On a nonlinear Robin problem with an absorption term on the boundary and L1 data

Francesco Della Pietra; Francescantonio Oliva; Sergio Segura de León

doi:10.1515/anona-2023-0118

Article Open Access

On a nonlinear Robin problem with an absorption term on the boundary and L¹ data

Francesco Della Pietra , Francescantonio Oliva and Sergio Segura de León

Published/Copyright: February 21, 2024

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From the journal Advances in Nonlinear Analysis Volume 13 Issue 1

Abstract

We deal with existence and uniqueness of nonnegative solutions to:

− Δ u = f ( x ) , in Ω , ∂ u ∂ ν + λ ( x ) u = g ( x ) u η , on ∂ Ω ,

where η ≥ 0 and f , λ , and g are the nonnegative integrable functions. The set Ω ⊂ R N ( N > 2 ) is open and bounded with smooth boundary, and ν denotes its unit outward normal vector. More generally, we handle equations driven by monotone operators of p -Laplacian type jointly with nonlinear boundary conditions. We prove the existence of an entropy solution and check that, under natural assumptions, this solution is unique. Among other features, we study the regularizing effect given to the solution by both the absorption and the nonlinear boundary term.

Keywords: nonlinear elliptic equations; Robin boundary conditions; singular lower-order term; entropy solutions; uniqueness

MSC 2010: 35J25; 35J60; 35J70; 35J75; 35A01; 35A02

1 Introduction

In this article, we analyze existence and uniqueness of nonnegative solutions to the following model problem:

(1.1) − Δ u = f ( x ) , in Ω , ∂ u ∂ ν + λ ( x ) u = g ( x ) u η , on ∂ Ω ,

where η ≥ 0 and f , λ and g are the nonnegative functions that can be even merely integrable. Here, Ω denotes an open bounded subset of R N ( N > 2 ) with smooth boundary, while ν denotes its unit outward normal vector.

The main interest in this problem is the nonlinear Robin boundary condition, which contains a blowing-up term.

To the best of our knowledge, issues such as (1.1) have not yet been discussed in the literature. When the singularity is, for example, volumetric, singular problems are extensively studied.

Let us just cite some historical articles [7,16] and also more recent articles [6,8,12,21] that investigate different aspects of this problem using various techniques.

The literature concerning nonlinear boundary conditions is more limited and focuses primarily on u ν + F ( u ) = 0 with F nondecreasing and finite at the origin [2,3,10,14,22]. Aside from this, in [19], the authors deal with a problem involving a function F blowing up at the origin in a concrete model case. It is proven here existence and nonexistence results in the presence of subcritical powers in both the interior and the boundary equation. It is also worth mentioning articles [11,13], in which the authors consider homogeneous Robin boundary conditions in conjunction with singular terms even dependent on the gradient of the solution itself when λ is a positive constant. In particular, they show the existence of solutions using variational methods and a sub- and super-solution technique. Let also mention that recent results can be found in [1,18] if g ≡ 0 and λ is a positive constant. Furthermore, if η = 0 and the principal operator is the p -Laplacian, one can also refer, for instance, to [9] which studies, among other things, the asymptotic behaviour of the solutions as p → 1 + .

In this article, we first show the existence of a weak solution to equation (1.1) when the data are regular enough (Theorem 2.2).

We use a comparison argument as well as the classical Hopf lemma to derive that the approximation sequence is bounded from below on ∂ Ω through an appropriate regularization technique on the data involved. Roughly speaking, this gives the problem a non-singular feature that allows us to easily pass to the limit in the approximation sequence. Let us explicitly stress that in the previous argument, we strongly need λ to be bounded and not null; in particular, this permits us to deduce that the sub-solution to the approximation sequence given by equation (2.11) is actually positive on the boundary of the domain.

Let also mention that the regularizing effect given by the singular boundary term is expressed by condition (2.2). If η > 0 , we obtain finite energy solutions for a broader class of data; for instance, if η ≥ 1 , we need g to be just an integrable function to obtain the solution u ∈ H 1 ( Ω ) . This phenomenon is caused by the degeneration at infinity of the nonlinear boundary term.

In the second part of this article, we deal with a generalization of problem (1.1) given by:

(1.2) − div ( a ( x , ∇ u ) ) = f , in Ω , u ≥ 0 , in Ω , a ( x , ∇ u ) ⋅ ν + λ σ ( u ) = h ( u ) g , on ∂ Ω ,

where a , σ , and h are the suitable generalizations of the functions involved in equation (1.1) satisfying Assumptions (3.2), (3.3), (3.4), (3.5), (3.6), and (3.7). Finally, λ and g are merely nonnegative integrable functions on ∂ Ω as well as 0 ≤ f ∈ L 1 ( Ω ) .

