Concavity properties for solutions to p-Laplace equations with concave nonlinearities

William Borrelli; Sunra Mosconi; Marco Squassina

doi:10.1515/acv-2021-0100

Article Open Access

Concavity properties for solutions to p-Laplace equations with concave nonlinearities

William Borrelli , Sunra Mosconi and Marco Squassina

Published/Copyright: July 22, 2022

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From the journal Advances in Calculus of Variations Volume 17 Issue 1

Abstract

We obtain new concavity results, up to a suitable transformation, for a class of quasi-linear equations in a convex domain involving the p-Laplace operator and a general nonlinearity satisfying concavity-type assumptions. This provides an extension of results previously known in the literature only for the torsion and the first eigenvalue equations. In the semilinear case p = 2 the results are already new since they include new admissible nonlinearities.

Keywords: Quasilinear problems; convexity of solutions; maximum principles

MSC 2010: 35J92; 35B50; 26B25

1 Introduction

1.1 Overview

Convexity properties of solutions to elliptic partial differential equations in a convex domain are a fascinating subject. One of the first results in this direction can be traced back to the work of Brascamp and Lieb in 1976 [5]. They proved that the logarithm function applied to the first eigenfunction ϕ 1 of the Laplace operator - Δ with zero Dirichlet boundary conditions in a convex domain is concave. It is readily seen that ϕ 1 itself is never concave in any convex domain, thus considering a transformation of the solution is necessary. Previously, in 1971, Makar-Limanov [30] proved that if u > 0 is the solution to the torsion equation Δ ⁢ u + 1 = 0 in a convex planar domain, then u is concave. Years later, at the beginning of the eighties, Korevaar [25, 24] and Kennington [23] derived these results from some general convexity properties (see also [22, 9, 8] for related seminal works in those years).

More precisely, given Ω ⊂ ℝ N convex and a function u on Ω ¯ , these are maximum principles for

( y , z , λ ) ↦ u ⁢ ( λ ⁢ y + ( 1 - λ ) ⁢ z ) - λ ⁢ u ⁢ ( y ) - ( 1 - λ ) ⁢ u ⁢ ( z )

for y , z ∈ Ω ¯ and λ ∈ [ 0 , 1 ] since positivity (negativity) in Ω ¯ × Ω ¯ × [ 0 , 1 ] corresponds to concavity (convexity) of u. As a byproduct, some results about concavity of positive solutions of semilinear problems can be obtained. For instance, if N ≥ 2 , q ∈ ( 0 , 1 ) , Ω is convex and u is a solution to

{ - Δ ⁢ u = u q in Ω , u > 0 in Ω , u = 0 on ∂ ⁡ Ω ,

then u 1 - q 2 is concave in Ω ¯ . Roughly speaking, some form of concavity on the nonlinear term forces a suitable power of the solution to be concave. In 1987 Sakaguchi treated, via a suitable approximation argument to handle lack of regularity of the solutions, the following problems involving the p-Laplacian operator

{ - Δ p ⁢ u = λ ⁢ u p - 1 in Ω , u > 0 in Ω , u = 0 on ∂ ⁡ Ω , { - Δ p ⁢ u = 1 in Ω , u > 0 in Ω , u = 0 on ∂ ⁡ Ω ,

where in the former, λ is (necessarily) the first eigenvalue. In [31], he indeed proved that log ⁡ u and u p - 1 p are concave, respectively.

1.2 Main results

Given the quasi-linear elliptic problem

(1.1) { - Δ p ⁢ u = f ⁢ ( u ) in Ω , u > 0 in Ω , u = 0 on ∂ ⁡ Ω

for p > 1 and Ω convex and bounded, our focus is to find the most general reaction term f ensuring that u is quasi-concave, i.e., its super-level sets { u > k } are convex for any k and to highlight the interplay between the reaction f and suitable concavity properties of u.

A natural method to obtain quasi-concavity is to find an increasing function φ : ℝ + → ℝ such that φ ⁢ ( u ) is concave. In this respect, by slightly modifying the proof of Sakaguchi [31], we find the following:

Theorem 1.1.

Let Ω ⊂ R N be a bounded convex domain with C 2 boundary, let f : R + → [ 0 , + ∞ ) be Hölder continuous with

M = inf ⁡ { t > 0 : f ⁢ ( t ) = 0 }

and let u ∈ W 0 1 , p ⁢ ( Ω ) solve (1.1). If

t ↦ f ⁢ ( t ) t p - 1 is non-increasing,
t ↦ e ( p - 1 ) ⁢ t f ⁢ ( e t ) is convex on ( - ∞ , log ⁡ M ) ,

then log ⁡ u is concave.

Notice that, given any concave increasing h : ℝ → ℝ , the concavity of φ ⁢ ( u ) (called φ-concavity of u) implies that h ⁢ ( φ ⁢ ( u ) ) is concave as well. Supposing both h and φ are smooth, by computing the second derivative of h ∘ φ one sees that h ∘ φ is “more concave” than φ, so that our interest is to determine the “less concave” φ such that φ ⁢ ( u ) is concave, as reasonable measure of the optimal quasi-concavity of u. For example, the less concave increasing functions φ are clearly the affine ones, and in this case φ-concavity reduces concavity.

To this end, suppose f ∈ C ⁢ ( ℝ + , [ 0 , + ∞ ) ) and let

F ⁢ ( t ) = ∫ 0 t f ⁢ ( u ) ⁢ 𝑑 u .

We will consider the function φ : ( 0 , + ∞ ) → ℝ defined by

(1.2) φ ⁢ ( t ) := ∫ 1 t 1 F 1 p ⁢ ( τ ) ⁢ 𝑑 τ

as detailed in the following theorem, which is the main result of the paper.

Theorem 1.2.

Let Ω ⊂ R N be a bounded convex domain with C 2 boundary, let f : R + → [ 0 , + ∞ ) be Hölder continuous with

M = inf ⁡ { t > 0 : f ⁢ ( t ) = 0 }

and let u ∈ W 0 1 , p ⁢ ( Ω ) solve (1.1). If

F 1 p is concave,
F f is convex on ( 0 , M ) ,

then φ ⁢ ( u ) is concave, where φ is defined in (1.2).

Let us make some remarks on these two statements.

Remark 1.3 (Motivations).

A conjecture of Lions [29] going back to the eighties states that any solution of (1.1) for p = 2 is quasi-concave, i.e., has convex super-level sets, as long as Ω is convex and f ≥ 0 . The latter statement (while true in the ball thanks to the celebrated Gidas-Ni-Nirenberg symmetry result) has been recently disproved in [16], thus revamping the question wether it is possible to select a large class of reactions f for which Lions’ statement turns out to be true.

The subject of the optimal concavity properties of u has been the object of recent research in the restricted framework of optimal power concavity (i.e., concavity properties of powers of u, for the highest possible power). It is known [23, item (iii) in Theorem 6.2 ] that if f ⁢ ( t ) ≡ 1 and p = 2 , the solution of (1.1) is such that u q is concave for q ≤ 1 2 , while for q > 1 2 there are convex domains for which u q fails to be concave. However, this phenomenon is known to happen only for convex domains having corners, while in the ball the torsion function is actually concave (thus, the optimal concavity exponent of the ball is 1). See [17] and the literature therein for further references on the general problem of linking the optimal concavity exponent with the smoothness of the domain.

Remark 1.4 (Relations between Theorems 1.1 and 1.2).

Condition (1) of Theorem 1.2 implies (1) of Theorem 1.1, but not the opposite, see Lemma 3.3 and Remark 3.4. For p = 2 the reaction

f ⁢ ( u ) = 1 + u

satisfies the assumptions of Theorem 1.1 but not (2) of Theorem 1.2. However, the concavity property asserted in Theorem 1.2 is in general stronger than the log -concavity provided by Theorem 1.1. Indeed, if ψ denotes the inverse of the function φ defined in (1.2), from

log ⁡ u = log ⁡ ψ ⁢ ( φ ⁢ ( u ) ) ,

we see that, as long as t ↦ log ⁡ ψ ⁢ ( t ) is concave, φ-concavity of u implies its log concavity. Lemma 3.5 below shows, under assumption (1) of Theorem 1.2, log ⁡ ψ is always concave. That φ-concavity is strictly stronger than log concavity is readily seen considering power-concavity, see below.

Remark 1.5 (Applications).

If f ∈ C 1 ⁢ ( ℝ + ) , conditions (1) and (2) of Theorem 1.2 read

F ⁢ f ′ f 2 ≤ 1 - 1 p , ( F ⁢ f ′ f 2 ) ′ ≤ 0

on ( 0 , M ) , so that actually the assumptions are equivalent to

F ⁢ f ′ f 2 ⁢ non-increasing on ( 0 , M ) and lim t → 0 + ⁡ F ⁢ f ′ f 2 ≤ 1 - 1 p

The assumptions in Theorem 1.2 are easily verified for f ⁢ ( t ) = t q with q ∈ [ 0 , p - 1 ) , giving, for its first instance, the concavity of u 1 p ⁢ ( p - q - 1 ) (briefly called α-concavity of u for α = 1 p ⁢ ( p - q - 1 ) ) of the solution u of

{ - Δ p ⁢ u = u q in Ω , u > 0 , in Ω , u = 0 on ∂ ⁡ Ω .

However, already in the semilinear case p = 2 , several meaningful reactions related to natural entropy models can be treated. We refer to section 2 for such examples of non-power like reactions.

Remark 1.6 (Assumptions on f).

