

###  2.2. Concavity function

Given a continuous function $v:\Omega\to\mathbb{R}$ with $\Omega$ convex, its
convexity function $c:\Omega\times\Omega\times[0,1]\to\mathbb{R}$ is defined
as

$c_{v}(x,y,t)=t\,v(x)+(1-t)\,v(y)-v\big{(}t\,x+(1-t)\,y\big{)}.$

Clearly, $v$ is concave in $\Omega$ if and only if $c_{v}\leq 0$ in its
domain. We recall the following fundamental properties of the concavity
function and its relation with solutions of PDE.

######  Proposition 2.1 ([27]).

Let $\Omega$ be bounded and convex in $\mathbb{R}^{N}$, $N\geq 2$ and $v\in
C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$ solve

$-{\rm Tr}\,\left(A(Dv)\,D^{2}v\right)=b(v,Dv)$

for $b\in C^{0}(\mathbb{R}\times\mathbb{R}^{N};\mathbb{R}_{+})$ and $A\in
C^{0}(\mathbb{R}^{N},\mathbb{R}^{N\times N})$ such that for some
$0<\lambda\leq\Lambda<\infty$ it holds

$\lambda\,|v|^{2}\leq\left(A(z)\,v,v\right)\leq\Lambda\,|v|^{2}\qquad\text{
for $z\in Dv(\Omega)$, $v\in\mathbb{R}^{N}$}.$

If

  1. (1)

$t\mapsto b(t,z)$ is non-increasing on $v(\Omega)$ for all $z\in Dv(\Omega)$

  2. (2)

$t\mapsto 1/b(t,z)$ is convex on $v(\Omega)$ for all $z\in Dv(\Omega)$.

Then $c_{v}$ cannot attain a positive interior maximum in
$\Omega\times\Omega\times[0,1]$.

In the following, given $\Omega\subseteq\mathbb{R}^{N}$ and $\delta>0$, we set

$\Omega_{\delta}=\\{x\in\Omega:\delta<{\rm dist}(x,\partial\Omega)\\}.$

######  Proposition 2.2 ([33], Lemma 2.4 ).

Suppose that $\Omega$ is smooth, bounded and strongly convex and $u\in
C^{1}(\overline{\Omega})\cap C^{2}(\overline{\Omega}\setminus\Omega_{\eta})$
for some $\eta>0$ is such that

(12) $u>0\quad\text{in $\Omega$},\qquad u=0\quad\text{on
$\partial\Omega$},\qquad\frac{\partial u}{\partial n}>0\quad\text{on
$\partial\Omega$}.$

If $\varphi\in C^{2}(\mathbb{R}_{+};\mathbb{R})$ fulfils

(13) $\lim_{t\to
0^{+}}\varphi^{\prime}(t)=+\infty,\qquad\varphi^{\prime\prime}<0<\varphi^{\prime}\
\text{near $0$},\qquad\lim_{t\to
0^{+}}\frac{\varphi(t)}{\varphi^{\prime}(t)}=\lim_{t\to
0^{+}}\frac{\varphi^{\prime}(t)}{\varphi^{\prime\prime}(t)}=0,$

set $v=\varphi(u)$. Then there exists $\delta\in\ ]0,\eta[$ such that

(14) $D^{2}v<0\quad\text{on }\Omega\setminus\Omega_{\delta}$

and for all $x_{0}\in\Omega\setminus\Omega_{\delta}$ it holds

(15) $v(x_{0})+(Dv(x_{0}),x-x_{0})>v(x)\qquad\text{in
$\overline{\Omega}\setminus\\{x_{0}\\}$}.$

###### Proof.

The proof of (14) is in [33, Lemma 2.4, fact 2]. Let then $\delta_{0}>0$ be
such that $D^{2}v<0$ in $\Omega\setminus\Omega_{\delta_{0}}$. We give another
proof of (15) since the last part of [33, Lemma 2.4] is a bit obscure. Let

$A_{x_{0}}=\left\\{x\in\Omega:v(x_{0})+(Dv(x_{0}),x-x_{0})\leq v(x)\right\\}.$

Fix $\delta_{1}=\delta_{1}(\delta_{0},\Omega)\in\ ]0,\delta_{0}[$ such that

(16) $x_{0}\in\Omega,\quad{\rm
dist}\,(x_{0},\partial\Omega)<\delta_{1}\quad\Longrightarrow\quad
B_{\delta_{1}}(x_{0})\cap\Omega\subseteq\Omega\setminus\Omega_{\delta_{0}}.$

We then claim that for sufficiently small $\delta<\delta_{1}$ (depending on
$u$ as well) it holds

(17) $x_{0}\in\Omega,\quad{\rm
dist}\,(x_{0},\partial\Omega)<\delta\quad\Longrightarrow\quad
A_{x_{0}}\subseteq B_{\delta_{1}}(x_{0})\cap\Omega.$

Before proving the claim, let us note that it implies (15) thanks to (16) and
the strict concavity of $v$ in the convex open set
$B_{\delta_{1}}(x_{0})\cap\Omega$ granted by (14), which force
$A_{x_{0}}=\\{x_{0}\\}$. In particular, (15) holds true for any
$x_{0}\in\Omega$ such that ${\rm dist}\,(x_{0},\Omega)<\delta$ and all
$x\in\Omega$. Moreover, from (13) we can readily infer that
$\varphi(t)\to-\infty$ as $t\to 0^{+}$ so that (15) is always trivially
satisfied when $x\in\partial\Omega$.

We prove (17) by contradiction, thus assuming that there exists
$\delta_{n}\downarrow 0$, $x_{n}\in\Omega\setminus\Omega_{\delta_{n}}$ and
$y_{n}\in\Omega$ such that

$|y_{n}-x_{n}|\geq\delta_{1},\qquad v(x_{n})+(Dv(x_{n}),y_{n}-x_{n})\leq
v(y_{n}).$

By compactness we can suppose $x_{n}\to\bar{x}\in\partial\Omega$ and
$y_{n}\to\bar{y}\in\Omega$, with $|\bar{y}-\bar{x}|\geq\delta_{1}$. Denoting
by $n(\bar{x})$ the interior normal to $\partial\Omega$ at $\bar{x}$, it holds

$\lim_{n}(Du(x_{n}),y_{n}-x_{n})=(Du(\bar{x}),\bar{y}-\bar{x})=|Du(\bar{x})|\,(n(\bar{x}),\bar{y}-\bar{x})>0$

by the strict convexity of $\partial\Omega$ and (12). Therefore there exists
$\theta>0$ such that for sufficiently large $n$ it holds

$(Du(x_{n}),y_{n}-x_{n})>\theta.$

Recalling the definition of $v$, we thus have

$\varphi^{\prime}(u(x_{n}))\left(\frac{\varphi(u(x_{n}))}{\varphi^{\prime}(u(x_{n}))}+(Du(x_{n}),y_{n}-x_{n})\right)=v(x_{n})+(Dv(x_{n}),y_{n}-x_{n})\leq
v(y_{n})=\varphi(u(y_{n}))$

so that for sufficiently large $n$

$\varphi^{\prime}(u(x_{n}))\left(\frac{\varphi(u(x_{n}))}{\varphi^{\prime}(u(x_{n}))}+\theta\right)\leq\varphi(u(y_{n})).$

However, since $u(x_{n})\to 0$, the left hand side goes to $+\infty$ by (13)
while the right and side is bounded from above (since $\varphi$ is smooth on
$]0,+\infty[$ and increasing near $0$). This proves claim (17) and then (15).
∎

The next proposition shows the rôle of condition (15) in analysing the
boundary behaviour of the convexity function. Notice that we will apply it for
convex domains slightly smaller than the domain of definition of the function
$v$ defined above, in order to ensure that it is smooth up to the boundary.

######  Proposition 2.3 ([33], Lemma 2.1 ).

Suppose that $\Omega$ is smooth, bounded and strongly convex and $\eta>0$. If
$v\in C^{1}(\overline{\Omega})$ fulfils (15) for all $x_{0}\in\partial\Omega$,
then $c_{v}$ cannot attain a positive maximum on
$\partial(\Omega\times\Omega)\times[0,1]$.

Clearly, condition (15) is not stable under $C^{2}$ convergence, but the
conjunction of (14) and (15) is, as has been observed in [11]. We report the
argument therein for sake of completeness.

######  Proposition 2.4.

Let $\Omega$ be smooth, bounded and strongly convex, $\eta>0$ and let
$v_{n}\in C^{1}(\overline{\Omega})\cap
C^{2}(\overline{\Omega}\setminus\Omega_{\eta})$ be such that $v_{n}\to v$ in
$C^{1}(\overline{\Omega})$ and in
$C^{2}(\overline{\Omega}\setminus\Omega_{\eta})$. If $v$ fulfils (14) for some
$\delta\in\ ]0,\eta]$ and (15) at all points $x_{0}\in\partial\Omega$, then
$v_{n}$ fulfils them as well for all sufficiently large $n$

###### Proof.

Clearly (14) holds true for sufficiently large $n$, which we’ll assume
henceforth. If (15) does not hold, for a (not relabelled) subsequence there
are points $x_{n}\in\partial\Omega$ and $y_{n}\in\overline{\Omega}$ such that
$x_{n}\to\bar{x}\in\partial\Omega$, $y_{n}\to\bar{y}\in\overline{\Omega}$ and

(18) $v_{n}(x_{n})+(Dv_{n}(x_{n}),y_{n}-x_{n})\geq v_{n}(y_{n}).$

By the smoothness and strong convexity of $\partial\Omega$ we can find $c>0$
such that
$B_{c\delta}(\bar{x})\cap\Omega\subseteq\Omega\setminus\Omega_{\delta}$. Let
$n$ be so large that $B_{c\delta/2}(x_{n})\subseteq B_{c\delta}(\bar{x})$.
Since $v_{n}$ is strictly concave on the convex open set
$B_{c\delta/2}(x_{n})\cap\Omega$, the point $y_{n}$ must fulfil

$|x_{n}-y_{n}|\geq c\,\delta/2,$

hence $|\bar{x}-\bar{y}|>0$. Taking the limit in (18), we reach

$v(\bar{x})+(Dv(\bar{x}),\bar{y}-\bar{x})\geq
v(\bar{y}),\qquad\overline{\Omega}\ni\bar{y}\neq\bar{x}\in\partial\Omega,$

contradicting assumption (15) for $v$.