Problem (1.2) is clearly non-variational, and here, the approximation process is strongly needed to show the existence of entropy solutions (Definition 3.8).

In this situation, a completely different strategy is required than in Theorem 2.2: we cannot establish that the approximation sequence is bounded from below on ∂ Ω since, among other things, λ is unbounded. In this case, we take advantage of suitable test functions to control the nonlinear (and possibly singular) boundary term.

Finally, Theorem 3.6 shows that the entropy solution to equation (1.2) is unique given natural monotonicity assumptions on the involved functions.

The goal of this study is as follows: in Section 2, we deal with the existence of a weak solution to equation (1.1). In Section 3, we prove the existence and uniqueness of entropy solutions to equation (1.2).

1.1 Notation and preliminaries

For the entire article, Ω is an open bounded set of R N ( N ≥ 2 ) with regular boundary.

For a given function v , we denote by v + = max ( v , 0 ) and by v − = − min ( v , 0 ) . Moreover, χ E denotes the characteristic function of a set E . For a fixed k > 0 , we define the truncation function T k : R → R as:

T k ( s ) ≔ max ( − k , min ( s , k ) ) .

We will also use the functions:

(1.3) V δ ( s ) ≔ 1 , s ≤ δ , 2 δ − s δ , δ < s < 2 δ , 0 , s ≥ 2 δ ,

and

(1.4) ϕ t , ε ( s ) ≔ 0 , s ≤ t , s − t ε , t < s < t + ε , 1 , s ≥ t + ε .

Fixed a nonnegative λ ∈ L N − 1 p − 1 ( Ω ) (not identically null), with 1 < p < N , we consider in W 1 , p ( Ω ) the norm defined by:

‖ v ‖ λ , p p = ∫ Ω ∣ ∇ v ∣ p + ∫ ∂ Ω λ ∣ v ∣ p d ℋ N − 1 , v ∈ W 1 , p ( Ω ) .

This norm turns out to be equivalent in W 1 , p ( Ω ) to the usual norm (see [20, Section 2.7]). As a consequence, classical embeddings that hold for W 1 , p ( Ω ) can be translated to this norm.

Let also recall the following well-known trace inequality (see [20, Theorem 4.2]). There exists C > 0 such that:

(1.5) ‖ v ‖ L ( N − 1 ) p N − p ( ∂ Ω ) ≤ C ‖ v ‖ W 1 , p ( Ω ) , ∀ v ∈ W 1 , p ( Ω ) .

It is worth mentioning that the previous immersion is also compact in L q ( ∂ Ω ) if q < ( N − 1 ) p N − p (see [20, Theorem 6.1]).

For any 0 < r < ∞ , by M r ( Ω ) , we denote the usual Marcinkiewicz (or weak Lebesgue) space of index r , which is the space of functions f such that there exists a positive constant C for which ∣ { ∣ f ∣ > t } ∣ ≤ C t − r , for any t > 0 . Let only recall that if ∣ Ω ∣ < ∞ , L r ( Ω ) ⊂ M r ( Ω ) ⊂ L r − ε ( Ω ) , for any ε > 0 . For an overview to these spaces, we refer to [15].

If no otherwise specified, we will denote by C several positive constants whose value may change from line to line and, sometimes, on the same line. These values will only depend on the data but they will never depend on the indexes of the sequences we will introduce.

Finally, we underline that if no ambiguity occurs, we will often use the following notation for the Lebesgue integral of a function f :

∫ Ω f ≔ ∫ Ω f ( x ) d x .

2 Case with regular data

In this section, under the assumption of Ω bounded open set with C 1 boundary, we prove the existence of solution to the following model problem:

(2.1) − Δ u = f , in Ω , u ≥ 0 , in Ω , ∂ u ∂ ν + λ u = g u η , on ∂ Ω ,

where η ≥ 0 and f ∈ L 2 N N + 2 ( Ω ) , λ ∈ L ∞ ( ∂ Ω ) (not identically null) and g ∈ L r ( ∂ Ω ) are nonnegative and

(2.2) r = max 2 ( N − 1 ) N + η ( N − 2 ) , 1 .

The main interesting fact in this section is that, under the aforementioned assumptions and through classical tools, the solution is far away from zero on ∂ Ω . Roughly speaking, this means that Problem (2.1) is non-singular.