The first assumptions of both Theorem 1.1 and Theorem 1.2 strengthen as p decreases, meaning that if the they hold for some p, then they do for any q ≥ p as well.

The non-negativity hypothesis on the reaction f is not essential and we made it to have a cleaner statement. It is possible to deal with functions obeying

f ⁢ ( t ) ≤ 0 for all ⁢ t ≥ M := inf ⁡ { t : f ⁢ ( t ) ≤ 0 } ,

since in this case an application of the weak maximum principle shows that u ≤ M (more details on this a-priori estimate can be found in Example 2.2 or at the beginning of Section 4). This, in turn, allows to require the whole assumption in (1) of Theorem 1.2 to be fulfilled on the interval ( 0 , M ) only. This restriction can sometimes be useful, as it allows to prove quasi-concavity of solutions to simple problems such as

{ - Δ ⁢ u + u = 1 in Ω , u > 0 in Ω , u = 0 on ∂ ⁡ Ω

in Example 2.2.

Let us also mention that the Hölder continuity of f can be weakened to Dini-continuity, since this suffices to obtain C 2 ⁢ ( Ω ) of solution of regular elliptic quasilinear problems with reaction f ⁢ ( u ) .

Remark 1.7 (Comparison with previous results).

Theorem 1.1 is folklore and we stated it only for completeness, since its proof follows the same approximation procedure of [31]. Regarding Theorem 1.2, the cases f ⁢ ( t ) ≡ 1 and f ⁢ ( t ) = t p - 1 have been treated in [31]. The power case f ⁢ ( t ) = t q with 0 < q < p - 1 is essentially contained in [19].

General reaction terms have been previously considered in [29, 25] in the semilinear case p = 2 , obtaining log -concavity of the positive solution of

- Δ ⁢ u = f ⁢ ( u ) , u = 0 on ∂ ⁡ Ω

for f ∈ C 1 ⁢ ( [ 0 , + ∞ ) , [ 0 , + ∞ ) ) fulfilling

(1.3) t ↦ f ⁢ ( t ) t ⁢ non-increasing and f ′′ ⁢ ( t ) ⁢ t 2 - f ′ ⁢ ( t ) ⁢ t + f ⁢ ( t ) ≤ 0 .

The second condition above fails for powers f ⁢ ( t ) = t q and is equivalent to the concavity of g ⁢ ( t ) = f ⁢ ( e t ) e t . Hypothesis (2) in Theorem 1.1 (in the case p = 2 ) is instead equivalent to the so-called harmonic concavity of g. It is well known that any concave function is harmonic concave, but the opposite is not true. For example, both Theorems 1.1 and 1.2 apply in the semilinear case p = 2 with f ⁢ ( t ) = t q with q < 1 (the latter providing, as already remarked, a stronger conclusion than the former), but (1.3) fails.

The monograph [22] reviews most of the techniques available to prove quasi-concavity of solutions to elliptic differential equation up to the mid-eighties. Apart from the Korevaar–Kennington approach, another fruitful method is to consider the concave envelope, as done in the seminal work [1] in the framework of viscosity solutions of fully nonlinear equations. This technique has been applied in [11, 12, 20, 21, 18, 27]. Other close approaches are the quasi-concave envelope method and its modifications [4, 10, 13] and the microscopic concavity principle initiated in [8] and developed in [26, 3]. The latter has been employed in [28] to prove that any solution in a convex domain of the plane of (1.1) for p = 2 and f ⁢ ( t ) = t q with q > 1 is 1 2 ⁢ ( q - 1 ) -concave. Unfortunately, none of these methods easily apply to (1.1) unless additional a-priori (and unexpected) regularity is assumed on the solution u.

1.3 Outline of the proof

The choice of the transformation (1.2) can be motivated, at least heuristically, by the following argument.

Let u be a solution to (1.1). Consider an increasing invertible function φ : ℝ + → ℝ with inverse ψ. We set

v := φ ⁢ ( u )

and compute the equation solved by v from the one for u, to obtain

(1.4) - ( Δ ⁢ v + p - 2 | ∇ ⁡ v | 2 ⁢ ∑ i , j = 1 N ∂ i ⁡ v ⁢ ∂ j ⁡ v ⁢ ∂ i ⁢ j 2 ⁡ v ) = f ⁢ ( ψ ⁢ ( v ) ) ψ ′ ⁢ ( v ) p - 1 ⁢ | ∇ ⁡ v | 2 - p + ( p - 1 ) ⁢ ψ ′′ ⁢ ( v ) ψ ′ ⁢ ( v ) ⁢ | ∇ ⁡ v | 2 = 0

In order to simplify (1.4) we force a factorization of the dependencies on v and on ∇ ⁡ v by requiring

(1.5) f ⁢ ( ψ ) ( ψ ′ ) p - 1 = p ⁢ ψ ′′ ψ ′

(the constant p is irrelevant of course, we chose that one for convenience). This amounts, by integration, to

ψ ′ ⁢ ( s ) = F ⁢ ( ψ ⁢ ( s ) ) 1 p , s > 0 ,

where F is the primitive of f that vanishes at ψ ⁢ ( 0 ) . This condition indeed gives φ given in (1.2).

By (1.5) we can rewrite (1.4) in the more tractable form

(1.6) Δ ⁢ v + p - 2 | ∇ ⁡ v | 2 ⁢ ∑ i , j = 1 n ∂ i ⁡ v ⁢ ∂ j ⁡ v ⁢ ∂ i ⁢ j 2 ⁡ v + b ⁢ ( v , ∇ ⁡ v ) = 0

where

b ⁢ ( v , ∇ ⁡ v ) := ψ ′′ ⁢ ( v ) ψ ′ ⁢ ( v ) ⁢ ( p ⁢ | ∇ ⁡ v | 2 - p + ( p - 1 ) ⁢ | ∇ ⁡ v | 2 ) ,

and we can require the assumptions on [23, Theorem 3.1] to apply Kennington theorem. The latter involve only the ratio ψ ′′ ψ ′ and, as it turns out, are equivalent to (1) and (2) of Theorem 1.2.

Of course, the problem is that equation (1.6) lacks sufficient smoothness of both the right- and the left-hand side to employ directly Kennington’s theorem, and the derivation depicted above seems quite rigid. The issue is then to find suitable regularizations of the original equation so that the previous computations makes sense and the requirements on the term ψ ′′ ψ ′ are stable in the limit.

Luckily, the expected condition (1) in Theorem 1.2 also implies uniqueness of the solution of the original equation (1.1). This is proved in Theorem 3.8 through a refined version of the celebrated Brezis–Oswald theorem [7]. This preliminary result is pivotal in ensuring the effectiveness of any approximation.

It turns out that a good approximation is given by the minimizers u ε of the functional

I ε ⁢ ( u ) := 1 p ⁢ ∫ Ω ( ε ⁢ F ⁢ ( u ) 2 p + | ∇ ⁡ u | 2 ) p 2 ⁢ 𝑑 x - ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x ,

as ε → 0 . Indeed, the aforementioned uniqueness ensures that u ε → u in all reasonable senses and, for sufficiently small ε, we are able to prove that φ ⁢ ( u ε ) is actually concave. The claimed concavity of φ ⁢ ( u ) then follows.

1.4 Structure of the paper and notations

In Section 2 we provide examples and application of Theorem 1.2. Section 3 is devoted to some preliminary material: we first recall the main tools we will use, essentially contained in [25, 23], then prove some properties of the functions f and φ, and conclude the section with the aforementioned Brezis–Oswald-type result. In Section 4 we focus on the proof of Theorems 1.1 and 1.2, dividing it in several steps.

Notations.

In the following Ω will always denote a bounded open convex domain, with interior normal usually denoted by n. The norm in L p ⁢ ( Ω ) of a function u : Ω → ℝ will be denoted by ∥ u ∥ p for any p ∈ [ 1 , + ∞ ] . We will use the same notation also for vector valued functions. Constants may change in value from line to line without changing symbol, as long as their dependance is clear from the context.

2 Examples and applications

In this section we provide some examples of application of Theorem 1.2 which were not available via known results.

Example 2.1.

The following examples provide non-power-like functions F fulfilling the assumptions of Theorem 1.2 for p = 2 (and thus for any p ≥ 2 ).

(1) The logarithmic entropy function

F ⁢ ( t ) = ( t + 1 ) ⁢ log ⁡ ( t + 1 ) - t

arises as the potential for the Dirichlet problem

(2.1) { - Δ ⁢ u = log ⁡ ( 1 + u ) in Ω , u = 0 on ∂ ⁡ Ω .

For t ∈ ℝ + , the function F verifies

( F 1 2 ) ′′ ( t ) = - F ⁢ ( t ) - 3 2 4 ⁢ ( t + 1 ) [ 2 t + ( t + 1 ) ( log ( t + 1 ) - 2 ) log ( t + 1 ) ] = : - F ⁢ ( t ) - 3 2 4 ⁢ ( t + 1 ) g ( t )

and g is non-negative on ℝ + , since

g ′ ⁢ ( 0 ) = 0 , g ′ ⁢ ( t ) = log 2 ⁡ ( t + 1 ) ,

implying that F 1 2 is concave. Finally,

( F f ) ′′ ⁢ ( t ) = ( t + 2 ) ⁢ log ⁡ ( t + 1 ) - 2 ⁢ t ( t + 1 ) 2 ⁢ log 3 ⁡ ( t + 1 )

whose numerator N ⁢ ( t ) fulfills

N ⁢ ( 0 ) = N ′ ⁢ ( 0 ) = 0 , N ′′ ⁢ ( t ) = t ( t + 1 ) 2 .