∎

The previous Propositions will eventually be applied to $v=\varphi(u)$, with
$\varphi$ defined as in (4). To this end, we recall the following elementary
facts from [11].

######  Lemma 2.5.

Let $f\in C^{0}([0,+\infty[,\mathbb{R})$ fulfill (10). If $F^{1/p}$ is concave
and $F/f$ is convex on $]0,M_{f}]\cap\mathbb{R}$, with $M_{f}$ as in (5), then

  1. (1)

The function $\varphi$ defined in (4) is invertible and fulfils (13) on
$]0,M[$.

  2. (2)

If $\psi=\varphi^{-1}$, then $\psi^{\prime\prime}/\psi^{\prime}$ is non-
increasing and $\psi^{\prime}/\psi^{\prime\prime}$ is convex on
$]0,\varphi(M)[$

##  3\. Critical point theory

In this section we give a meaning to problem (3), which at the moment is oddly
defined as $H$ may fail to be differentiable, by using the notion of energy
critical point. We then study the existence and uniqueness of the
corresponding energy critical points.

###  3.1. Energy critical points

Given a convex $H\in C^{0}(\mathbb{R}^{N})$ fulfilling (8) and an even $f\in
C^{0}(\mathbb{R})$ obeying (10), the corresponding functional $J$ will be
defined as

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

Note that $J$ is always well defined and $J(u)>-\infty$ for all $u\in
W^{1,p}_{0}(\Omega)$, as $(F(v))_{+}\in L^{1}(\Omega)$ for any $v\in
W^{1,p}_{0}(\Omega)$ (thanks to (10), the finite measure assumption on
$\Omega$ and Poincaré inequality) but, as (8) and (10) are only one-sided, the
resulting $J$ may assume the value $+\infty$. Note that under assumption (10),
$J$ is anyway sequentially l. s. c. in the weak topology of
$W^{1,p}_{0}(\Omega)$, as

$-\int_{\Omega}F(v)\,dx=\int_{\Omega}(F(v))_{-}\,dx-\int_{\Omega}(F(v))_{+}\,dx$

and the first term is l. s. c. by Fatou lemma while the second is continuous
by (10) and the finite measure assumption on $\Omega$. As we are interested in
non-negative solutions of (3), we let

$\big{(}W^{1,p}_{0}(\Omega)\big{)}_{+}:=\left\\{v\in W^{1,p}_{0}(\Omega):v\geq
0\right\\}.$

and define $u\in\big{(}W^{1,p}_{0}(\Omega)\big{)}_{+}$ to be a non-negative
energy critical point for $J$ on $\big{(}W^{1,p}_{0}(\Omega)\big{)}_{+}$ if
$u$ minimises the corresponding semi-linearized convex functional

(19) $v\mapsto J_{u}(v):=\int_{\Omega}\frac{1}{p}\,H(Dv)-f(u)\,v\,dx$

on $\big{(}W^{1,p}_{0}(\Omega)\big{)}_{+}$. Alternatively, an energy critical
point $u$ for $J$ on the full $W^{1,p}_{0}(\Omega)$ is a minimiser of (19)
over

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

These different definitions of critical point, taken from [23], deserve some
comment. Note that $J_{u}$, as defined in (19) is always well defined on
$\big{(}W^{1,p}_{0}(\Omega)\big{)}_{+}$ under assumption (10), thanks to the
same argument as before and the condition $v\geq 0$. This is not the case for
$J_{u}$ on the whole $W^{1,p}_{0}(\Omega)$ and in this case the requirement
$\big{(}f(u)\,v\big{)}_{+}\in L^{1}(\Omega)$ is needed.

Note, moreover, that as before (10) ensures that $J_{u}(u)>-\infty$ while
trivially $J_{u}(0)=0$, so that

(20) $\text{$u\in C_{J}$ or $C_{J}^{+}$}\qquad\Longrightarrow\qquad H(Du)\in
L^{1}(\Omega),\quad f(u)\,u\in L^{1}(\Omega).$

The set of all energy critical points for $J$ on
$\big{(}W^{1,p}_{0}(\Omega)\big{)}_{+}$ and $W^{1,p}_{0}(\Omega)$,
respectively, will be denoted by $C_{J}^{+}$ and $C_{J}$, respectively. By
using the nonlinearity $f_{+}$ instead of $f$, we can define the functional

$J_{+}(v)=\int_{\Omega}\frac{1}{p}\,H(Dv)-F_{+}(v)\,dx$

and the consider corresponding critical points $C_{J_{+}}$ and
$C_{J_{+}}^{+}$.

Under quite general assumptions on $f$, all the previuos notions of critical
points coincide.

######  Lemma 3.1.

Let $\Omega\subseteq\mathbb{R}^{N}$ have finite measure, $H$ be convex and
obey (8) and $f\in C^{0}(\mathbb{R})$ be even, fulfil (10), and

(21) $f(t)>0\quad\text{for $|t|<M_{f}$},\quad f(t)\leq 0\quad\text{for
$|t|\geq M_{f}$}$

for some $M\geq 0$, with possibly $M_{f}=+\infty$. Then

(22) $C_{J}=C_{J}^{+}=C_{J_{+}}^{+}=C_{J_{+}}$

and any energy critical points fulfils $0\leq u\leq M_{f}$

###### Proof.

Denote by $C$ any one of the set $C_{J},C_{J_{+}},C_{J}^{+}$ or
$C_{J_{+}}^{+}$. We can assume that $f$ does not vanishes identically,
otherwise the claim is trivial as all the previous sets are $\\{0\\}$. In this
case, $M_{f}>0$ in (21). We claim that

(23) $u\in C\quad\Longrightarrow\quad|u|\leq M.$

Recalling (20), we test the minimality of $u$ with $v=\min\\{u,M\\}$ to get

(24)
$\begin{split}\int_{\Omega}\frac{1}{p}\,H(Du)-f(u)\,u\,dx&\leq\int_{\Omega}\frac{1}{p}\,H(Dv)-f(u)\,v\,dx\\\
&=\int_{\\{u\leq
M\\}}\frac{1}{p}\,H(Du)-f(u)\,u\,dx-\int_{\\{u>M\\}}f(u)\,M\,dx.\end{split}$

Note from (21) and (20) that it holds

$0\leq-\int_{\\{u>M\\}}f(u)\,M\,dx\leq-\int_{\\{u>M\\}}f(u)\,u\,dx$

so that all terms in (24) are finite and rearranging we get

$\int_{\\{u>M\\}}\frac{1}{p}\,H(Du)-f(u)\,(u-M)\,dx\leq 0$

proving that $u\leq M_{f}$ thanks to (8) and (21). The same conclusion holds
if $f$ is replaced by $f_{+}$, hence (23) is proved if
$u\in\big{(}W^{1,p}_{0}(\Omega)\big{)}_{+}$, so it suffices to consider the
case $u\in C=C_{J}$ or $C_{J_{+}}$. But in this case, $\hat{u}(x)=-u(-x)\in
C$, where $\Omega$ is replaced by $-\Omega$, so that the previous argument
ensures $\hat{u}\leq M_{f}$, i. e. $u\geq-M_{f}$, which concludes the proof of
(23) in all cases. Therefore $f(u)=f_{+}(u)\geq 0$ for any $u\in C$, hence the
functionals in (19) are the same for $f$ or $f_{+}$. It follows that

$C_{J_{+}}=C_{J},\qquad C_{J_{+}}^{+}=C_{J}^{+}.$

Moreover, by [36, Corollary 3.6]111Note that only (8) is used in the proof of
point 1 therein., any $u\in C_{J_{+}}$ is non-negative in $\Omega$, so that
$C_{J_{+}}\subseteq\big{(}W^{1,p}_{0}(\Omega)\big{)}_{+}$. Finally, again from
$f(u)\geq 0$, we get that minimisation over
$\big{(}W^{1,p}_{0}(\Omega)\big{)}_{+}$ of the functional in (19) is
equivalent to minimisation over the whole $W^{1,p}_{0}(\Omega)$, since

$\int_{\Omega}\frac{1}{p}\,H(Dv)-f(u)\,v\,dx\geq\int_{\Omega}\frac{1}{p}\,H(Dv_{+})-f(u)\,v_{+}\,dx.$

so that $C_{J_{+}}^{+}=C_{J_{+}}$ and the proof is complete.

∎

By [36, Theorem 3.2 and Theorem 3.5], if $\Omega$ is bounded, $\partial\Omega$
is Lipschitz and $H$ fulfils the two-sided bound (9), any of the corresponding
energy critical point belongs to $C^{\alpha}(\overline{\Omega})$, with
$\alpha\in\ ]0,1[$ and its $C^{\alpha}(\overline{\Omega})$ norm depends on
$\|u\|_{p}$, $\Omega$ and the structural constants appearing the bounds for
$H$ and $f$. Moreover, on each connected component of $\Omega$ either $u$
vanishes identically or it is strictly positive in $\Omega$.

A particular reaction $f$ and the corresponding energy critical points provide
the notion of first positive Dirichlet eigenfunction. More precisely, the
number

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

is called the first positive Dirichlet eigenvalue and the corresponding
minimisers are the normalised first positive Dirichlet eigenfunctions. The
latter are the non-negative energy critical points for $J$ with
$f(t)=\lambda_{1,H}^{+}(\Omega)\,t^{p-1}$.

We finally prove the following form of the weak comparison principle, which
holds true under very mild assumptions on $H$ and $\Omega$. It basically
ensures that if $u$ solves (3) in $\Omega$ and $\underline{u}$ solves $-{\rm
div}\,(DH(D\underline{u}))=0$ in $A\subseteq\Omega$, then $\underline{u}\leq
u$ in $A$ as long as the inequality holds true on $\partial A$. While usually
this is deduced through a strong form of convexity for $H$, here we deal with
possibly non-strictly convex $H$.

######  Proposition 3.2.

Let $\Omega$, $H$ and $f$ be as in the previous lemma and $u\in C_{J}^{+}$.
For $A\subseteq\Omega$ open, let $\underline{u}\in W^{1,p}(A)$ be a minimiser
of

$\underline{u}+W^{1,p}_{0}(A)\ni v\mapsto\int_{A}H(Dv)\,dx$

such that $(\underline{u}-u)_{+}\in W^{1,p}_{0}(A)$. Then $u\geq\underline{u}$
in $A$.