Let us firstly precise what we mean by a weak solution.

Definition 2.1

A function u ∈ H 1 ( Ω ) is a weak solution to equation (2.1) if g u − η ∈ L 1 ( ∂ Ω ) and if it satisfies

(2.3) ∫ Ω ∇ u ⋅ ∇ φ + ∫ ∂ Ω λ u φ d ℋ N − 1 = ∫ Ω f φ + ∫ ∂ Ω g φ u η d ℋ N − 1 ,

for all φ ∈ H 1 ( Ω ) ∩ L ∞ ( ∂ Ω ) .

Let us state the existence result for this section.

Theorem 2.2

Let 0 ≤ f ∈ L 2 N N + 2 ( Ω ) , let 0 ≤ λ ∈ L ∞ ( ∂ Ω ) be not identically null, and let 0 ≤ g ∈ L r ( ∂ Ω ) with r satisfying (2.2). Then, there exists a weak solution to equation (2.1).

Remark 2.3

Let us stress that the previous existence result concerns nonnegative solutions. Anyway, simple basic examples show that, in general, changing sign solutions exist as shown in Example 1. Roughly speaking, here we are formally dealing with existence of solutions to:

− Δ u = f , in Ω , ∂ u ∂ ν + λ u = g ∣ u ∣ η , on ∂ Ω ,

which are nonnegative. Let us also underline that the study of problems as in equation (2.1), where f and g are not necessarily positive, is the object of a forthcoming article. Obviously, in this case, nonnegative solutions are not always expected to exist.

Example 1

Let B 1 ( 0 ) be the unit ball in R 2 , and let us consider the following problem:

(2.4) − Δ u = 0 , in B 1 ( 0 ) , ∂ u ∂ ν + λ u = g u , on ∂ B 1 ( 0 ) ,

In what follows, we use polar coordinates 0 ≤ r ≤ 1 and − π < θ ≤ π if we fix the nonnegative functions

λ ( θ ) = 1 ∣ θ ∣ α ,

with 0 ≤ α < 1 , and

g ( θ ) = sin 2 θ 1 + 1 ∣ θ ∣ α ,

which give λ ∈ L 1 ( ∂ B 1 ( 0 ) ) and g ∈ L ∞ ( ∂ B 1 ( 0 ) ) Then, it is simple to convince that u ( r , θ ) = r sin θ is a solution to equation (2.4).

Let us observe that u vanishes on the boundary at θ = 0 and θ = π . At θ = 0 , the function λ exhibits a singularity. However, at θ = π , the weight λ is bounded. Moreover, when α = 0 , λ is bounded but both zeros remain.

2.1 Approximation scheme and proof of the existence result

In order to prove the aforementioned theorem, we work by approximation through the following problems:

(2.5) − Δ u n = f n , in Ω , u n ≥ 0 , in Ω , ∂ u n ∂ ν + λ u n = g n ∣ u n ∣ + 1 n η , on ∂ Ω ,

where f n ≔ T n ( f ) and g n ≔ T n ( g ) . We first show the existence of a weak solution to equation (2.5), namely a function u n ∈ H 1 ( Ω ) satisfying

(2.6) ∫ Ω ∇ u n ⋅ ∇ φ + ∫ ∂ Ω λ u n φ d ℋ N − 1 = ∫ Ω f n φ + ∫ ∂ Ω g n φ ∣ u n ∣ + 1 n η d ℋ N − 1 ,

for all φ ∈ H 1 ( Ω ) .

Lemma 2.4

Let 0 ≤ f ∈ L 1 ( Ω ) , let 0 ≤ λ ∈ L ∞ ( ∂ Ω ) not identically null, and let 0 ≤ g ∈ L 1 ( ∂ Ω ) . Then, there exists a nonnegative weak solution u n to equation (2.5).

Proof

In order to show the existence of a solution to equation (2.5), let us consider

(2.7) − Δ w = f n , in Ω , ∂ w ∂ ν + λ w = g n ∣ v ∣ + 1 n η , on ∂ Ω ,

where v ∈ L 2 ( ∂ Ω ) . The existence of a solution w ∈ H 1 ( Ω ) to equation (2.7) follows, for example, from the classical results contained in [17]; moreover, it is easy to deduce that w is actually nonnegative and, from a classical argument by Stampacchia, it is also bounded. In order to deduce the existence of a solution u n to equation (2.5), we aim to show that the application T : L 2 ( ∂ Ω ) ↦ L 2 ( ∂ Ω ) such that T ( v ) = w ∣ ∂ Ω admits a fixed point. Hence, it will be sufficient to show that T is invariant, compact, and continuous to apply the Schauder fixed point theorem in order to conclude the proof.