Applying Taylor’s theorem with integral remainder, we thus see that N is non-negative on ℝ + . Therefore, the unique positive solution of (2.1) has convex super-level sets. This conclusion cannot be reached through the results in [1, 25, 23] or [29].

(2) Consider the function F : ℝ + → ℝ given by

F ⁢ ( t ) = t ⁢ log ⁡ ( 1 + t ) ,

arising from the Dirichlet problem

(2.2) { - Δ ⁢ u = log ⁡ ( 1 + u ) + u 1 + u in Ω , u = 0 on ∂ ⁡ Ω .

It holds

( F 1 2 ) ′′ ⁢ ( t ) = - F ⁢ ( t ) - 3 2 4 ⁢ ( t + 1 ) 2 ⁢ [ ( t - log ⁡ ( t + 1 ) ) 2 + ( ( t + 1 ) 2 - 1 ) ⁢ log 2 ⁡ ( t + 1 ) ]

which is non-positive for t ≥ 0 . Moreover,

( F f ) ′′ ⁢ ( t ) = ( t + 1 ) ⁢ ( t ⁢ ( t + 4 ) - 2 ⁢ ( t + 1 ) ) ⁢ log ⁡ ( t + 1 ) - t 2 ⁢ ( t + 2 ) ( t + 1 ) ⁢ ( t + ( t + 1 ) ⁢ log ⁡ ( t + 1 ) ) 3 ,

so that, in order to prove that F f is convex, it suffices to consider the numerator N ⁢ ( t ) . Using log ⁡ ( t + 1 ) ≤ t , we get

N ( t ) ≥ g ( t ) := ( t + 1 ) ( t ( t + 4 ) - 2 t ) ) log ( t + 1 ) - t 2 ( t + 2 ) ,

and the auxiliary function g obeys

g ( k ) ⁢ ( 0 ) = 0 for k = 0 , 1 , 2 , g ( 3 ) ⁢ ( 0 ) = 6 , g ( 4 ) ⁢ ( t ) = 2 ⁢ ( 3 ⁢ t 2 + 6 ⁢ t + 2 ) ( t + 1 ) 3 .

Reasoning as before, we see that g ≥ 0 on [ 0 , + ∞ ) , proving that F f is convex. Theorem 1.2 therefore ensures that the unique positive solution of (2.2) is quasi-concave, which again cannot be deduced from previously known results.

(3) In general, observe that if a function F satisfies the assumptions of Theorem 1.2 for p = 2 , then, given q > 1 , F q 2 satisfies the assumptions for p = q as

( F q 2 ) 1 p = F 1 2 , F q 2 ( F q 2 ) ′ = 2 q ⁢ F f .

We conclude with an example showing what is the rôle of the restriction to the image of the conditions in Theorem 1.2

Example 2.2.

Consider the problem

(2.3) { - Δ ⁢ u + u = 1 in Ω , u = 0 on ∂ ⁡ Ω , u > 0 in Ω

for a convex Ω. A unique, non-negative solution is immediately obtained via minimization of the strictly convex functional

J ⁢ ( v ) = ∫ Ω | ∇ ⁡ v | 2 + v 2 - 2 ⁢ v ⁢ d ⁢ x ,

but Theorem 1.2 does not naively apply, since the reaction term fails to be non-negative. However, since v ⁢ ( x ) ≡ 1 solves (2.3), an application of the comparison principle implies that u ≤ 1 in Ω. Therefore, problem (2.3) can be rephrased as

{ - Δ ⁢ u = ( 1 - u ) + in Ω u = 0 on ∂ ⁡ Ω u > 0 in Ω .

Let us denote with F the primitive of ( 1 - t ) + vanishing at 0. A simple calculation shows that F 1 2 is concave on ℝ + and thanks to the a-priori bound u ≤ 1 , it suffices to check the convexity of F f only on the interval ( 0 , 1 ) , where it holds

( F f ) ′′ ⁢ ( t ) = 1 ( 1 - t ) 3 > 0 .

So, the solution of (2.3) has convex super-level sets. Notice that, looking at the original problem (2.3), with

f ~ ⁢ ( t ) = 1 - t , F ~ ⁢ ( t ) = t - t 2 2 ,

the function F ~ f ~ turns out to be concave for t > 1 .

3 Preliminaries

In this section we collect various results that will be useful in the proof of Theorem 1.2.

3.1 The convexity function

We recall here the main tools developed by Korevaar and Kennigton to deal with concavity properties of solution to (1.1). Given a continuous function v : Ω → ℝ with Ω convex, its convexity function c : Ω × Ω → ℝ is defined as

c ⁢ ( x , y ) = v ⁢ ( x ) + v ⁢ ( y ) 2 - v ⁢ ( x + y 2 ) .

Clearly, v is concave in Ω if and only if c ≤ 0 in its domain. The main result of Kennington, generalized in [15], is the following; that assumptions (1) and (2) below can be checked only on the image of v, while not explicitly stated, follows from inspecting the proof of [15].

Proposition 3.1 ([15]).

Let Ω be bounded and convex in R N , N ≥ 2 , and assume that v ∈ C 2 ⁢ ( Ω ) ∩ C 0 ⁢ ( Ω ¯ ) solves

∑ i , j = 1 N a i , j ⁢ ( D ⁢ v ) ⁢ ∂ i ⁢ j ⁡ v + b ⁢ ( v , D ⁢ v ) = 0

for a i ⁢ j ∈ C 0 ⁢ ( R N ) such that ( a i ⁢ j ) is uniformly elliptic and b ∈ C 0 ⁢ ( R × R N ; R + ) . If

t ↦ b ⁢ ( t , z ) is non-increasing on v ⁢ ( Ω ) for all z ∈ ℝ N ,
t ↦ b ⁢ ( t , z ) harmonic concave on v ⁢ ( Ω ) for all z ∈ ℝ N ,

then the convexity function of v cannot attain a positive maximum in Ω × Ω .

We refer to [15] for a general definition of harmonic concavity. For our purposes, the function b will be positive on v ⁢ ( Ω ) and in this case harmonic concavity coincides with the convexity of t ↦ 1 b ⁢ ( t , z ) .

Korevaar singled out a class of transformations allowing to exclude that the maximum of the convexity function is attained at the boundary ∂ ⁡ ( Ω × Ω ) . The following proposition has been obtained in [25, Lemmas 2.1 and 2.4] for u ∈ C 2 ⁢ ( Ω ¯ ) . Recall that a strongly convex set is a smooth convex set such that the principal curvatures of ∂ ⁡ Ω are positive. Given such an Ω we set

S δ = { x ∈ Ω : δ 2 ≤ dist ⁢ ( x , ∂ ⁡ Ω ) ≤ 2 ⁢ δ } , Ω δ = { x ∈ Ω : δ < dist ⁢ ( x , ∂ ⁡ Ω ) } .

Proposition 3.2 ([25, Lemmas 2.1 and 2.4]).

Suppose that Ω is smooth, bounded and strongly convex, N is a neighborhood of ∂ ⁡ Ω and let u ∈ C 1 ⁢ ( Ω ¯ ) ∩ C 2 ⁢ ( N ) be such that

(3.1) u > 0 in Ω , u = 0 on ∂ ⁡ Ω , ∂ ⁡ u ∂ ⁡ n > 0 on ∂ ⁡ Ω .

If φ ∈ C 2 ⁢ ( R + ; R ) fulfils

(3.2) lim t → 0 + ⁡ φ ′ ⁢ ( t ) = + ∞ , φ ′′ < 0 < φ ′ near 0 , lim t → 0 + ⁡ φ ⁢ ( t ) φ ′ ⁢ ( t ) = lim t → 0 + ⁡ φ ′ ⁢ ( t ) φ ′′ ⁢ ( t ) = 0 ,

and v := φ ⁢ ( u ) , then, for any sufficiently small δ, it holds

(3.3) D 2 ⁢ v ⁢ ( x ) < 0 in ⁢ S δ ,

and

(3.4) ∀ x ¯ ∈ ∂ ⁡ Ω δ v ⁢ ( x ) < L v , x ¯ ⁢ ( x ) in ⁢ Ω δ ∖ { x ¯ } ,

where L v , x ¯ ⁢ ( x ) = v ⁢ ( x ¯ ) + ∇ ⁡ v ⁢ ( x ¯ ) ⁢ ( x - x ¯ ) is the tangent plane to the graph of v at x ¯ .

Moreover, if (3.3) and (3.4) hold for some given v ∈ C 2 ⁢ ( Ω δ / 2 ) , its convexity function cannot attain a positive maximum on ∂ ⁡ ( Ω δ × Ω δ ) .

3.2 The transformation φ

Since we are interested in positive solutions of (1.1), we will assume henceforth that f is extended to ℝ as an even function, so that F is odd. First observe that being F 1 p concave on [ 0 , + ∞ ) , it has sublinear growth, implying the estimate

(3.5) F ⁢ ( t ) ≤ C ⁢ ( 1 + t p ) , t ≥ 0 .

Applying again the concavity assumption, we also infer

(3.6) f ⁢ ( t ) ≤ C ⁢ ( 1 + t p - 1 ) , t ≥ 0 ,

therefore solutions of (1.1) are critical points of the C 1 functional on W 0 1 , p ⁢ ( Ω ) defined by

(3.7) J ⁢ ( u ) = ∫ Ω 1 p ⁢ | ∇ ⁡ u | p - F ⁢ ( u ) ⁢ d ⁢ x .