###### Proof.

Fix a representative of $u$ and $\underline{u}$. By the previous Lemma we can
use $f_{+}$ instead of $f$ in the definition of $J$, hence we can assume
$f\geq 0$. By the previous Lemma it holds $0\leq u\leq M_{f}$, with $M_{f}$
given in (21), possibly infinite. From
$(\underline{u}-M)_{+}\leq(\underline{u}-u)_{+}\in W^{1,p}_{0}(A)$ we infer
that $\min\\{\underline{u},M\\}\in\underline{u}+W^{1,p}_{0}(A)$, hence by the
minimality of $\underline{u}$ againts the competitor
$\min\\{\underline{u},M\\}$ we find

$\int_{A}H(D\underline{u})\,dx\leq\int_{\\{\underline{u}<M\\}}H(D\underline{u})\,dx$

therefore

$\int_{\\{\underline{u}\geq M\\}}H(D\underline{u})\,dx\leq 0$

and thus $\underline{u}\leq M_{f}$ in $A$. Since $W^{1,p}(A)\subseteq
W^{1,p}(\Omega)$ we can test the minimality of $u$ against
$\max\\{u,\underline{u}\\}=u+(\underline{u}-u)_{+}\in W^{1,p}_{0}(\Omega)$ for
the functional in (19). This gives

$\frac{1}{p}\,\int_{\Omega}H(Du)-f(u)\,u\,dx\leq\frac{1}{p}\,\int_{\\{\underline{u}>u\\}}H(D\underline{u})-f(u)\,\underline{u}\,dx+\frac{1}{p}\,\int_{\\{\underline{u}\leq
u\\}}H(Du)-f(u)\,u\,dx$

so that, recalling (20),

(25)
$\frac{1}{p}\,\int_{\\{\underline{u}>u\\}}H(Du)-f(u)\,u\,dx\leq\frac{1}{p}\,\int_{\\{\underline{u}>u\\}}H(D\underline{u})-f(u)\,\underline{u}\,dx.$

On the other hand, by the minimality of $\underline{u}$ against the competitor
$\min\\{u,\underline{u}\\}=\underline{u}-(u-\underline{u})_{+}\in
W^{1,p}_{0}(A)$, we have

$\int_{A}H(D\underline{u})\,\leq\int_{\\{\underline{u}\leq
u\\}}H(D\underline{u})\,dx+\int_{\\{\underline{u}>u\\}}H(Du)\,dx$

so that

$\int_{\\{\underline{u}>u\\}}H(D\underline{u})\,dx\leq\int_{\\{\underline{u}>u\\}}H(Du)\,dx.$

Inserting the latter into (25) and rearranging we obtain

$\int_{\\{\underline{u}>u\\}}f(u)\,(u-\underline{u})\,dx\leq 0.$

Since $0\leq u\leq M_{f}$ and $f(t)>0$ for $t\in\ ]0,M[$, the previous
inequality forces $f(u)\,(u-\underline{u})_{+}=0$ a. e. on
$\\{\underline{u}>u\\}$ and that equality holds in (25). In particular

$\left|\big{(}\\{u=0\\}\cup\\{u=M\\}\big{)}\setminus\\{\underline{u}>u\\}\right|=0.$

Being $\underline{u}\leq M_{f}$ a. e. it must hold

$\left|\\{u=0\\}\setminus\\{\underline{u}>u\\}\right|=0.$

The equality in (25) then reads

$0=\int_{\\{\underline{u}>u\\}}H(D\underline{u})\,dx=\int_{A}H(D(\underline{u}-u)_{+})\,dx$

which implies that $\underline{u}\leq u$ by (8). ∎

###  3.2. Existence and uniqueness

The existence of non-negative energy critical points can be variationally
characterised for $H$ being positively $p$-homogeneous and $f$ fulfilling
(11). Indeed, in this case [36, Theorem 5.3, point 1] provides

(26) $C_{J}^{+}={\rm Argmin}\,J$

where, by (22), the minimisation problem on the right can be equivalently
settled on either $\big{(}W^{1,p}_{0}(\Omega)\big{)}_{+}$ or
$W^{1,p}_{0}(\Omega)$.

Necessary and sufficient conditions on $f$ for the existence of positive
critical points under assumption (11) have been derived for $\partial\Omega$
of class $C^{2}$ and $H(z)=|z|^{p}$ (see [11, Proposition 3.8]) through the
Hopf Lemma. Here we consider the case of possibly non-smooth $H$ and $\Omega$,
so that the Hopf lemma does not hold.

######  Proposition 3.3.

Let $\Omega\subseteq\mathbb{R}^{N}$ be an open, connected set with finite
measure, $H\in C^{0}(\mathbb{R}^{N})$ be convex, positively $p$-homogeneous
and vanishing only at the origin and $f\in C^{0}(\mathbb{R})$ be even and such
that (11) holds true. Then either $C_{J}^{+}\setminus\\{0\\}$ consists of
first positive Dirichlet eigenfunctions or

(27)
$C_{J}^{+}\setminus\\{0\\}\neq\emptyset\quad\iff\quad\lim_{t\to+\infty}\frac{f(t)}{t^{p-1}}<\lambda_{1,H}^{+}(\Omega)<\lim_{t\to
0^{+}}\frac{f(t)}{t^{p-1}}.$

###### Proof.

Suppose that

$\mu_{\infty}:=\lim_{t\to+\infty}\frac{f(t)}{t^{p-1}}<\lambda_{1,H}^{+}(\Omega)<\lim_{t\to
0^{+}}\frac{f(t)}{t^{p-1}}=:\mu_{0}$

holds true. Then for a fixed $\varepsilon\in\
]0,\lambda_{1,H}^{+}(\Omega)-\mu_{\infty}[$ there exists $t_{0}>0$ such that

$F(t)\leq(\lambda_{1,H}^{+}(\Omega)-\varepsilon)\,\frac{t^{p}}{p}\qquad\text{for
$t>t_{0}$}$

hence for any $v\in\big{(}W^{1,p}_{0}(\Omega)\big{)}_{+}$ it holds

$J(v)\geq\int_{\Omega}\frac{1}{p}\,H(Dv)\,dx-\int_{v\leq
t_{0}}F(v)\,dx-(\lambda_{1,H}^{+}(\Omega)-\varepsilon)\int_{\Omega}\frac{v^{p}}{p}\,dx$

so that by the definition of $\lambda_{1,H}^{+}(\Omega)$

$J(v)\geq\frac{\varepsilon}{p\,\lambda_{1,H}^{+}(\Omega)}\int_{\Omega}H(Dv)\,dx-\sup_{[0,t_{0}]}F\,|\Omega|$

which implies coercivity of $J$ on $\big{(}W^{1,p}_{0}(\Omega))_{+}$ thanks to
(9) (which still holds true under the present, weaker assumption on $H$).
Therefore $J$ admits a minimiser $u\in C_{J}^{+}(\Omega)$. To prove that
$u\neq 0$, note that from $\mu_{0}>\lambda_{1,H}^{+}(\Omega)$ we infer that
for some positive $\varepsilon$ and $t_{1}$

$F(t)\geq(\lambda_{1,H}^{+}(\Omega)+\varepsilon)\,\frac{t^{p}}{p}\qquad\text{for
$t\in[0,t_{1}]$}.$

Therefore, if $w$ is a first positive normalised Dirichlet eigenfunction
(which is bounded), for sufficiently small $t>0$ we have

$J(t\,w)\leq\frac{t^{p}}{p}\int_{\Omega}H(Dw)-(\lambda_{1,H}^{+}(\Omega)+\varepsilon)\,w^{p}\,dx=-\varepsilon\,\frac{t^{p}}{p}<0.$

Thus $J(u)<0$ and $u\in C_{J}^{+}\setminus\\{0\\}$.

Vice-versa, suppose that $u\in C_{J}^{+}\setminus\\{0\\}$ and that $u$ is not
a first positive Dirichlet eigenfunction. By [36, Corollary 2.7] it holds

$\int_{\Omega}H(Du)\,dx=\int_{\Omega}f(u)\,u\,dx$

so by the definition of $\lambda_{1,H}^{+}(\Omega)$ and (11)

$\lambda_{1,H}^{+}(\Omega)\,\int_{\Omega}u^{p}\,dx\leq\int_{\Omega}H(Du)\,dx=\int_{\Omega}f(u)\,u\,dx=\int_{\Omega}\frac{f(u)}{u^{p-1}}\,u^{p}\,dx\leq\mu_{0}\,\int_{\Omega}u^{p}\,dx,$

proving that $\mu_{0}\geq\lambda_{1,H}^{+}(\Omega)$ and furthermore that
$\mu_{0}>\lambda_{1,H}^{+}(\Omega)$, since otherwise the previous inequalities
are all equalities, forcing

$f(t)=\lambda_{1,H}^{+}(\Omega)\,t^{p}\qquad\text{ on $u(\Omega)$},$

i. e. that $u$ is a first positive Dirichlet eigenfunction. To prove that
$\mu_{\infty}<\lambda_{1,H}^{+}(\Omega)$, recall that for any bounded
$v\in\big{(}W^{1,p}_{0}(\Omega)\big{)}_{+}$ it holds (see [36, eq. (5.7)])

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

and in particular the integrand on the right is in $L^{1}(\Omega)$. Choosing
$v$ to be a first positive Dirichlet eigenfunction in the previous inequality,
we obtain by (11)

$\lambda_{1,H}^{+}(\Omega)\,\int_{\Omega}v^{p}\,dx=\int_{\Omega}H(v)\,dx\geq\int_{\Omega}\frac{f(u)}{u^{p-1}}\,v^{p}\,dx\geq\mu_{\infty}\,\int_{\Omega}v^{p}\,dx$

so that $\mu_{\infty}\leq\lambda_{1,H}^{+}(\Omega)$ and, as before, equality
holds if and only if $u$ is a first positive Dirichlet eigenfunction.

∎

Regarding uniqueness, we recall the following result from [36, Theorem 5.3].

######  Proposition 3.4.

Under the assumptions of the previous proposition, let $u\in
C_{J}^{+}\setminus\\{0\\}$. Then either $u$ is a first positive Dirichlet
eigenfunction or $u$ is the unique positive energy critical point for $J$ in
the following cases:

  1. (1)

$H$ is strictly convex

  2. (2)

$t\mapsto f(t)/t^{p-1}$ is strictly decreasing on $u(\Omega)$.

Moreover, if $H$ is strictly convex the first positive Dirichlet eigenvalue is
simple, meaning that any other first positive Dirichlet eigenfunction is a
positive scalar multiple of $u$.