We start by proving that T is invariant; to this end, let us take w as a test function in the weak formulation in equation (2.7) deducing that (recall f n , g n ≤ n )

∫ Ω ∣ ∇ w ∣ 2 + ∫ ∂ Ω λ w 2 d ℋ N − 1 ≤ n ∫ Ω w + n η + 1 ∫ ∂ Ω w d ℋ N − 1 .

Now observe that on the left-hand side, our norm appears, while on the right-hand side, we may apply Young’s inequality with weights ( ε 1 , C ε 1 ) and ( ε 2 , C ε 2 ) (where ε 1 , ε 2 > 0 to be chosen), which leads to:

‖ w ‖ λ , 2 2 ≤ n ε 1 ∫ Ω w 2 + n η + 1 ε 2 ∫ ∂ Ω w 2 d ℋ N − 1 + C ε 1 n ∣ Ω ∣ + C ε 2 n η + 1 ℋ N − 1 ( ∂ Ω ) .

Then, applying (1.5), one simply obtains

‖ w ‖ λ , 2 2 ≤ n ε 1 C 1 ‖ w ‖ λ , 2 2 + n η + 1 ε 2 C 2 ‖ w ‖ λ , 2 2 + C ε 1 n ∣ Ω ∣ + C ε 2 n η + 1 ℋ N − 1 ( ∂ Ω ) ,

where we also used that ‖ ⋅ ‖ λ , 2 and ‖ ⋅ ‖ H 1 ( Ω ) are equivalent norms. Then, fixing ε 2 and satisfying n η + 1 ε 2 C 2 < 1 2 and ε 1 such that n ε 1 C 1 < 1 4 , one deduces that

‖ w ‖ λ , 2 2 ≤ C n ,

where C n is a positive constant that depends on n but it is independent on w . Applying again equation (1.5), we obtain that a ball in L 2 ( ∂ Ω ) (let us say of radius R n ) is invariant for T .

Moreover, since C n does not depend on w , the compactness of the trace embedding and the aforementioned argument show that T ( A ) ¯ is compact for any A subset of the ball of radius R n contained in L 2 ( ∂ Ω ) .

For the continuity, we let v k ∈ L 2 ( ∂ Ω ) , which converges to v in L 2 ( ∂ Ω ) as k → ∞ and we consider T ( v k ) = w k ∣ ∂ Ω , i.e. w k satisfies

(2.8) − Δ w k = f n , in Ω , ∂ w k ∂ ν + λ w k = g n ∣ v k ∣ + 1 n η , on ∂ Ω .

Reasoning as for the proof of the invariance, one deduces that w k is bounded in H 1 ( Ω ) with respect to k . This is sufficient to pass to the limit any term in the weak formulation of equation (2.8) using weak convergence in H 1 ( Ω ) and strong convergence in L 2 ( ∂ Ω ) of w k to a function w as k → ∞ . Since the equation has a unique solution, this proves that T ( v ) = w ∣ ∂ Ω , i.e., the continuity of T .

As already mentioned, we are now able to deduce from the Schauder fixed point theorem that there exists a solution u n to equation (2.5). It follows by taking u n − that u n ≥ 0 almost everywhere in Ω . This concludes the proof.□

Remark 2.5

Let us just underline that, instead of finding the fixed point, one could have proven Lemma 2.4 by minimizing a suitable functional, which in the case, η = 1 is given by:

I ( u ) = 1 2 ∫ Ω ∣ ∇ u ∣ 2 + 1 2 ∫ ∂ Ω λ u 2 d ℋ N − 1 − ∫ Ω f n u − ∫ ∂ Ω g n log 1 n + u + d ℋ N − 1 , u ∈ H 1 ( Ω ) .

Let us now show that the sequence u n is nondecreasing in n . The proof is similar to the one contained in [6].

Lemma 2.6

Under the assumptions of Lemma 2.4, let u n be a solution to equation (2.5). Then, the sequence u n is nondecreasing with respect to n. Moreover, there exists c ¯ > 0 such that

(2.9) u n ( x ) ≥ c ¯ > 0 , for ℋ N − 1 a l m o s t e v e r y x ∈ ∂ Ω and f o r a n y n ∈ N .