A more refined and useful estimate than (3.6) is contained in the following lemma.

Lemma 3.3.

Let F : [ 0 , + ∞ ) → [ 0 , + ∞ ) be differentiable and such that F ⁢ ( 0 ) = 0 . If F 1 p is concave for some p ≥ 1 , then t ↦ F ′ ⁢ ( t ) t p - 1 is non-increasing.

Proof.

Let G ⁢ ( t ) = F 1 p ⁢ ( t ) , and observe that G ′ ≥ 0 on [ 0 , + ∞ ) , since G ′ ⁢ ( t 0 ) < 0 implies by concavity

G ⁢ ( t ) ≤ G ⁢ ( t 0 ) + G ′ ⁢ ( t 0 ) ⁢ ( t - t 0 ) → - ∞ ,

contradicting G ≥ 0 . If

H ⁢ ( s ) = G p ⁢ ( s 1 p ) ,

we claim that H is concave. Indeed,

H ′ ⁢ ( s ) = G p - 1 ⁢ ( s 1 p ) ⁢ G ′ ⁢ ( s 1 p ) ⁢ s 1 - p p = ( G ⁢ ( s 1 p ) s 1 p ) p - 1 ⁢ G ′ ⁢ ( s 1 p )

and s ↦ H ′ ⁢ ( s ) is non-increasing on ℝ + if and only if so is t ↦ H ′ ⁢ ( t p ) , i.e., if and only if

(3.8) t ↦ ( G ⁢ ( t ) t ) p - 1 ⁢ G ′ ⁢ ( t )

is non-increasing. Since both

t ↦ G ′ ⁢ ( t ) and t ↦ G ⁢ ( t ) t = G ⁢ ( t ) - G ⁢ ( 0 ) t - 0

are non-negative, and non-increasing by the concavity of G, so is (3.8), being the product of non-negative, non-increasing functions. It follows that H ⁢ ( s ) = F ⁢ ( s 1 p ) is concave, so that its derivative is non-increasing. Since

H ′ ⁢ ( s ) = 1 p ⁢ F ′ ⁢ ( s 1 p ) ⁢ s 1 - p p ,

gives the claim by the monotone increasing change of variable t = s 1 p . ∎

Remark 3.4.

The opposite implication in the previous assertion fails to be true. When p = 2 , the function

F ⁢ ( t ) = 1 + t + t 2 , t ≥ 0 ,

is such that F ′ ⁢ ( t ) t is non-increasing on ℝ + , but F 1 2 is convex for t ≥ 0 .

We next provide the elementary proof of a property, mentioned in the introduction, of the inverse ψ = φ - 1 of the function φ given (1.2). In that framework, we apply the lemma with G = F 1 p .

Lemma 3.5.

Suppose that G ∈ C 0 ⁢ ( [ 0 , + ∞ ) ; R + ) is concave with G ⁢ ( 0 ) = 0 and let

φ ⁢ ( t ) = ∫ 1 t 1 G ⁢ ( τ ) ⁢ 𝑑 τ , ψ ⁢ ( s ) = φ - 1 ⁢ ( s ) .

Then s ↦ log ⁡ ψ ⁢ ( s ) is concave.

Proof.

We compute

( log ⁡ ψ ) ′ = ψ ′ ψ = G ⁢ ( ψ ) ψ

and, since ψ is increasing, the claim is equivalent to the fact that t ↦ G ⁢ ( t ) t is non-increasing. But this follows from the assumed concavity of G together with G ⁢ ( 0 ) = 0 . ∎

In order to apply Proposition 3.2, we point out the following.

Lemma 3.6.

Let f ∈ C 0 ⁢ ( R + , [ 0 , + ∞ ) ) with

M = inf ⁡ { t > 0 : f ⁢ ( t ) = 0 } > 0 .

If the corresponding F is such that F 1 p is concave and F f is convex in ( 0 , M ) , the function

φ ⁢ ( t ) = ∫ 1 t F ⁢ ( s ) - 1 p ⁢ 𝑑 s , where ⁢ F ⁢ ( s ) = ∫ 0 s f ⁢ ( τ ) ⁢ 𝑑 τ , t , s > 0 ,

fulfils (3.2) on ( 0 , M ) .

Proof.

By the definition of M, for any t ∈ ( 0 , M ) it holds

φ ′′ ⁢ ( t ) = - f ⁢ ( t ) p ⁢ F ⁢ ( t ) 1 + 1 p < 0 ,

and from φ ′ ⁢ ( t ) = F ⁢ ( t ) - 1 p and F ≥ 0 we readily have φ ′ ⁢ ( t ) → + ∞ for t → 0 + . By construction F ⁢ ( t ) ≤ C ⁢ t in ( 0 , M ) and from the concavity of F 1 p we infer F 1 p ⁢ ( t ) ≥ c ⁢ t for t ∈ ( 0 , M ) , c > 0 . Therefore, for t ∈ ( 0 , M ) ,

0 ≤ φ ⁢ ( t ) φ ′ ⁢ ( t ) = F 1 p ⁢ ( t ) ⁢ ∫ t 1 1 F 1 p ⁢ ( t ) ⁢ 𝑑 t ≤ C 1 p c ⁢ t 1 p ⁢ log ⁡ t → 0 .

It remains to prove that

lim t → 0 + ⁡ φ ′ ⁢ ( t ) φ ′′ ⁢ ( t ) = lim t → 0 + ⁡ p ⁢ F ⁢ ( t ) f ⁢ ( t ) = 0 .

By the convexity of F f the limit exists, as does, by monotonicity, the limit

l = lim t → 0 + ⁡ φ ⁢ ( t ) ≤ 0 .

Hence the claim follows from the previous limit by de l’Hôpital rule, either applied to φ φ ′ if l = - ∞ or to φ - l φ ′ if l is finite. ∎

3.3 A Brezis–Oswald-type result

It is known that the functional J in (3.7) is convex in the variable w = u p and this implies the monotonicity of the operator

w ↦ - Δ p ⁢ w 1 p w p - 1 p

as first remarked for p = 2 by Benguria, Brezis and Lieb [2], see also [6].

An useful consequence (see e.g. [14]) of this convexity property is the following:

Lemma 3.7.

For i = 1 , 2 let w i ∈ L ∞ ⁢ ( Ω ) ∩ W 1 , p ⁢ ( Ω ) be such that

w i > 0 , Δ p ⁢ w i ∈ L ∞ ⁢ ( Ω ) , w 1 - w 2 ∈ W 0 1 , p ⁢ ( Ω ) , w 2 w 1 , w 1 w 2 ∈ L ∞ ⁢ ( Ω ) .

Then there holds

(3.9) ∫ Ω ( - Δ p ⁢ w 1 w 1 p - 1 - - Δ p ⁢ w 2 w 2 p - 1 ) ⁢ ( w 1 p - w 2 p ) ⁢ 𝑑 x ≥ 0 .

We also recall the definition of the first eigenvalue of the p-Laplacian

λ 1 , p := inf ⁡ { ∫ Ω | ∇ ⁡ v | p ⁢ 𝑑 x : v ∈ W 0 1 , p ⁢ ( Ω ) , ∫ Ω | v | p ⁢ 𝑑 x = 1 } .

The next proposition is a Brezis–Oswald-type of result [7]. The proof follows Díaz and Saá [14] but here we deal with the case when f ⁢ ( t ) t p - 1 is only monotone and not (as assumed in [14]) strictly monotone. Notice that it holds, more generally, when Ω is connected but not necessarily convex.

Proposition 3.8.

Let Ω ⊆ R N be connected with C 2 boundary, and let f ∈ C 0 ⁢ ( R + , R ) be such that t ↦ f ⁢ ( t ) t p - 1 is non-increasing on ( 0 , + ∞ ) . If u ∈ W 0 1 , p ⁢ ( Ω ) ∩ C 1 , α ⁢ ( Ω ¯ ) solves (1.1), then either u is a λ 1 , p -eigenfunction or

(3.10) lim t → + ∞ ⁡ f ⁢ ( t ) t p - 1 < λ 1 , p < lim t → 0 + ⁡ f ⁢ ( t ) t p - 1

and u is the unique solution of (1.1), which also minimizes J on W 0 1 , p ⁢ ( Ω ) where, in the definition of J, f is evenly extended on R .

Proof.

Suppose that u is not a λ 1 , p -eigenfunction. Then, being positive in Ω, it is not an eigenfunction at all, so that we may assume that

(3.11) f is not of the form k ⁢ t p - 1 on [ 0 , ∥ u ∥ ∞ ] ,

By [14, Theorem 2] the solvability of (1.1) implies that

(3.12) lim t → + ∞ f ⁢ ( t ) t p - 1 = : μ ∞ ≤ λ 1 , p .

Indeed, observe that by the monotonicity assumption on t ↦ f ⁢ ( t ) t p - 1 , it holds

(3.13) f ⁢ ( u ) u p - 1 ≥ f ⁢ ( ∥ u ∥ ∞ ) ∥ u ∥ ∞ p - 1 ≥ μ ∞ .