##  4\. Regularised problems

In this section we construct regular problems approximating (3) and derive the
relevant regularity properties of the solutions, together with their boundary
behaviour.

###  4.1. Approximation scheme

######  Lemma 4.1.

Let $G\in C^{\infty}(\mathbb{R}^{N})$ be such that

$G(0)=0,\qquad D^{2}G(z)\geq\lambda\,{\rm Id}\quad\forall z\in\mathbb{R}^{N},$

for fixed $\lambda>0$ and set

$\Phi(z)=\inf\left\\{t>0:G(z/t)\leq 1\right\\}.$

Then $\Phi\in C^{\infty}(\mathbb{R}^{N}\setminus\\{0\\})$, is positively
$1$-homogeneous and

(28)
$\left(D^{2}\Phi(z)\,v,v\right)\geq\widehat{\lambda}\,|v|^{2},\qquad\forall
z,v\ \text{such that $\Phi(z)=1$, $\left(D\Phi(z),v\right)=0$}$

for $\widehat{\lambda}>0$ depending on $G$ and $\lambda$.

###### Proof.

The assumptions on $G$ ensure that the latter is strongly convex, so that
$\Phi$, being the Minkowski functional of $\\{G\leq 1\\}$, is automatically
$C^{\infty}(\mathbb{R}^{N}\setminus\\{0\\})$ and positively $1$-homogeneous.
By construction it holds $\\{G=1\\}=\\{\Phi=1\\}$ and

$G\left(z/\Phi(z)\right)=1\qquad\forall z\neq 0$

which, differentiated, gives

$\frac{DG(z/\Phi(z))}{\Phi(z)}-\frac{D\Phi(z)\,\left(DG(z/\Phi(z)),z\right)}{\Phi^{2}(z)}=0,$

or

(29) $\Phi(z)\,DG(z/\Phi(z))=D\Phi(z)\left(DG(z/\Phi(z)),z\right).$

Differentiating once more, we obtain

$\begin{split}&D\Phi(z)\otimes DG(z/\Phi(z))+D^{2}G(z/\Phi(z))\,\left({\rm
Id}-\frac{z\otimes
D\Phi(z)}{\Phi(z)}\right)=\left(DG(z/\Phi(z)),z\right)\,D^{2}\Phi(z)\\\
&\quad+D\Phi(z)\otimes\left(DG(z/\Phi(z))+z\,D^{2}G(z/\Phi(z))\left(\frac{{\rm
Id}}{\Phi(z)}-\frac{z\otimes D\Phi(z)}{\Phi^{2}(z)}\right)\right).\end{split}$

For $\Phi(z)=1$ we thus have

$\begin{split}&D\Phi(z)\otimes DG(z)+D^{2}G(z)\,\left({\rm Id}-z\otimes
D\Phi(z)\right)=\left(DG(z),z\right)\,D^{2}\Phi(z)\\\
&\quad+D\Phi(z)\otimes\left(DG(z)+z\,D^{2}G(z)\left({\rm Id}-z\otimes
D\Phi(z)\right)\right).\end{split}$

For such $z$’s, if $v$ obeys $\left(D\Phi(z),v\right)=0$, we have

$D^{2}G(z)\,v=\left(DG(z),z\right)\,D^{2}\Phi(z)\,v+D\Phi(z)\,\left(z\,D^{2}G(z),v\right)$

and taking the scalar product with $v$ provides by $(D\Phi(z),v)=0$

$\left(D^{2}G(z)\,v,v\right)=\left(DG(z),z\right)\,\left(D^{2}\Phi(z)\,v,v\right).$

The claim is thus proved with

$\widehat{\lambda}=\frac{\lambda}{\sup_{\\{G=1\\}}\left(DG(z),z\right)}.$

∎

######  Proposition 4.2.

Let $\Omega\subseteq\mathbb{R}^{N}$ be open, connected and with finite
measure. Suppose that

  1. (1)

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

  2. (2)

$f\in C^{0}(\mathbb{R})$ is even and fulfils (11)

and that either the convexity of $H$ or the monotonicity of $\mathbb{R}_{+}\ni
t\mapsto f(t)/t^{p-1}$ are strict. If $u\in C_{J}^{+}\setminus\\{0\\}$ is not
a first positive Dirichlet eigenfunction, there exists a sequence of convex,
positively $p$-homogeneous $H_{n}\in C^{1}(\mathbb{R}^{N})\cap
C^{\infty}(\mathbb{R}^{N}\setminus\\{0\\})$ such that

(30)
$\lambda_{n}\,|v|^{2}\,|z|^{p-2}\leq\left(D^{2}H_{n}(z)\,v,v\right)\leq\Lambda_{n}\,|v|^{2}\,|z|^{p-2}$

for $0<\lambda_{n}\leq\Lambda_{n}$ and corresponding $u_{n}\in C_{J_{n}}^{+}$
with

$J_{n}(v)=\int_{\Omega}\frac{1}{p}\,H_{n}(Dv)-F(v)\,dx$

such that $u_{n}\rightharpoonup u$ in $W^{1,p}_{0}(\Omega)$.

###### Proof.

Fix an even $\varphi\in C^{\infty}_{c}(B_{1};[0,\infty[)$ such that
$\|\varphi\|_{1}=1$ and set, for a sequence $\varepsilon_{n}\downarrow 0$,
$\varphi_{n}(z)=\varepsilon_{n}^{-N}\varphi(z/\varepsilon_{n})$. By Jensen
inequality it holds $\varphi_{n}*H\geq H$ and for each $n$ the function
$\varphi_{n}*H$ is smooth and convex. We can then set

$G_{n}(z)=\varphi_{n}*H(z)+\varepsilon_{n}\,\frac{|z|^{2}}{2}$

so that each $G_{n}$ is convex as well and it holds

(31) $G_{n}\geq H,\qquad G_{n}\to H\quad\text{in $C^{0}_{\rm
loc}(\mathbb{R}^{N})$}.$

Define the convex sets

$K_{n}=\\{z:G_{n}(z)\leq 1\\}\subseteq\\{H\leq 1\\}=K.$

Thanks to (31) and (9) there exists $\delta>0$ such that for sufficiently
large $n$ it holds

(32) $B_{\delta}(0)\subseteq K_{n}$

so that for such $n$’s we can then let $\Phi_{n}$ be the Minkowski functional
of $K_{n}$ and similarly define $\Phi$ as the Minkowski functional of
$K:=\\{H\leq 1\\}$ (notice that then $H=\Phi^{p}$). Being $K_{n}\subseteq K$
it holds $\Phi_{n}\geq\Phi$ and from (32) we infer

(33) $\Phi_{n}(z)\leq|z|/\delta$

for all sufficiently large $n$. Given $z\neq 0$, from

$1=G_{n}(z/\Phi_{n}(z))$

and the local uniform convergence of $G_{n}$ to $H$, we infer that any limit
point $\mu\in\mathbb{R}$ of the bounded sequence $(\Phi_{n}(z))$ fulfils
$H(z/\mu)=1$, i. e. $\mu=\Phi(z)$. Therefore, a sub-subsequence argument
ensures that $\Phi_{n}\to\Phi$ point-wise and thus, being each $\Phi_{n}$, as
well as $\Phi$, convex and finite, locally uniformly on $\mathbb{R}^{N}$.
Finally set

$H_{n}(z)=\Phi_{n}^{p}(z),$

which, by (33), satisfies

(34) $H_{n}(z)\leq\frac{|z|^{p}}{\delta^{p}}\qquad\forall z\in\mathbb{R}^{N}$

for any sufficiently large $n$. By construction, each $H_{n}$ is smooth on
$\mathbb{R}^{N}\setminus\\{0\\}$, strictly convex and positively
$p$-homogeneous. Since $D^{2}G_{n}\geq\varepsilon_{n}\,{\rm Id}$, by Lemma 4.1
$\Phi_{n}$ has strongly convex unit ball in the sense that (28) holds true, so
that (30) follows from [21, Proposition 3.1]. Moreover, by the previous
analysis of the sequence $\Phi_{n}$, it holds

$H_{n}\geq H,\qquad H_{n}\to H\quad\text{in $C^{0}_{\rm
loc}(\mathbb{R}^{N})$}$

so that in particular $J_{n}\geq J$. By Proposition 3.3, (27) holds true,
hence $J_{n}$ has a nontrivial non-negative energy critical point $u_{n}$. By
the variational characterisation (26) and Proposition 3.4, $u_{n}$ is the
unique minimiser for $J_{n}$ on $\big{(}W^{1,p}_{0}(\Omega)\big{)}_{+}$, hence
in particular $J_{n}(u_{n})\leq 0$. By (27), for a suitable $\theta\in\ ]0,1[$
there exists $L>0$ such that

$f(t)\leq\theta\,\lambda_{1,H}^{+}(\Omega)t^{p-1}\qquad\text{for $t>L$},$

so that for $t>0$ it holds

$F(t)\leq\theta\,\lambda_{1,H}^{+}(\Omega)\frac{t^{p}}{p}+L\,\sup_{[0,L]}f.$

By $H_{n}\geq H$, the definition of $\lambda^{+}_{1,H}(\Omega)$ and (9) we
thus infer

$\begin{split}0\geq J_{n}(u_{n})&\geq
J(u_{n})\geq\frac{1}{p}\,\int_{\Omega}H(Du_{n})\,dx-\frac{\theta\,\lambda_{1,H}^{+}(\Omega)}{p}\,\int_{\Omega}u_{n}^{p}\,dx-C\,|\Omega|\\\
&\geq\frac{1-\theta}{p}\,\int_{\Omega}H(Du_{n})\,dx-C\,|\Omega|\geq\frac{1-\theta}{C\,p}\,\int_{\Omega}|Du_{n}|^{p}\,dx-C\,|\Omega|.\end{split}$

Therefore $(u_{n})$ is bounded in $W^{1,p}_{0}(\Omega)$. Suppose, up to
subsequences, that
$u_{n}\rightharpoonup\bar{u}\in\big{(}W^{1,p}_{0}(\Omega)\big{)}_{+}$. From
the lower semicontinuity of $J$, $J_{n}\geq J$ and the minimality of $u_{n}$
we get

(35)
$J(\bar{u})\leq\varliminf_{n}J(u_{n})\leq\varliminf_{n}J_{n}(u_{n})\leq\varliminf_{n}J_{n}(u).$

Finally, by (34) and dominated convergence

$\lim_{n}\int_{\Omega}H_{n}(Du)\,dx=\int_{\Omega}H(Du)\,dx$

so that

$J(\bar{u})\leq\varliminf_{n}J_{n}(u)=J(u).$

Since by (26) $u$ is a minimiser for $J$, so is $\bar{u}$, and using again
Proposition 3.4 grants $\bar{u}=u$. ∎

We also provide a similar approximation scheme for first Dirichlet positive
eigenfunctions.

######  Proposition 4.3.

Let $\Omega\subseteq\mathbb{R}^{N}$ be open, connected and with finite measure
and $H:\mathbb{R}^{N}\to[0,+\infty[$ be strictly convex and positively
$p$-homogeneous. If $u$ is a first positive Dirichlet eigenfunction for
$\lambda_{1,H}^{+}(\Omega)$, there exists a sequence of convex, positively
$p$-homogeneous $H_{n}\in C^{1}(\mathbb{R}^{N})\cap
C^{\infty}(\mathbb{R}^{N}\setminus\\{0\\})$ obeying (30) and corresponding
first positive Dirichlet eigenfunctions $u_{n}$ such that
$u_{n}\rightharpoonup u$ in $W^{1,p}_{0}(\Omega)$.

###### Proof.

By Proposition 3.4 the strict convexity of $H$ ensures that the eigenvalue
$\lambda_{1,H}^{+}(\Omega)$ is simple, so we can normalise and suppose that
$\|u\|_{p}=1$. For $H_{n}:\mathbb{R}^{N}\to\mathbb{R}$ defined as in the
previous proof, consider the first positive Dirichlet eigenfunctions $u_{n}$
associated to $H_{n}$, normalised with unitary $L^{p}$ norm. Fix $v\in
C^{\infty}_{c}(\Omega)$ such that $\|v\|_{p}=1$ and recall that $H_{n}\to H$
in $C^{0}_{\rm loc}(\mathbb{R}^{N})$, hence

$\lim_{n}\int_{\Omega}H_{n}(Dv)\,dx=\int_{\Omega}H(Dv)\,dx$

Since by the definition of $u_{n}$ it holds

$\int_{\Omega}H_{n}(Dv)\,dx\geq\int_{\Omega}H_{n}(Du_{n})\,dx\geq\frac{1}{C}\,\int_{\Omega}|Du_{n}|^{p}\,dx$

hence $(u_{n})$ is bounded in $W^{1,p}_{0}(\Omega)$ and we can suppose that
$u_{n}\rightharpoonup\bar{u}$ and $\|\bar{u}\|_{p}=1$ by the compactness of
$W^{1,p}_{0}(\Omega)\hookrightarrow L^{p}(\Omega)$. The chain of inequalities
(35) still holds true, proving that $\bar{u}$ is a first positive Dirichlet
eigenfunction and then that $\bar{u}=u$ by the simplicity of
$\lambda_{1,H}^{+}(\Omega)$ stated in Proposition 3.4. ∎

In [17] is proved an Hopf Lemma for a class of anisotropic operators, where
the function giving the anisotropy is even. In the sequel we extend the result
to the case of a not even anisotropy.

###  4.2. Hopf Lemma

Let $\Phi\in C^{\infty}(\mathbb{R}^{N}\setminus\\{0\\})$ be convex, positively
$1$-homogeneous and such that

(36) $\\{\Phi\leq 1\\}\qquad\text{is bounded and strongly convex}.$

The polar of $\Phi$ is

$\Phi^{\circ}(z)=\sup\left\\{(\xi,z):\Phi(\xi)\leq 1\right\\}$

and $\Phi^{\circ}$ has the same regularity of $\Phi$ and still fulfils (36)
(see for instance [39]).

Moreover, for any $x\neq 0$ it holds [7]:

(37) $\Phi(D\Phi^{\circ}(x))\equiv 1\equiv\Phi^{\circ}(D\Phi(x))\,,$ (38)
$\Phi(x)\,D\Phi^{\circ}(D\Phi(x))=x=\Phi^{\circ}(x)\,D\Phi(D\Phi^{\circ}(x))\,.$

######  Lemma 4.4.

For $\Phi$ as given and $r>0$ set

(39) $A_{r}=\\{x\in\mathbb{R}^{N}:r>\check{\Phi}^{\circ}(x)>r/2\\},$

where

$\check{\Phi}^{\circ}(x)=\Phi^{\circ}(-x).$

For any $m>0$ there exists $\underline{u}\in C^{2}(\overline{A_{r}})$ such
that

$\begin{cases}{\rm
div}\,(\Phi^{p-1}(D\underline{u})\,D\Phi(D\underline{u}))=0&\text{ in
$A_{r}$}\\\ \underline{u}=m&\text{on $\\{\check{\Phi}^{\circ}=r/2\\}$}\\\
\underline{u}=0,\quad\partial_{n}\underline{u}>0&\text{on
$\\{\check{\Phi}^{\circ}=r\\}$}\end{cases}$

where $n$ is the inner normal to $\\{\check{\Phi}^{\circ}\leq r\\}$.

###### Proof.

We choose $\underline{u}(x)=w(\check{\Phi}^{\circ}(x))$ for a suitable,
decreasing $w\in C^{2}([r/2,r])$. Since
$D\check{\Phi}^{\circ}(x)=-D\Phi^{\circ}(-x)$, we have, for $x\neq 0$ (which
can be assumed henceforth as $0\notin A_{r}$)

(40)
$D\underline{u}(x)=w^{\prime}(\check{\Phi}^{\circ}(x))\,D\check{\Phi}^{\circ}(x)=-w^{\prime}(\check{\Phi}^{\circ}(x))\,D\Phi^{\circ}(-x).$

Therefore, as we are assuming $w^{\prime}<0$, (37) and (38) and the
$0$-positive homogeneity of $D\check{\Phi}^{\circ}$ give

$\Phi(D\underline{u}(x))=-w^{\prime}(\check{\Phi}^{\circ}(x))\,\Phi(D\Phi^{\circ}(-x))=-w^{\prime}(\check{\Phi}^{\circ}(x)).$
$D\Phi(D\underline{u}(x))=D\Phi(D\Phi^{\circ}(-x))=\frac{-x}{\Phi^{\circ}(-x)}=-\frac{x}{\check{\Phi}^{\circ}(x)}$

so that

$\begin{split}-{\rm
div}\,\big{(}\Phi^{p-1}(D\underline{u}(x))\,D\Phi(D\underline{u}(x))\big{)}&={\rm
div}\,\left(x\,\frac{(-w^{\prime}(\check{\Phi}^{\circ}(x))^{p-1}}{\check{\Phi}^{\circ}(x)}\right)\\\
&=\left(x,D\frac{(-w^{\prime}(\check{\Phi}^{\circ}(x))^{p-1}}{\check{\Phi}^{\circ}(x)}\right)+N\frac{(-w^{\prime}(\check{\Phi}^{\circ}(x))^{p-1}}{\check{\Phi}^{\circ}(x)}\end{split}$

We compute

$\begin{split}D\frac{(-w^{\prime}(\check{\Phi}^{\circ}(x))^{p-1}}{\check{\Phi}^{\circ}(x)}=&-\frac{(p-1)\,(-w^{\prime}(\check{\Phi}^{\circ}(x))^{p-2}\,w^{\prime\prime}(\check{\Phi}^{\circ}(x))\,D\check{\Phi}^{\circ}(x)}{\check{\Phi}^{\circ}(x)}\\\
&\quad-\frac{(-w^{\prime}(\check{\Phi}^{\circ}(x))^{p-1}\,D\check{\Phi}^{\circ}(x)}{(\check{\Phi}^{\circ}(x))^{2}}\end{split}$

and note that, by the $1$-positive homogeneity of $\check{\Phi}^{\circ}$

$\left(x,D\check{\Phi}^{\circ}(x)\right)=\check{\Phi}^{\circ}(x),$

hence

${\rm
div}\,\big{(}\Phi^{p-1}(D\underline{u})\,D\Phi(D\underline{u})\big{)}=(p-1)\,(-w^{\prime}(\check{\Phi}^{\circ})^{p-2}\,w^{\prime\prime}(\check{\Phi}^{\circ})+(N-1)\,\frac{(-w^{\prime}(\check{\Phi}^{\circ})^{p-1}}{\check{\Phi}^{\circ}}.$

It thus suffices to choose a decreasing $w$ so that

$(p-1)\,(-w^{\prime}(t))^{p-2}\,w^{\prime\prime}(t)+(N-1)\,\frac{(-w^{\prime}(t)^{p-1}}{t}=0$

which is equivalent to

$\left((-w^{\prime}(t))^{p-1}\,t^{N-1}\right)^{\prime}=0.$

All the strictly decreasing solutions on $\mathbb{R}_{+}$ of the latter ODE
are given by

$w(t)=\begin{cases}\frac{A}{p-N}\,t^{(p-N)/(p-1)}+B&\text{if $p\neq N$}\\\
A\,\log t+B&\text{if $p=N$}\end{cases}$

for arbitrary $A<0,B\in\mathbb{R}$, and if $m>0$ it is readily verified that
we can indeed choose $A<0$, $B\in\mathbb{R}$ in such a way that

$w(r/2)=m,\qquad w(r)=0$

and therefore also $w^{\prime}<0$. For such a choice, the corresponding
$\underline{u}=w(\check{\Phi}^{\circ})$ fulfils all the requirements, as the
interior normal to $\\{\check{\Phi}^{\circ}\leq r\\}$ is
$-D\check{\Phi}^{\circ}/|D\check{\Phi}^{\circ}|$ and by (40)

$\partial_{n}\underline{u}=-\left(\frac{D\check{\Phi}^{\circ}}{|D\check{\Phi}^{\circ}|},w^{\prime}(\check{\Phi}^{\circ})\,D\check{\Phi}^{\circ}\right)=-w^{\prime}(\check{\Phi}^{\circ})\,|D\check{\Phi}^{\circ}|>0$

∎

######  Proposition 4.5.

Suppose $\Omega$ is bounded and connected with $C^{2}$ boundary, $H\in
C^{1}(\mathbb{R}^{N})\cap C^{\infty}(\mathbb{R}^{N}\setminus\\{0\\})$ is
positively $p$-homogeneous and fulfils (30), and $f\in
C^{0}(\mathbb{R},[0,+\infty[)$ obeys (10). Then any critical point $u$ for $J$
is $C^{1,\alpha}(\overline{\Omega})$, $u>0$ in $\Omega$ and

(41) $\frac{\partial u}{\partial n}>0\qquad\text{on $\partial\Omega$}$

where $n$ is the interior normal to $\partial\Omega$.