Let φ ∈ C 1 , α ( Ω ) ¯ be a positive λ 1 , p -eigenfunction. Any positive multiple of φ is still an eigenfunction so that, using the Hopf Lemma and C 1 , α ⁢ ( Ω ¯ ) of both u and φ, we can choose k > 0 such that k ⁢ φ > u on Ω. Notice that all the assumption of Lemma 3.7 hold for w 1 = k ⁢ φ and w 2 = u on Ω, so that by (3.9) and (3.13), we obtain

0 ≤ ∫ Ω ( - Δ p ⁢ ( k ⁢ φ ) ( k ⁢ φ ) p - 1 - - Δ p ⁢ u u p - 1 ) ⁢ ( ( k ⁢ φ ) p - u p ) ⁢ 𝑑 x = ∫ Ω ( λ 1 , p - f ⁢ ( u ) u p - 1 ) ⁢ ( k p ⁢ φ p - u p ) ⁢ 𝑑 x ≤ ( λ 1 , p - μ ∞ ) ⁢ ∫ Ω ( k p ⁢ φ p - u p ) ⁢ 𝑑 x

giving (3.12) by the positivity of the integral. By the previous chain of inequalities, μ ∞ = λ 1 , p implies that f ⁢ ( u ) = λ 1 , p ⁢ u p on Ω, contradicting (3.11), so that the inequality in (3.13) is strict. Consider now

μ 0 := lim t → 0 + ⁡ f ⁢ ( t ) t p - 1 .

By the properties of the first eigenvalue and the monotonicity of t ↦ f ⁢ ( t ) t p - 1 , there holds

λ 1 , p ⁢ ∫ Ω u p ⁢ 𝑑 x ≤ ∫ Ω | ∇ ⁡ u | p ⁢ 𝑑 x ≤ ∫ Ω f ⁢ ( u ) ⁢ u ⁢ 𝑑 x = ∫ Ω f ⁢ ( u ) u p - 1 ⁢ u p ⁢ 𝑑 x ≤ μ 0 ⁢ ∫ Ω u p ⁢ 𝑑 x .

The latter chain of inequalities ensures that λ 1 , p ≤ μ 0 and that equality holds if and only if f ⁢ ( u ) = μ 0 ⁢ u p , again contradicting (3.11) and proving that λ 1 , p < μ 0 .

To prove the assertion on J, observe that the first inequality in (3.10) ensures by standard methods that J is coercive on W 0 1 , p ⁢ ( Ω ) and possesses a minimum u ¯ ∈ W 0 1 , p ⁢ ( Ω ) ∩ C 1 , α ⁢ ( Ω ¯ ) . Being F odd, it holds J ⁢ ( | v | ) ≤ J ⁢ ( v ) for any v ∈ W 0 1 , p ⁢ ( Ω ) , so that we can assume that u ¯ ≥ 0 . Moreover, again by standard methods, the second inequality in (3.10) implies

inf W 0 1 , p ⁢ ( Ω ) ⁡ J < 0 .

We conclude that u ¯ is nontrivial, and therefore strictly positive by the strong minimum principle.

It remains to show that u = u ¯ . Applying again Lemma 3.7 we have

0 ≤ ∫ Ω ( f ⁢ ( u ) u p - 1 - f ⁢ ( u ¯ ) u ¯ p - 1 ) ⁢ ( u p - u ¯ p ) ⁢ 𝑑 x .

By the monotonicity assumption, the two factors in the integrand have opposite sign, so that we infer

(3.14) f ⁢ ( u ) u p - 1 = f ⁢ ( u ¯ ) u ¯ p - 1 in Ω

Next recall that the pointwise Picone inequality [6]

(3.15) | ∇ ⁡ v | p - 2 ⁢ ∇ ⁡ v ⋅ ∇ ⁡ w p v p - 1 ≤ | ∇ ⁡ w | p , v , w > 0 ,

becomes an equality in a connected set if and only if v = k ⁢ w . Applying the latter for v = u ¯ and w = u and integrating, we get

∫ Ω f ⁢ ( u ¯ ) u ¯ p - 1 ⁢ u p ⁢ 𝑑 x = ∫ Ω | ∇ ⁡ u ¯ | p - 2 ⁢ ∇ ⁡ u ¯ ⋅ ∇ ⁡ u p u ¯ p - 1 ≤ ∫ Ω | ∇ ⁡ u | p ≤ ∫ Ω f ⁢ ( u ) ⁢ u ⁢ 𝑑 x = ∫ Ω f ⁢ ( u ) u p - 1 ⁢ u p ⁢ 𝑑 x .

Since u is positive, (3.14) implies that equality is attained in (3.15) for v = u ¯ and w = u everywhere in Ω. Hence

(3.16) u = k ⁢ u ¯ for some k > 0 .

Let g ⁢ ( t ) := f ⁢ ( t ) t p - 1 and assume that k > 1 in (3.16). By continuity, for any n ∈ ℕ there exists x n ∈ Ω such that

u ⁢ ( x n ) = ∥ u ∥ ∞ k n

and thus (3.14) and (3.16) give

g ⁢ ( ∥ u ∥ ∞ ) = g ⁢ ( u ⁢ ( x 0 ) ) = g ⁢ ( u ¯ ⁢ ( x 0 ) ) = g ⁢ ( u ⁢ ( x 0 ) k ) = g ⁢ ( u ⁢ ( x 1 ) ) = … = g ⁢ ( ∥ u ∥ ∞ k n )

for any n ≥ 1 . Therefore g, being non-increasing, is constant on ( 0 , ∥ u ∥ ∞ ) , contradicting (3.11). Similarly, if 0 < k < 1 , we infer that g is constant on

( 0 , ∥ u ¯ ∥ ∞ ) = ( 0 , ∥ u ∥ ∞ k ) ⊇ ( 0 , ∥ u ∥ ∞ ) ,

again contrary to (3.11). Therefore k = 1 and u = u ¯ , as claimed. ∎

4 Proof of the main result

Our proof consists in a two-step approximation. Following [31], we first show that it suffices to deal with strictly convex domains. Then, under such an assumption, we introduce a regularized problem for which we can prove the claim of Theorem 1.2. The solutions of the regularized problem converge uniformly to that of the original problem, thus proving Theorem 1.2.

In light of the result of [31], we will suppose henceforth that u is not a λ 1 , p -eigenfunction, so that inequalities (3.10) are in place, ensuring that the functional J in (3.7) has a unique positive minimizer on W 0 1 , p ⁢ ( Ω ) by Proposition 3.8 and Lemma 3.3. Notice that this holds for any smooth Ω, justifying some of the following arguments.

4.1 Restricting to strictly convex domains

We start by showing that there is no loss of generality considering strictly convex domains. To this aim, consider a sequence of strongly convex smooth domains ( Ω k ) k ∈ ℕ such that

Ω ¯ k ⊆ Ω k + 1 for all ⁢ k ∈ ℕ , Ω k → Ω ,

where the last convergence is meant in the Hausdorff sense. Let u k ∈ W 0 1 , p ⁢ ( Ω k ) be the unique solution to (1.1) in Ω k , which can be extended to Ω setting u k = 0 on Ω ∖ Ω k . Observe that u k is the unique positive minimizer of

J k ⁢ ( v ) = ∫ Ω k 1 p ⁢ | ∇ ⁡ v | p - F ⁢ ( v ) ⁢ d ⁢ x ,

and by minimality

J ⁢ ( u k + 1 ) = J k + 1 ⁢ ( u k + 1 ) ≤ J k + 1 ⁢ ( u k ) = J ⁢ ( u k ) ,

and then the coercivity of J ensures that ( u k ) k ∈ ℕ is bounded in W 0 1 , p ⁢ ( Ω ) . Thus, up to subsequences, we can assume that

u k ⇀ u ~ weakly in W 0 1 , p ⁢ ( Ω )

u k → u ~ strongly in L s ⁢ ( Ω ) , for all s ∈ [ 1 , p ]

u k → u ~ a.e.

for some u ~ ∈ W 0 1 , p ⁢ ( Ω ) . The continuity of the Nemitski operator associated to F gives

lim n → ∞ ⁡ ∫ Ω F ⁢ ( u k ) ⁢ 𝑑 x = ∫ Ω F ⁢ ( u ~ ) ⁢ 𝑑 x

and by the semicontinuity of v ↦ ∥ ∇ ⁡ v ∥ p p , it holds

(4.1) J ⁢ ( u ~ ) ≤ lim ¯ k → + ∞ ⁡ J ⁢ ( u k ) = lim ¯ k → + ∞ ⁡ J k ⁢ ( u k ) .

Set

ε k = 2 ⁢ sup Ω ∖ Ω k ⁡ u .

Since u ∈ C 0 ⁢ ( Ω ¯ ) , u > 0 in Ω and Ω k → Ω in Hausdorff distance, it holds ε k ↓ 0 . Moreover, ( u - ε k ) + ∈ W 0 1 , p ⁢ ( Ω k ) and there holds ( u - ε k ) + → u in W 0 1 , p ⁢ ( Ω ) , so that

lim k → + ∞ ⁡ J k ⁢ ( ( u - ε k ) + ) = J ⁢ ( u ) .

Observe that we have used again the fact that

J k ⁢ ( ( u - ε k ) + ) = J ⁢ ( ( u - ε k ) + ) for all ⁢ k ∈ ℕ

as ( u - ε k ) + vanishes on Ω ∖ Ω k . By the minimality of u k and (4.1), we infer

J ⁢ ( u ~ ) ≤ lim ¯ k → + ∞ ⁡ J k ⁢ ( u k ) ≤ lim ¯ k → + ∞ ⁡ J k ⁢ ( ( u - ε k ) + ) = J ⁢ ( u ) ,

ensuring that u ~ = u by Proposition 3.8. Then assume that we proved the claim of Theorem 1.2 for u k , for all k ∈ ℕ , that is, φ ⁢ ( u k ) is concave on Ω k . The convexity functions c k of φ ⁢ ( u k ) are therefore non-positive on Ω k × Ω k × [ 0 , 1 ] and converge pointwise a.e. on Ω × Ω × [ 0 , 1 ] to the convexity function c of φ ⁢ ( u ) . By the continuity of u, we thus infer that c ≤ 0 on Ω k 0 × Ω k 0 × [ 0 , 1 ] for any k 0 ∈ ℕ , thus φ ⁢ ( u ) is concave on Ω. Therefore we will suppose in the following that Ω is strictly convex.