###### Proof.

Any critical point for $J$ (which, under the stated assumption is a $C^{1}$
functional) is a weak solution of (3) with non-negative and subcritical left
hand side. The boundedness and positivity of $u$ has already been discussed
and its regularity up to the boundary follows from (30) and [34]. Let $\Phi$
be the Minkowski functional of $\\{H\leq 1\\}$, so that $H=\Phi^{p}$, and let
$A_{r}$ be given in (39). Since $\partial\Omega$ is $C^{2}$ and
$\check{\Phi}^{\circ}\in C^{2}(\mathbb{R}^{N}\setminus\\{0\\})$, for any
$x_{0}\in\partial\Omega$ there exists $x_{1}\in\Omega$ and $r>0$ such that

$x_{1}+A_{r}\subseteq\Omega,\qquad\left(x_{1}+\overline{A_{r}}\right)\cap\partial\Omega=\\{x_{0}\\}.$

Let

$m=\inf\\{u(x):\check{\Phi}^{\circ}(x-x_{1})=r/2\\}>0$

and choose the corresponding $\underline{u}$ given in the previous Lemma. The
weak comparison principle in Proposition 3.2 ensures that
$u\geq\underline{u}$, which in turn implies (41). ∎

##  5\. Proof of the main result

$\bullet$ Step 1

Given $u\in C_{J}^{+}$, by Lemma 3.1 we can assume that $F>0$ on
$]0,+\infty[$. Consider the sequence $(u_{n})$ given in Propositions 4.2 and
4.3, solving the corresponding regularised problems. Suppose we can prove that
$c_{\varphi(u_{n})}\leq 0$. Since $u_{n}\to u$ almost everywhere in $\Omega$ ,
so does $\varphi(u_{n})$ to $\varphi(u)$, hence being $u$ (and thus
$\varphi(u)$) continuous in $\Omega$, we will have $c_{\varphi(u)}\leq 0$.

Therefore we can assume that $H\in C^{1}(\mathbb{R}^{N})\cap
C^{\infty}(\mathbb{R}^{N}\setminus\\{0\\})$ fulfils (52) for some given
$0<\lambda\leq\Lambda$ (and is therefore strictly convex). Moreover,
proceeding as in [11, Section 4.1] (with obvious modifications in the case $u$
is a first positive Dirichlet eigenfunction), we can assume that $\Omega$ is
strongly convex with smooth boundary. In particular, the strong minimum
principle and the Hopf Lemma apply, so we can choose $\eta>0$, $\beta\in\
]0,1[$ such that $u\in C^{1,\beta}(\overline{\Omega})$ and furthermore

(42) $\inf_{\Omega_{\eta}}u>0,\qquad\frac{\partial u}{\partial n}>0\text{ on
$\partial\Omega$},\qquad\inf_{\Omega\setminus\Omega_{\eta}}|Du|>0.$

We aim at proving that for any given $\delta\in\ ]0,\eta[$ (which we’ll assume
henceforth),

(43) $c_{\varphi(u)}\leq 0\qquad\text{in
$\Omega_{\delta/2}\times\Omega_{\delta/2}\times[0,1]$},$

which will prove the theorem. In doing so, we can also assume that $\delta$ is
so small that $\partial\Omega_{\delta/2}$ is strongly convex and smooth.

$\bullet$ Step 2

Denoting $a^{2/p}=(a^{2})^{1/p}$ for any $a\in\mathbb{R}$, we define the
family of integrands (recall that $F$ is oddly extended to $\mathbb{R}$)

$G_{\varepsilon}(t,z)=\frac{1}{p}\left[\varepsilon\,F(t)^{\frac{2}{p}}+H^{\frac{2}{p}}(z)\right]^{\frac{p}{2}}-F(t)$

and corresponding auxiliary functionals

(44)
$I_{\varepsilon}(u)=\frac{1}{p}\int_{\Omega}\left[\varepsilon\,F(u)^{\frac{2}{p}}+H^{\frac{2}{p}}(Du)\right]^{\frac{p}{2}}dx-\int_{\Omega}F(u)dx$

If $u$ is not a first positive Dirichlet eigenfunction, by [11, Lemma 4.1],
for any sufficiently small $\varepsilon$ problem

$\inf\left\\{I_{\varepsilon}(u):u\in W^{1,p}_{0}(\Omega)\right\\}$

admits a minimiser $u_{\varepsilon}$ with the property that

(45) $u_{\varepsilon}\to u\qquad\text{in $C^{\beta}(\overline{\Omega})$}.$

Indeed, the proof of [11, Lemma 4.1] can be repeated verbatim when $u$ is not
a first positive Dirichlet eigenfunction, since only the uniqueness of the
minimiser $u$ of $J$ is used therein and the latter is granted by the strict
convexity of $H$ (due to the previous point) and Lemma 3.1, (26) and
Proposition 3.4. When $u$ is a normalised first positive Dirichlet
eigenfunction we instead consider

$\inf\left\\{I_{\varepsilon}(u):u\in W^{1,p}_{0}(\Omega),u\geq
0,\|u\|_{p}=1\right\\}$

and proceed in the same way, using again Proposition 3.4 to ensure weak
convergence of $u_{\varepsilon}$ to $u$, as well as (45) by uniform
$C^{\beta}(\overline{\Omega})$ bounds.

$\bullet$ Step 3

We now improve the convergence of $u_{\varepsilon}\to u$ beyond the
$C^{\beta}$ level. Any $u_{\varepsilon}$ defined above satisfies weakly the
Euler-Lagrange equation for $I_{\varepsilon}$,

$-{\rm
div}\,(D_{z}G_{\varepsilon}(u_{\varepsilon},Du_{\varepsilon}))+\partial_{t}G_{\varepsilon}(u_{\varepsilon},Du_{\varepsilon})=0$

which is more explicitly computed as

(46) $\begin{split}-\operatorname{div}&\left(\big{(}\varepsilon
F(u_{\varepsilon})^{\frac{2}{p}}+H^{\frac{2}{p}}(Du_{\varepsilon})\big{)}^{\frac{p-2}{2}}D\frac{H^{\frac{2}{p}}}{2}(Du_{\varepsilon})\right)\\\
&\quad=f(u_{\varepsilon})\left[1-\frac{\varepsilon}{p}\,\big{(}\varepsilon\,F(u_{\varepsilon})^{\frac{2}{p}}+H^{\frac{2}{p}}(Du_{\varepsilon})\big{)}^{\frac{p-2}{2}}F(u_{\varepsilon})^{\frac{2-p}{p}}\right].\end{split}$

Note that (45) and the positivity of $u$ ensure that given any $\delta\in\
]0,\eta[$, there exists $C>0$ such that for a sufficiently small $\varepsilon$
(which will be assumed henceforth) it holds

(47) $\frac{1}{C}\leq u_{\varepsilon}\leq C\qquad\text{ in
$\Omega_{\delta/4}$}$

and therefore, $F(u_{\varepsilon})$ is uniformly bounded from above and below
by a positive constant. Thanks to Lemma A.1, point 2, Tolskdorff local
regularity theory [43] applies , ensuring that

(48) $\|u_{\varepsilon}\|_{C^{1,\beta}(\overline{\Omega_{\delta/3}})}\leq C$

for a $\beta\in\ ]0,1[$ (possibly different from the one in (45)) depending on
$\Omega,H,N,p$ and $C>0$ with the same dependencies and additionally on
$\delta$ and $\|u\|_{L^{\infty}(\Omega)}$, but none of them depending on
$\varepsilon$. We’ll assume henceforth that $\beta<\alpha$, the Hölder
continuity coefficient of $f$. In particular It follows that
$u_{\varepsilon}\to u$ in $C^{1}(\overline{\Omega_{\delta/3}})$ by Ascoli-
Arzelá. Further regularity can be obtained noting that by (47) and the
boundedness of $H(Du_{\varepsilon})$, the matrix $D^{2}_{z}G_{\varepsilon}$ is
strongly elliptic in $\Omega_{\delta/4}$ thanks to Lemma A.1, point 2. The
difference quotient method yields $u_{\varepsilon}\in W^{2,2}_{\rm
loc}(\Omega_{\delta/4})$, with any partial derivative
$w=\partial_{i}u_{\varepsilon}$, $i=1,\dots,N$ obeying

$-{\rm
div}\,\left(D_{z}^{2}G_{\varepsilon}(u_{\varepsilon},Du_{\varepsilon})\,Dw\right)={\rm
div}\,\left(\partial_{u}D_{z}G_{\varepsilon}(u_{\varepsilon},Du_{\varepsilon})\,w+e_{i}\,\partial_{u}G_{\varepsilon}(u_{\varepsilon},Du_{\varepsilon})\right)$

weakly in $\Omega_{\delta/4}$. Using the $C^{1,\beta}$ regularity of
$u_{\varepsilon}$, we see that

$D_{z}^{2}G_{\varepsilon}(u_{\varepsilon},Du_{\varepsilon})\qquad\text{and}\qquad\partial_{u}D_{z}G_{\varepsilon}(u_{\varepsilon},Du_{\varepsilon})\,w+e_{i}\,\partial_{u}G_{\varepsilon}(u_{\varepsilon},Du_{\varepsilon})$

are $\beta$-Hölder continuous in $\Omega_{\delta/3}$, so that local linear
regularity theory ensures $w\in C^{1,\beta}(\Omega_{\delta/3})$, i. e.
$u_{\varepsilon}\in C^{2,\beta}(\Omega_{\delta/3})$. As $\varepsilon\to 0$,
the $C^{2,\beta}(\Omega_{\delta/3})$ norm of $u_{\varepsilon}$ may blow up,
but by (42), for sufficiently small $\varepsilon$ it holds

$\inf_{\Omega\setminus\Omega_{\eta}}|Du_{\varepsilon}|>0,$

therefore, by looking at (54), we see that $D^{2}_{z}G_{\varepsilon}$ is
strongly elliptic in $\Omega\setminus\Omega_{\eta}$, with ellipticity
constants uniformly bounded from below and above as $\varepsilon\to 0$. We
conclude by local elliptic regularity theory that, given $\delta\in\
]0,\eta[$, for any sufficiently small $\varepsilon$ it holds

(49)
$\|u_{\varepsilon}\|_{C^{2,\beta}(\overline{\Omega_{\delta/2}}\setminus\Omega_{\delta})}\leq
C$

with $C$ depending only on $\Omega,H,N,p,\delta,\eta$ and $\|u\|_{\infty}$,
but not on $\varepsilon$.