4.2 A regularized problem

Let us now turn to the approximation procedure. We introduce a regularized problem for which we can prove the claim of Theorem 1.2. In the following we can assume that u is not a λ 1 , p -eigenfunction, so that (3.10) holds true.

Given ε > 0 , consider the variational problem

(4.2) inf v ∈ W 0 1 , p ⁢ ( Ω ) I ε ( v ) = : λ ε ,

where

I ε ⁢ ( v ) := 1 p ⁢ ∫ Ω ( ε ⁢ ( G ⁢ ( v ) 2 ) 1 p + | ∇ ⁡ v | 2 ) p 2 ⁢ 𝑑 x - ∫ Ω F ⁢ ( v ) ⁢ 𝑑 x

for some odd G ∈ C 1 ⁢ ( ℝ ) that will later be chosen. In the following we will let g := G ′ .

Lemma 4.1.

Suppose that f ∈ C 0 ⁢ ( R ) is even, obeys (3.10) and t ↦ f ⁢ ( t ) t p - 1 is non-increasing on R + . Then, for any sufficiently small ε > 0 , problem (4.2) admits a non-negative, nontrivial solution u ε ∈ W 0 1 , p ⁢ ( Ω ) such that, for a fixed α > 0 , u ε → u in C 1 , α ⁢ ( Ω ) as ε → 0 , where u is the unique positive solution to (1.1).

Proof.

The equi-coercivity of I ε follows from the one of J and the obvious inequality I ε ≥ J . Furthermore, if u is as in the statement, we have by (3.10)

J ⁢ ( u ) = inf W 0 1 , p ⁢ ( Ω ) ⁡ J < 0 ,

so that by continuity I ε ⁢ ( u ) < 0 for sufficiently small ε > 0 . Thus (4.2) has a nontrivial solution, which is non-negative by I ε ⁢ ( | v | ) ≤ I ε ⁢ ( v ) . In order to prove the convergence of u ε to u, first observe that by dominated convergence it is readily checked that for any w ∈ W 0 1 , p ⁢ ( Ω ) .

(4.3) lim ε → 0 + ⁡ I ε ⁢ ( w ) = J ⁢ ( w ) .

Since

J ⁢ ( u ) ≤ J ⁢ ( u ε ) ≤ I ε ⁢ ( u ε ) = λ ε ≤ I ε ⁢ ( 0 ) = 0 ,

it follows that u ε is bounded in W 0 1 , p ⁢ ( Ω ) , uniformly for 0 < ε < 1 . Hence there exists v ∈ W 0 1 , p ⁢ ( Ω ) such that, up to subsequences, u ε k ⇀ v weakly in W 0 1 , p ⁢ ( Ω ) , as k → ∞ .

From the weak lower semicontinuity of J, the minimality of u ε k and (4.3), we get

J ⁢ ( v ) ≤ lim ¯ k ⁡ J ⁢ ( u ε k ) ≤ lim ¯ k ⁡ I ε k ⁢ ( u ε k ) ≤ lim ¯ k ⁡ I ε k ⁢ ( u ) = J ⁢ ( u )

so that v is a minimizer for J as well, forcing u = v by the uniqueness proved in Proposition 3.8. The previous display forces J ⁢ ( u ε k ) → J ⁢ ( u ) but, as already noted,

∫ Ω F ⁢ ( u ε k ) ⁢ 𝑑 x → ∫ Ω F ⁢ ( u ) ⁢ 𝑑 x

so that we infer that ∥ ∇ ⁡ u ε k ∥ p p → ∥ ∇ ⁡ u ∥ p p , giving the strong convergence u ε k → u in W 0 1 , p ⁢ ( Ω ) by uniform convexity. A standard sub-subsequence argument allow to conclude that u ε → u in W 0 1 , p ⁢ ( Ω ) . Arguing as in [31, Proposition 6.2], we infer from (3.5) and the uniform bound on ∥ ∇ ⁡ u ε ∥ p a uniform bound on ∥ u ε ∥ ∞ for ε ∈ ( 0 , 1 ) . Standard regularity theory then ensures that ( u ε ) is uniformly bounded in C 1 , β ⁢ ( Ω ¯ ) , β > 0 and thus converges to u in C 1 , β 2 ⁢ ( Ω ¯ ) by the Ascoli–Arzelá theorem. ∎

We collect next some regularity properties of the minimizers u ε just constructed.

Lemma 4.2.

Let u ε be the solutions constructed in Lemma 4.1 under its assumptions, and suppose that G > 0 on R + .

Each u ε satisfies in Ω the Euler–Lagrange equation

(4.4) - div ⁡ ( ( ε ⁢ G ⁢ ( u ε ) 2 p + | ∇ ⁡ u ε | 2 ) p - 2 2 ⁢ ∇ ⁡ u ε ) = f ⁢ ( u ε ) - ε p ⁢ ( ε ⁢ G ⁢ ( u ε ) 2 p + | ∇ ⁡ u ε | 2 ) p - 2 2 ⁢ G ⁢ ( u ε ) 2 - p p ⁢ g ⁢ ( u ε ) .
For sufficiently small ε , we have

u ε > 0 in Ω , ∂ ⁡ u ε ∂ ⁡ n > 0 on ∂ ⁡ Ω

where n is the interior normal to ∂ ⁡ Ω .
If f and g are α -Hölder continuous, then u ε ∈ C 2 , α ⁢ ( Ω ) and there exists η 0 > 0 such that u ε is uniformly bounded in C 2 , α ⁢ ( S η 0 ) , where

S η := { x ∈ Ω : η 2 ≤ dist ⁢ ( x , ∂ ⁡ Ω ) ≤ 2 ⁢ η } .

Proof.

The first assertion is a simple calculation. Recall that u itself satisfies

u > 0 in Ω , ∂ ⁡ u ∂ ⁡ n > 0 on ∂ ⁡ Ω

by the Hopf lemma, so that the second assertion follows from the convergence u ε → u in C 1 , α ⁢ ( Ω ¯ ) . Regarding the last regularity property, it suffice to observe that the operator on the left-hand side of (4.4) is uniformly elliptic with Hölder continuous coefficients since, from the previous point, there exists c 0 > 0 such that either | ∇ ⁡ u ε | ≥ c 0 near ∂ ⁡ Ω , or F ⁢ ( u ) ≥ c 0 away from ∂ ⁡ Ω . The right-hand side of (4.4) is Hölder continuous away from ∂ ⁡ Ω , so that the nonlinear Schauder estimates give the claims in (3). ∎

Theorems 1.2 and 1.1 will now follow from the next statement, through a simple approximation argument on the concavity function.

Proposition 4.3.

Under the assumptions of Theorem 1.2, let φ be defined in (1.2). For any sufficiently small δ > 0 , there exist ε δ such that if ε < ε δ , then v = φ ⁢ ( u ε ) is concave on

Ω δ = { x ∈ Ω : dist ⁢ ( x , ∂ ⁡ Ω ) > δ } .

The same statement holds for v = log ⁡ u .

4.3 Non-positivity at the boundary

From the arguments given in the proof of Lemma 4.2 we can fix δ 0 so that the solution u of (1.1) belongs to C 2 ⁢ ( Ω ∖ Ω δ 0 ) . Moreover, (3.1) hold true for u and, thanks to Lemma 3.6 (and a direct computation in the case of log ⁡ t ), Proposition 3.2 applies for both the transformations φ defined in (1.2) an φ ~ ⁢ ( t ) = log ⁡ t . In the next argument, with v we denote for simplicity both φ ⁢ ( u ) and φ ~ ⁢ ( u ) , and v ε will stand for both φ ⁢ ( u ε ) and φ ~ ⁢ ( u ε ) . It follows that (3.3) and (3.4) both hold for v and for any δ ≤ δ 1 ≤ δ 0 .

If η 0 is given in (3) of Lemma 4.2, we next choose δ < 1 2 ⁢ min ⁡ { η 0 , δ 1 } so that v ε is uniformly bounded in C 2 , α ⁢ ( S δ ) , and therefore converges to u in C 2 ⁢ ( S δ ) . Since u ε is uniformly bounded from below by a positive constant and both φ and φ ~ are C 2 ⁢ ( ℝ + ) , we infer that v ε → v in C 2 ⁢ ( S δ ) . Therefore (3.3) continues to hold for any sufficiently small ε, so that v ε is locally strictly concave on S δ .

We claim that (3.4) holds as well for any (possibly smaller) ε. Suppose not, so that there are

ε n → 0 , x ¯ n ∈ ∂ ⁡ Ω δ , x n ∈ Ω δ ∖ { x ¯ n }

such that

(4.5) L v ε n , x ¯ n ⁢ ( x n ) ≥ v ε n ⁢ ( x n ) .

By compactness we can assume that x ¯ n → x ¯ , x n → x ∈ Ω ¯ δ . By eventually lowering δ 0 , we see by the smoothness and strong convexity of Ω that there is a constant c = c ⁢ ( δ , Ω ) > 0 such that

B c ⁢ δ ⁢ ( x ¯ ) ⊆ S δ for all ⁢ x ¯ ∈ ∂ ⁡ Ω δ .

Since v ε n is strictly concave on B c ⁢ δ ⁢ ( x ¯ n ) , the points x n fulfilling (4.5) must be at least c ⁢ δ away from x ¯ n . Therefore the limit point x must be at least c ⁢ δ away from x ¯ . By taking the limit in (4.5), we find that

L v , x ¯ ⁢ ( x ) ≥ v ⁢ ( x ) , x ≠ x ¯ ,

contradicting the established validity of (3.3) for v. Therefore both (3.3) and (3.4) hold for v ε , if ε is sufficiently small, and the last statement of Proposition 3.1 ensures that the convexity function of v ε cannot assume positive values on ∂ ⁡ ( Ω δ × Ω δ ) .

We next show that, for any δ found in the previous point and any correspondingly small ε, the convexity function of φ ⁢ ( u ε ) (or log ⁡ u ε , in the proof of Theorem 1.1) cannot have a positive interior maximum on Ω × Ω . In order to do so, we will choose accordingly the function G in the approximation procedure given in (4.2) (which, so far, played no rôle).

4.4 The transformed equation

We first consider the number

M = inf ⁡ { t > 0 : f ⁢ ( t ) = 0 } .

In both Theorems 1.1 and 1.2, the function t ↦ f ⁢ ( t ) t p - 1 is non-increasing (by Lemma 3.3 in the second case), hence f vanishes identically on [ M , + ∞ ) . Clearly we may assume that M > 0 , for otherwise problem (1.1) have no solutions at all. We thus assume that

(4.6) G ⁢ ( t ) = G ⁢ ( M ) for all ⁢ t ≥ M

and claim that, under this assumption,

(4.7) sup Ω ⁡ u ε ≤ M .

Indeed, if this is not the case, the function w ε = u ε - M is non-negative on the open set { u ε > M } , vanishes on its boundary, and attains a positive maximum inside. By (4.6) and f ⁢ ( t ) = 0 for all t ≥ M , we infer from (4.4) that w ε solves

- div ⁡ ( ( ε ⁢ G ⁢ ( M ) 2 p + | ∇ ⁡ w ε | 2 ) p - 2 2 ⁢ ∇ ⁡ w ε ) = 0 .

The operator on the left-hand side is monotone, therefore by comparison we obtain w ε ≡ 0 and thus a contradiction, proving (4.7). Notice that the truncation prescribed in (4.6), despite lowering the regularity of G, does not affect the validity of (3) of Lemma 4.2, due to (4.7).

For a, still to be determined, increasing diffeomorphism φ ∈ C 1 ⁢ ( ℝ + , ℝ ) with inverse ψ, set

(4.8) G ⁢ ( t ) = ( ψ ′ ⁢ ( φ ⁢ ( t ) ) ) p , t ∈ ( 0 , M ) ,

so that the oddness condition together with (4.6) defines G on the whole ℝ . For the corresponding solutions u ε of (4.4) we set

v ε := φ ⁢ ( u ε ) ,

and compute the equation solved by v ε . By construction, ψ satisfies

(4.9) G ⁢ ( ψ ⁢ ( s ) ) = ψ ′ ⁢ ( s ) p , g ⁢ ( ψ ⁢ ( s ) ) = p ⁢ ψ ′ ⁢ ( s ) p - 2 ⁢ ψ ′′ ⁢ ( s )

for s ∈ v ε ⁢ ( Ω ) and moreover ∇ ⁡ u ε = ψ ′ ⁢ ( v ε ) ⁢ ∇ ⁡ v ε . It follows that

ε ⁢ G ⁢ ( u ε ) 2 p + | ∇ ⁡ u ε | 2 = ψ ′ ⁢ ( v ε ) 2 ⁢ ( ε + | ∇ ⁡ v ε | 2 )

so that

(4.10) ε p ( ε G ( u ε ) 2 p + | ∇ u ε | 2 ) p - 2 2 G ( u ε ) 2 - p p g ( u ε ) = ε ( ε + | ∇ v ε | 2 ) p - 2 2 ψ ′ ( v ( ε ) p - 2 ψ ′′ ( v ε ) .

Regarding the left-hand side of (4.4), from

D 2 ⁢ u ε = ψ ′′ ⁢ ( v ε ) ⁢ ∇ ⁡ v ε ⊗ ∇ ⁡ v ε + ψ ′ ⁢ ( v ε ) ⁢ D 2 ⁢ v ε

we compute

(4.11) div ⁡ ( ( ε ⁢ G ⁢ ( u ε ) 2 p + | ∇ ⁡ u ε | 2 ) p - 2 2 ⁢ ∇ ⁡ u ε ) = ψ ′ ⁢ ( v ε ) p - 1 ⁢ div ⁡ ( ( ε + | ∇ ⁡ v ε | 2 ) p - 2 2 ⁢ ∇ ⁡ v ε ) + ( p - 1 ) ⁢ ψ ′ ⁢ ( v ε ) p - 2 ⁢ ψ ′′ ⁢ ( v ε ) ⁢ ( ε + | ∇ ⁡ v ε | 2 ) p - 2 2 ⁢ | ∇ ⁡ v ε | 2 .

Putting (4.10) and (4.11) into (4.4), we obtain

(4.12) - div ⁡ ( ( ε + | ∇ ⁡ v ε | 2 ) p - 2 2 ⁢ ∇ ⁡ v ε ) = f ⁢ ( ψ ⁢ ( v ε ) ) ( ψ ′ ⁢ ( v ε ) ) p - 1 + ψ ′′ ⁢ ( v ε ) ψ ′ ⁢ ( v ε ) ⁢ ( ε + | ∇ ⁡ v ε | 2 ) p - 2 2 ⁢ ( ( p - 1 ) ⁢ | ∇ ⁡ v ε | 2 - ε ) .

Now we split the proof into two cases, choosing ψ (or, equivalently, G) accordingly.

4.5 Non-positivity in the interior

We finally choose the transformation. For the first statement of Proposition 4.3, let φ defined by (1.2), which then fulfils (4.8) on [ 0 , M ] for G = F , i.e.,

(4.13) ψ ′ ⁢ ( s ) = F 1 p ⁢ ( ψ ⁢ ( s ) ) .

Differentiating this relation we obtain

f ⁢ ( ψ ⁢ ( s ) ) ( ψ ′ ⁢ ( s ) ) p - 1 = p ⁢ ψ ′′ ⁢ ( s ) ψ ′ ⁢ ( s ) ,

which, inserted into (4.12) gives

- div ⁡ ( ( ε + | ∇ ⁡ v ε | 2 ) p - 2 2 ⁢ ∇ ⁡ v ε ) = ψ ′′ ⁢ ( v ε ) ψ ′ ⁢ ( v ε ) ⁢ b ⁢ ( ∇ ⁡ v ε ) ,

where

(4.14) b ⁢ ( ∇ ⁡ v ε ) := p + ( ( p - 1 ) ⁢ | ∇ ⁡ v ε | 2 - ε ) ⁢ ( ε + | ∇ ⁡ v ε | 2 ) p - 2 2 .

Consider the function

h ⁢ ( t ) = p + ( ( p - 1 ) ⁢ t - ε ) ⁢ ( ε + t ) p - 2 2 , t ≥ 0

An elementary computation shows that h is increasing, so that its minimum is h ⁢ ( 0 ) = p - ε p 2 . It follows that the function b defined in (4.14) is positive whenever ε < p 2 p , which we will assume henceforth.

In order to apply Proposition 3.1, we have to check that

(4.15) ψ ′′ ψ ′ ⁢ is non-increasing and harmonic concave on v ε ⁢ ( Ω ) .

From (4.7) we infer sup Ω ⁡ v ε ≤ φ ⁢ ( M ) and clearly ψ ⁢ ( s ) > 0 for all s ∈ v ε ⁢ ( Ω ) , since 0 < u ε = ψ ⁢ ( v ε ) . Therefore

(4.16) ψ ⁢ ( v ε ⁢ ( Ω ) ) ⊆ [ 0 , M ] .

For the first assertion in (4.15), recall from (4.13) that

ψ ′′ ψ ′ = ( F 1 p ∘ ψ ) ′ ψ ′ = ( F 1 p ) ′ ∘ ψ .

Since F 1 p is assumed to be concave on ( 0 , M ) and (4.16) holds true, this expression is non-increasing being the composition of a non-increasing function with an increasing one.

Finally, we claim that ψ ′ ψ ′′ is convex. We first compute

ψ ′ ⁢ ( s ) ψ ′′ ⁢ ( s ) = p ⁢ F ⁢ ( ψ ⁢ ( s ) ) f ⁢ ( ψ ⁢ ( s ) ) ⁢ F ⁢ ( ψ ⁢ ( s ) ) - 1 p .