$\bullet$ Step 4

Let $v_{\varepsilon}=\varphi(u_{\varepsilon})$. We claim that, for any
sufficiently small $\delta\in\ ]0,\eta[$ and, correspondingly, sufficiently
small $\varepsilon$, $c_{v_{\varepsilon}}$ cannot assume a positive maximum on
$\partial(\Omega_{\delta/2}\times\Omega_{\delta/2})\times[0,1]$. Indeed,
Proposition 2.2 applies to $u$, so that $v=\varphi(u)$ fulfils (15) for any
sufficiently sufficiently small $\delta$. From the uniform bounds (48) and
(49) proved in the previous step, we infer by Ascoli-Arzelá that

$u_{\varepsilon}\to u\qquad\text{ in $C^{1}(\overline{\Omega_{\delta/2}})\cap
C^{2}(\overline{\Omega_{\delta/2}}\setminus\Omega_{\delta})$}.$

Since $u_{\varepsilon}$ fulfils (47) for sufficiently small $\varepsilon$ and
since $\varphi\in C^{2,\alpha}_{\rm loc}(\mathbb{R}_{+})$, the previous
convergence holds true for $v_{\varepsilon}$ as well. Proposition 2.4, applied
to the family $v_{\varepsilon}$ and $v$ on the strictly convex set
$\Omega_{\delta/2}$, then ensures that for any sufficiently small
$\varepsilon>0$ (15) holds true for $v_{\varepsilon}$ at all points
$x_{0}\in\partial\Omega_{\delta/2}$. Therefore, Proposition 2.3, applied to
such functions $v_{\varepsilon}$ on $\Omega_{\delta/2}$, proves the claim.

$\bullet$ Step 5

For $\varepsilon$ and $v_{\varepsilon}$ as above and $M_{f}$ as in (5), we
look at the equation fulfilled by $v_{\varepsilon}$. The proof of the last
statement of Lemma 3.1 still holds for the functional $I_{\varepsilon}$ and
its minimiser $u_{\varepsilon}$, showing that

$M_{\varepsilon}=\sup_{\Omega}u_{\varepsilon}\leq M.$

Let $\psi=\varphi^{-1}$, so that for $t\in\ ]0,M_{\varepsilon}]$ it holds

$\varphi^{\prime}(t)=F^{-\frac{1}{p}}(t),\qquad\psi^{\prime}(s)=F^{\frac{1}{p}}(\psi(s)).$

We thus have $F(u_{\varepsilon})=F(\psi(v_{\varepsilon}))$ and

(50)
$H(Du_{\varepsilon})=H(\psi^{\prime}(v_{\varepsilon})\,Dv_{\varepsilon})=F(\psi(v_{\varepsilon}))\,H(Dv_{\varepsilon})$

where we used the positive $p$-homogeneity of $H$ in the last step. Similarly,
by the positive $1$-homogeneity of $DH^{2/p}$

$(DH^{\frac{2}{p}})(Du_{\varepsilon})=\psi^{\prime}(v_{\varepsilon})\,(DH^{\frac{2}{p}})(Dv_{\varepsilon})=F^{\frac{1}{p}}(\psi(v_{\varepsilon}))\,(DH^{\frac{2}{p}})(Dv_{\varepsilon}).$

We look at equation (46) for $v_{\varepsilon}$. For the left hand side we
compute

$\begin{split}{\rm div}\,&\left(\big{(}\varepsilon
F(u_{\varepsilon})^{\frac{2}{p}}+H^{\frac{2}{p}}(Du_{\varepsilon})\big{)}^{\frac{p-2}{2}}D\frac{H}{2}^{\frac{2}{p}}(Du_{\varepsilon})\right)\\\
&={\rm
div}\,\left(F^{1-\frac{1}{p}}(\psi(v_{\varepsilon}))\,\big{(}\varepsilon+H^{\frac{2}{p}}(Dv_{\varepsilon})\big{)}^{\frac{p-2}{2}}\,D\frac{H^{\frac{2}{p}}}{2}(Dv_{\varepsilon})\right)\\\
&=\left(1-\frac{1}{p}\right)F^{-\frac{1}{p}}(\psi(v_{\varepsilon}))\,f(\psi(v_{\varepsilon}))\,\psi^{\prime}(v_{\varepsilon})\,\big{(}\varepsilon+H^{\frac{2}{p}}(Dv_{\varepsilon})\big{)}^{\frac{p-2}{2}}\,\left(D\frac{H^{\frac{2}{p}}}{2}(Dv_{\varepsilon}),Dv_{\varepsilon}\right)\\\
&\quad+F^{1-\frac{1}{p}}(\psi(v_{\varepsilon}))\,{\rm
div}\,\left(\big{(}\varepsilon+H^{\frac{2}{p}}(Dv_{\varepsilon})\big{)}^{\frac{p-2}{2}}\,D\frac{H^{\frac{2}{p}}}{2}(Dv_{\varepsilon})\right)\\\
&=\left(1-\frac{1}{p}\right)f(\psi(v_{\varepsilon}))\,\big{(}\varepsilon+H^{\frac{2}{p}}(Dv_{\varepsilon})\big{)}^{\frac{p-2}{2}}\,H^{\frac{2}{p}}(Dv_{\varepsilon})\\\
&\quad+F^{1-\frac{1}{p}}(\psi(v_{\varepsilon}))\,{\rm
div}\,\left(\big{(}\varepsilon+H^{\frac{2}{p}}(Dv_{\varepsilon})\big{)}^{\frac{p-2}{2}}\,D\frac{H^{\frac{2}{p}}}{2}(Dv_{\varepsilon})\right)\end{split}$

where we used $\psi^{\prime}(s)=F^{1/p}(\psi(s))$ and that, being $H^{2/p}$
positively $2$-homogeneous,

$\left(DH^{\frac{2}{p}}(Dv_{\varepsilon}),Dv_{\varepsilon}\right)=2\,H^{\frac{2}{p}}(Dv_{\varepsilon}).$

The right-hand side of (46) is, again by (50),

$f(u_{\varepsilon})\left[1-\frac{\varepsilon}{p}\,\big{(}\varepsilon\,F(u_{\varepsilon})^{\frac{2}{p}}+H^{\frac{2}{p}}(Du_{\varepsilon})\big{)}^{\frac{p-2}{2}}F(u_{\varepsilon})^{\frac{2-p}{p}}\right]=f(\psi(v_{\varepsilon}))\left[1-\frac{\varepsilon}{p}\,\big{(}\varepsilon+H^{\frac{2}{p}}(Dv_{\varepsilon})\big{)}^{\frac{p-2}{2}}\right].$

Hence $v_{\varepsilon}$ satisfies

$\begin{split}-F^{1-\frac{1}{p}}(\psi(v_{\varepsilon}))\,{\rm
div}\,&\left(\big{(}\varepsilon+H^{\frac{2}{p}}(Dv_{\varepsilon})\big{)}^{\frac{p-2}{2}}\,D\frac{H^{\frac{2}{p}}}{2}(Dv_{\varepsilon})\right)=f(\psi(v_{\varepsilon}))\left[1-\frac{\varepsilon}{p}\,\big{(}\varepsilon+H^{\frac{2}{p}}(Dv_{\varepsilon})\big{)}^{\frac{p-2}{2}}\right]\\\
&+\left(1-\frac{1}{p}\right)\,f(\psi(v_{\varepsilon}))\,\big{(}\varepsilon+H^{\frac{2}{p}}(Dv_{\varepsilon})\big{)}^{\frac{p-2}{2}}\,H^{\frac{2}{p}}(Dv_{\varepsilon})\end{split}$

which rewrites as

(51) $-{\rm
div}\,\left(DH_{\varepsilon}(Dv_{\varepsilon})\right)=\frac{f(\psi(v_{\varepsilon}))}{F^{1-\frac{1}{p}}(\psi(v_{\varepsilon}))}b_{\varepsilon}(Dv_{\varepsilon})$

where

$\begin{split}H_{\varepsilon}(z)&=\big{(}\varepsilon+H^{\frac{2}{p}}(z)\big{)}^{\frac{p}{2}}\\\
b_{\varepsilon}(z)&=p+\big{(}(p-1)\,H^{\frac{2}{p}}(z)-\varepsilon\big{)}\big{(}\varepsilon+H^{\frac{2}{p}}(z)\big{)}^{\frac{p-2}{2}}.\end{split}$

$\bullet$ Step 6

We finally exclude that, for $\varepsilon$ and $v_{\varepsilon}$ as above,
$c_{v_{\varepsilon}}$ attains a positive maximum on
$\Omega_{\delta/2}\times\Omega_{\delta/2}\times[0,1]$. Note that

$b_{\varepsilon}(z)\geq b(0)=p-\varepsilon^{\frac{p}{2}}$

which is positive for sufficiently small $\varepsilon$. On the other hand, for
$s\in v_{\varepsilon}(\Omega_{\delta/2})\subseteq\
]0,\varphi(M_{\varepsilon})]$ it holds

$\frac{f(\psi(s))}{F^{1-\frac{1}{p}}(\psi(s))}=\big{(}F^{\frac{1}{p}}\big{)}^{\prime}(\psi(s))$

which is non-increasing since $\psi$ is non-decreasing and $F^{1/p}$ is
concave, while

$\psi^{\prime\prime}(s)=\big{(}F^{\frac{1}{p}}(\psi(s))\big{)}^{\prime}=\frac{1}{p}\,F^{1-\frac{1}{p}}(\psi(s))\,f(\psi(s))\,\psi^{\prime}(s)=\frac{F(\psi(s))}{f(\psi(s))}$

so that

$\frac{F^{1-\frac{1}{p}}(\psi(s))}{f(\psi(s))}=\frac{\psi^{\prime\prime}(s)}{\psi^{\prime}(s)}$

which is convex by Lemma 2.5, point 2. Finally, since $v_{\varepsilon}\in
C^{2}(\Omega_{\delta/2})$, (51) rewrites as

$-{\rm
Tr}\,\big{(}D^{2}H_{\varepsilon}(Dv_{\varepsilon})\,D^{2}v_{\varepsilon})=\frac{f(\psi(v_{\varepsilon}))}{F^{1-\frac{1}{p}}(\psi(v_{\varepsilon}))}b_{\varepsilon}(Dv_{\varepsilon})$

and since $H(Dv_{\varepsilon})$ is bounded in $\Omega_{\delta/2}$, Lemma A.1,
point 2, grants the strong ellipticity of $D^{2}H_{\varepsilon}(z)$ for $z\in
Dv_{\varepsilon}(\Omega_{\delta/2})$. Therefore Proposition 2.1 ensures that
$c_{v_{\varepsilon}}$ cannot attain a positive maximum on
$\Omega_{\delta/2}\times\Omega_{\delta/2}\times[0,1]$. By step 4 we conclude
that, given a sufficiently small $\delta$, for any sufficiently small
$\varepsilon>0$, $c_{v_{\varepsilon}}\leq 0$ on
$\Omega_{\delta/2}\times\Omega_{\delta/2}\times[0,1]$ and taking the limit for
$\varepsilon\downarrow 0$, we infer (43), proving the theorem.