The function F f is assumed to be convex on ( 0 , M ) and then it is differentiable, except at most at countable set A ⊆ [ 0 , M ] . If B = φ ⁢ ( A ) and s ∈ v ε ⁢ ( Ω ) ∖ B , then ψ ⁢ ( s ) ∈ [ 0 , M ] ∖ A due to (4.16). Thus F f is differentiable at ψ ⁢ ( s ) and it holds

( ψ ′ ⁢ ( s ) ψ ′′ ⁢ ( s ) ) ′ = p ⁢ ( F ⁢ ( ψ ⁢ ( s ) ) f ⁢ ( ψ ⁢ ( s ) ) ) ′ ⁢ F - 1 p ⁢ ( ψ ⁢ ( s ) ) - F ⁢ ( ψ ⁢ ( s ) ) f ⁢ ( ψ ⁢ ( s ) ) ⁢ F - 1 - 1 p ⁢ ( ψ ⁢ ( s ) ) ⁢ f ⁢ ( ψ ⁢ ( s ) ) ⁢ ψ ′ ⁢ ( s ) = p ⁢ ( F ⁢ ( ψ ⁢ ( s ) ) f ⁢ ( ψ ⁢ ( s ) ) ) ′ ⁢ 1 ψ ′ ⁢ ( s ) - 1 ,

where we used (4.13) in the last step. Therefore

( ψ ′ ψ ′′ ) ′ = p ⁢ ( F f ) ′ ∘ ψ - 1 on v ε ⁢ ( Ω ) ∖ B ,

which is non-decreasing as composition of non-decreasing functions. Since B is at most countable, the convexity of ψ ′ ψ ′′ follows by well-known characterizations of convex function of the real line.

Hence Proposition 3.1 applies, and the convexity function of φ ⁢ ( u ε ) cannot attain a positive maximum on Ω × Ω and a fortiori on Ω δ × Ω δ .

Regarding the second statement of Proposition 4.3, we choose in (4.12) the function G ⁢ ( t ) = t p , which corresponds to ψ ⁢ ( s ) = e s , φ ⁢ ( t ) = log ⁡ t . In this case (4.12) reads

div ⁡ ( ( ε + | ∇ ⁡ v ε | 2 ) p - 2 2 ⁢ ∇ ⁡ v ε ) = f ⁢ ( e v ε ) e ( p - 1 ) ⁢ v ε + ( ε + | ∇ ⁡ v ε | 2 ) p - 2 2 ⁢ ( ( p - 1 ) ⁢ | ∇ ⁡ v ε | 2 - ε )

and the condition required in Theorem 1.2 allow to apply Proposition 3.1 to exclude the positivity of the convexity function of v ε in Ω × Ω .

Communicated by Frank Duzaar

Funding statement: The authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Sunra Mosconi is partially supported by project PIACERI – Linea 2 and 3 of the University of Catania, and grant PRIN n. 2017AYM8XW of the Ministero dell’Istruzione, Università e Ricerca. Part of this work has been developed during a visit of the second author at the Catholic University of Brescia, funded by project “Manipolazione di immagini”. The hosting institution is gratefully acknowledged.

Acknowledgements

Part of the paper was developed during a visit of the second author at the Department of Mathematics and Physics of the Catholic University of Sacred Hearth, Brescia, Italy. The hosting institution is gratefully acknowledged.

References

[1] O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl. (9) 76 (1997), no. 3, 265–288. 10.1016/S0021-7824(97)89952-7Search in Google Scholar

[2] R. Benguria, H. Brézis and E. H. Lieb, The Thomas–Fermi–von Weizsäcker theory of atoms and molecules, Comm. Math. Phys. 79 (1981), no. 2, 167–180. 10.1007/BF01942059Search in Google Scholar

[3] B. Bian and P. Guan, A microscopic convexity principle for nonlinear partial differential equations, Invent. Math. 177 (2009), no. 2, 307–335. 10.1007/s00222-009-0179-5Search in Google Scholar

[4] M. Bianchini and P. Salani, Power concavity for solutions of nonlinear elliptic problems in convex domains, Geometric Properties for Parabolic and Elliptic PDE’s, Springer INdAM Ser. 2, Springer, Milan (2013), 35–48. 10.1007/978-88-470-2841-8_3Search in Google Scholar

[5] H. J. Brascamp and E. H. Lieb, On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal. 22 (1976), no. 4, 366–389. 10.1016/0022-1236(76)90004-5Search in Google Scholar

[6] L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J. 37 (2014), no. 3, 769–799. 10.2996/kmj/1414674621Search in Google Scholar

[7] H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986), no. 1, 55–64. 10.1016/0362-546X(86)90011-8Search in Google Scholar

[8] L. A. Caffarelli and A. Friedman, Convexity of solutions of semilinear elliptic equations, Duke Math. J. 52 (1985), no. 2, 431–456. 10.1215/S0012-7094-85-05221-4Search in Google Scholar

[9] L. A. Caffarelli and J. Spruck, Convexity properties of solutions to some classical variational problems, Comm. Partial Differential Equations 7 (1982), no. 11, 1337–1379. 10.1080/03605308208820254Search in Google Scholar

[10] A. Colesanti and P. Salani, Quasi-concave envelope of a function and convexity of level sets of solutions to elliptic equations, Math. Nachr. 258 (2003), 3–15. 10.1002/mana.200310083Search in Google Scholar

[11] G. Crasta and I. Fragalà, A C 1 regularity result for the inhomogeneous normalized infinity Laplacian, Proc. Amer. Math. Soc. 144 (2016), no. 6, 2547–2558. 10.1090/proc/12916Search in Google Scholar

[12] G. Crasta and I. Fragalà, The Brunn–Minkowski inequality for the principal eigenvalue of fully nonlinear homogeneous elliptic operators, Adv. Math. 359 (2020), Article ID 106855. 10.1016/j.aim.2019.106855Search in Google Scholar

[13] P. Cuoghi and P. Salani, Convexity of level sets for solutions to nonlinear elliptic problems in convex rings, Electron. J. Differential Equations 2006 (2006), Paper No. 124. Search in Google Scholar

[14] J. I. Díaz and J. E. Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Math. Acad. Sci. Paris 305 (1987), no. 12, 521–524. Search in Google Scholar

[15] A. Greco and G. Porru, Convexity of solutions to some elliptic partial differential equations, SIAM J. Math. Anal. 24 (1993), no. 4, 833–839. 10.1137/0524052Search in Google Scholar

[16] F. Hamel, N. Nadirashvili and Y. Sire, Convexity of level sets for elliptic problems in convex domains or convex rings: Two counterexamples, Amer. J. Math. 138 (2016), no. 2, 499–527. 10.1353/ajm.2016.0012Search in Google Scholar

[17] A. Henrot, C. Nitsch, P. Salani and C. Trombetti, Optimal concavity of the torsion function, J. Optim. Theory Appl. 178 (2018), no. 1, 26–35. 10.1007/s10957-018-1302-9Search in Google Scholar

[18] F. A. Høeg, Concave power solutions of the dominative p-Laplace equation, NoDEA Nonlinear Differential Equations Appl. 27 (2020), no. 2, Paper No. 19. 10.1007/s00030-020-0622-2Search in Google Scholar

[19] H. Hu and S. Zhou, Brunn–Minkowski inequality for variational functional involving the p-Laplacian operator, Acta Math. Sci. Ser. B (Engl. Ed.) 29 (2009), no. 5, 1143–1154. 10.1016/S0252-9602(09)60092-8Search in Google Scholar

[20] P. Juutinen, Concavity maximum principle for viscosity solutions of singular equations, NoDEA Nonlinear Differential Equations Appl. 17 (2010), no. 5, 601–618. 10.1007/s00030-010-0071-4Search in Google Scholar

[21] P. Juutinen, Convexity of solutions to boundary blow-up problems, Commun. Pure Appl. Anal. 12 (2013), no. 5, 2267–2275. 10.3934/cpaa.2013.12.2267Search in Google Scholar

[22] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math. 1150, Springer, Berlin, 1985. 10.1007/BFb0075060Search in Google Scholar

[23] A. U. Kennington, Power concavity and boundary value problems, Indiana Univ. Math. J. 34 (1985), no. 3, 687–704. 10.1512/iumj.1985.34.34036Search in Google Scholar

[24] N. J. Korevaar, Capillary surface convexity above convex domains, Indiana Univ. Math. J. 32 (1983), no. 1, 73–81. 10.1512/iumj.1983.32.32007Search in Google Scholar

[25] N. J. Korevaar, Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 32 (1983), no. 4, 603–614. 10.1512/iumj.1983.32.32042Search in Google Scholar

[26] N. J. Korevaar and J. L. Lewis, Convex solutions of certain elliptic equations have constant rank Hessians, Arch. Ration. Mech. Anal. 97 (1987), no. 1, 19–32. 10.1007/BF00279844Search in Google Scholar

[27] M. Kühn, Power- and log-concavity of viscosity solutions to some elliptic Dirichlet problems, Commun. Pure Appl. Anal. 17 (2018), no. 6, 2773–2788. 10.3934/cpaa.2018131Search in Google Scholar

[28] C. S. Lin, Uniqueness of least energy solutions to a semilinear elliptic equation in 𝐑 2 , Manuscripta Math. 84 (1994), no. 1, 13–19. 10.1007/BF02567439Search in Google Scholar

[29] P.-L. Lions, Two geometrical properties of solutions of semilinear problems, Appl. Anal. 12 (1981), no. 4, 267–272. 10.1080/00036818108839367Search in Google Scholar

[30] L. G. Makar-Limanov, The solution of the Dirichlet problem for the equation Δ ⁢ u = - 1 in a convex region, Mat. Zametki 9 (1971), 89–92. 10.1007/BF01405053Search in Google Scholar

[31] S. Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 14 (1987), no. 3, 403–421. Search in Google Scholar

Received: 2021-11-29

Accepted: 2022-06-01

Published Online: 2022-07-22

Published in Print: 2024-01-01

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https://doi.org/10.1515/acv-2021-0100

Keywords for this article

Quasilinear problems; convexity of solutions; maximum principles

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