##  Appendix A Ellipticity estimates

In this appendix we prove strong ellipticity estimates for the auxiliary
integrands constructed during the proof of Theorem 1.1, starting from a
smooth, positively $p$-homogeneous $H$ obeying

(52)
$\lambda\,|z|^{p-2}\,|v|^{2}\leq\left(D^{2}H(z)\,v,v\right)\leq\Lambda\,|z|^{p-2}\,|v|^{2}\qquad\forall
z\in\mathbb{R}^{N}\setminus\\{0\\},v\in\mathbb{R}^{N}.$

Their proofs are variations of [20, Appendix A], where (52) is shown to be a
consequence of the strong concavity of $\\{H\leq 1\\}$.

######  Lemma A.1.

Suppose $H\in C^{1}(\mathbb{R}^{N})\cap C^{2}(\mathbb{R}^{N}\setminus\\{0\\})$
is positively $p$-homogeneous and fulfils for $0<\lambda\leq\Lambda$ the
ellipticity estimate (52).

Then

  1. (1)

$H^{2/p}$ is strongly elliptic in the sense that there exists positive
$\widehat{\lambda}$, $\widehat{\Lambda}$ depending only $H$ and $p$ such that
for any $z,v\in\mathbb{R}^{N}$ it holds

(53)
$\widehat{\lambda}\,|v|^{2}\leq\left(D^{2}H^{2/p}(z)\,v,v\right)\leq\widehat{\Lambda}\,|v|^{2}$

  2. (2)

For any $\theta>0$ the function

$H_{\theta}(z)=\left(\theta+H^{\frac{2}{p}}(z)\right)^{\frac{p}{2}}$

fulfils

(54)
$\widetilde{\lambda}\,\left(\theta+H^{\frac{2}{p}}(z)\right)^{\frac{p-2}{2}}\,|v|^{2}\leq\left(D^{2}H_{\theta}(z)\,v,v\right)\leq\widetilde{\Lambda}\,\left(\theta+H^{\frac{2}{p}}(z)\right)^{\frac{p-2}{2}}\,|v|^{2}$

for all $z,v\in\mathbb{R}^{N}$, where
$\widetilde{\lambda},\widetilde{\Lambda}$ are positive numbers depending on
$H$ and $p$ but not on $\theta$.

###### Proof.

From the positive $p$-homogeneity of $H$ we get

(55) $p\,H(z)=\left(DH(z),z\right),\qquad
p\,(p-1)H(z)=\left(D^{2}H(z)\,z,z\right)$

as well as

(56) $D^{2}H(z)\,z=(p-1)DH(z).$

Let

$n_{z}=\frac{DH(z)}{|DH(z)|}$

be the exterior normal to the level sets of $H$. Then by (55) for any $z\neq
0$

$(z,n_{z})=p\,\frac{H(z)}{|DH(z)|}\geq c\,|z|$

where

$c=c(H,p)=\inf_{z\neq
0}p\,\frac{H(z)}{|DH(z)|\,|z|}=\inf_{|z|=1}p\,\frac{H(z)}{|DH(z)|}$

(where we used the $p$-positive homogeneity of $H$ and of $|DH(z)|\,|z|$),
which is finite and positive. In particular any $v\in\mathbb{R}^{N}$ can be
uniquely written as

(57) $v=k\,z+t\qquad\text{ with $k\in\mathbb{R}$, $t\in\mathbb{R}^{N}$,
$(t,n_{z})=0$}.$

We clearly have

$|v|^{2}\leq 2\left(|t|^{2}+k^{2}\,|z|^{2}\right).$

On the other hand, by Schwartz inequality

$|v|\geq|(v,n_{z})|=|k|\,(z,n_{z})\geq c\,|k|\,|z|,$

while by triangle inequality and the latter estimate

$|t|\leq|v|+|k|\,|z|\leq\left(1+\frac{1}{c}\right)|v|.$

All in all, we have found a constant $C=C(H,p)$ such that for any $z\neq 0$
and all $v\in\mathbb{R}^{N}$ decomposed as in (57), it holds

(58) $\frac{1}{C}\,(k^{2}\,|z|^{2}+|t|^{2})\leq|v|^{2}\leq
C\,(k^{2}\,|z|^{2}+|t|^{2}).$

Decomposition (57) allows the following computations for $z\neq 0$. Thanks to
(55) and $(DH(z),t)=0$, we have

(59) $(DH(z)\,v,v)^{2}=k^{2}(DH(z),z)^{2}=p^{2}\,H^{2}(z)\,k^{2}.$

On the other hand,

(60)
$\begin{split}(D^{2}H(z)\,v,v)&=k^{2}\,(D^{2}H(z)\,z,z)+2\,k\,(D^{2}H(z)\,z,t)+(D^{2}H(z)\,t,t)\\\
&=p\,(p-1)\,H(z)\,k^{2}+(D^{2}H(z)\,t,t)\end{split}$

thanks to (55), (56) and again $(DH(z),t)=0$.

With these tools at hand, let us prove assertion (1) of the Lemma. Being
$H^{2/p}$ a positively $2$-homogeneous function, $D^{2}H^{2/p}$ is
$0$-homogeneous, hence it suffices to consider the case $z\in\\{H=1\\}$. We
compute

$D^{2}H^{\frac{2}{p}}(z)=\frac{2}{p}\,H^{\frac{2-p}{p}}(z)\left[\frac{2-p}{p}\,H^{-1}(z)\,DH(z)\otimes
DH(z)+D^{2}H(z)\right]$

so that for a given $z\in\\{H=1\\}$

(61)
$\left(D^{2}H^{\frac{2}{p}}(z)\,v,v\right)=\frac{2}{p}\left[\frac{2-p}{p}\,(DH(z)\,v,v)^{2}+(D^{2}H(z)\,v,v)\right].$

Inserting (59) and (60) in (61) gives, for any $z\in\\{H=1\\}$,

$\left(D^{2}H^{\frac{2}{p}}(z)\,v,v\right)=2\,k^{2}+\frac{2}{p}\,(D^{2}H(z)\,t,t)$

and using (52) we obtain

$2\,k^{2}+\frac{2\,\lambda}{p}\,|z|^{p-2}\,|t|^{2}\leq\left(D^{2}H^{\frac{2}{p}}(z)\,v,v\right)\leq
2\,k^{2}+\frac{2\,\Lambda}{p}\,|z|^{p-2}\,|t|^{2}.$

Since $|z|$ is uniformly bounded from above and below on $\\{H=1\\}$, the two-
sided estimate (58) provides (53) for $z\in\\{H=1\\}$, and thus for all
$z\in\mathbb{R}^{N}$ by $0$-homogeneity.

To prove assertion (2), we set $\widehat{H}=H^{2/p}$, which is $2$ homogeneous
and satisfies (53), so that

$H_{\theta}(z)=\left(\theta+\widehat{H}(z)\right)^{\frac{p}{2}}.$

A standard computation gives

$D^{2}H_{\theta}(z)=\frac{p}{2}\left(\theta+\widehat{H}(z)\right)^{\frac{p-2}{2}}\left[\frac{p-2}{2\,(\theta+\widehat{H}(z))}D\widehat{H}(z)\otimes
D\widehat{H}(z)+D^{2}\widehat{H}(z)\right].$

For $z\neq 0$ and $v\in\mathbb{R}^{N}$, we consider again the decomposition
(57), so that (59) and (60) applied to $\widehat{H}$ (so that $p=2$ therein),
give

$\begin{split}\left(D^{2}H_{\theta}(z)\,v,v\right)&=\frac{p}{2}\left(\theta+\widehat{H}(z)\right)^{\frac{p-2}{2}}\left[\frac{p-2}{2\,(\theta+\widehat{H}(z))}\,4\,\widehat{H}^{2}(z)\,k^{2}+2\,\widehat{H}(z)\,k^{2}+\left(D^{2}H(z)\,t,t\right)\right]\\\
&=p\left(\theta+\widehat{H}(z)\right)^{\frac{p-2}{2}}\left[\frac{\theta+(p-1)\,\widehat{H}(z)}{\theta+\widehat{H}(z)}\,\widehat{H}(z)\,k^{2}+\frac{1}{2}\left(D^{2}\widehat{H}(z)\,t,t\right)\right].\end{split}$

Next note that

$a_{p}:=\min\\{1,p-1\\}=\min_{t\geq 0}\frac{\theta+(p-1)\,t}{\theta+t},\qquad
b_{p}:=\max\\{1,p-1\\}=\max_{t\geq 0}\frac{\theta+(p-1)\,t}{\theta+t}$

which are positive and independent of $\theta$, hence (53) provides

(62)
$\begin{split}p\left(\theta+\widehat{H}(z)\right)^{\frac{p-2}{2}}\left[a_{p}\,\widehat{H}(z)\,k^{2}+\frac{\widehat{\lambda}}{2}\,|t|^{2}\right]&\leq\left(D^{2}H_{\theta}(z)\,v,v\right)\\\
&\leq
p\left(\theta+\widehat{H}(z)\right)^{\frac{p-2}{2}}\left[b_{p}\,\widehat{H}(z)\,k^{2}+\frac{\widehat{\Lambda}}{2}\,|t|^{2}\right]\end{split}$

where $\widehat{\lambda}$ and $\widehat{\Lambda}$ are given in (53), hence are
independent of $\theta$. Since $\widehat{H}$ is positively $2$-homogeneous and
vanishes only at the origin, we readily have

$\frac{1}{C}\,|z|^{2}\leq\widehat{H}(z)\leq C\,|z|^{2}$

for a constant $C=C(H,p)$, and using these inequalities in (62) together with
(58) provides (54) with the stated dependencies. ∎

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