Proof

Let us take ( u n − u n + 1 ) + as a test function in the difference of the weak formulations solved, respectively, by u n and by u n + 1 . Then, one yields

∫ Ω ∣ ∇ ( u n − u n + 1 ) + ∣ 2 + ∫ ∂ Ω λ ( ( u n − u n + 1 ) + ) 2 d ℋ N − 1 ≤ ∫ Ω ( f n − f n + 1 ) ( u n − u n + 1 ) + + ∫ ∂ Ω g n u n + 1 n η − g n + 1 ( u n + 1 + 1 n + 1 ) η ( u n − u n + 1 ) + d ℋ N − 1 ,

which implies that

(2.10) ∫ Ω ∣ ∇ ( u n − u n + 1 ) + ∣ 2 + ∫ ∂ Ω λ ( ( u n − u n + 1 ) + ) 2 d ℋ N − 1 ≤ ∫ ∂ Ω g n + 1 1 ( u n + 1 n + 1 ) η − 1 ( u n + 1 + 1 n + 1 ) η ( u n − u n + 1 ) + d ℋ N − 1 ≤ 0 .

Equation (2.10) gives that ‖ ( u n − u n + 1 ) + ‖ λ , 2 = 0 , which means that u n + 1 ≥ u n ℋ N − 1 almost everywhere on ∂ Ω and almost everywhere in Ω .

To prove equation (2.9), let us observe that it follows from classical results that there exists v ∈ C 1 ( Ω ¯ ) nonnegative solution to:

(2.11) − Δ v = f 1 , in Ω , ∂ v ∂ ν + ‖ λ ‖ L ∞ ( ∂ Ω ) v = 0 , on ∂ Ω .

A contradiction argument, using the Hopf lemma [23, Theorem 2], shows that v > 0 in Ω ¯ . Moreover, analogously to the monotonicity’s proof of u n in n , one can show that

u n ≥ v , for ℋ N − 1 almost every x ∈ ∂ Ω and for any n ∈ N .

Since v is continuous and strictly positive on ∂ Ω , this shows that

□ u n ≥ v > min ∂ Ω v = c ¯ for ℋ N − 1 almost every x ∈ ∂ Ω and for any n ∈ N .

Let us explicitly underline that in the previous proof, the fact that λ is bounded and not identically null plays an essential role.

Let us show some a priori estimates on u n with respect to n .

Lemma 2.7

Let 0 ≤ f ∈ L 2 N N + 2 ( Ω ) , 0 ≤ λ ∈ L ∞ ( ∂ Ω ) not identically null and 0 ≤ g ∈ L r ( ∂ Ω ) with r satisfying (2.2). Let u n be a solution to equation (2.5), then u n is bounded in H 1 ( Ω ) with respect to n .

Proof

Let us take u n as a test function in equation (2.6), obtaining

(2.12) ∫ Ω ∣ ∇ u n ∣ 2 + ∫ ∂ Ω λ u n 2 d ℋ N − 1 = ∫ Ω f n u n + ∫ ∂ Ω g n u n u n + 1 n η d ℋ N − 1 .

For the first term on the right-hand of equation (2.12), it follows from the Hölder and Sobolev inequalities that

(2.13) ∫ Ω f n u n ≤ ‖ f ‖ L 2 N N + 2 ( Ω ) ‖ u n ‖ L 2 N N − 2 ( Ω ) ≤ S 2 ‖ f ‖ L 2 N N + 2 ( Ω ) ‖ u n ‖ H 1 ( Ω ) ,

where S 2 is the best constant in the Sobolev inequality for functions in H 1 ( Ω ) . For the second term in the right-hand of equation (2.12), we observe that, if η ≥ 1 , one can simply estimate as:

(2.14) ∫ ∂ Ω g n u n u n + 1 n η d ℋ N − 1 ≤ ‖ g ‖ L 1 ( ∂ Ω ) c ¯ η − 1 .

Otherwise, if η < 1 it follows from the Hölder inequality and from the choice of r that

∫ ∂ Ω g n u n u n + 1 n η d ℋ N − 1 ≤ ‖ g ‖ L r ( ∂ Ω ) ‖ u n ‖ L ( 1 − η ) r r − 1 ( ∂ Ω ) 1 − η = ‖ g ‖ L r ( ∂ Ω ) ‖ u n ‖ L 2 ( N − 1 ) N − 2 ( ∂ Ω ) 1 − η ,

which, applying (1.5), gives

(2.15) ∫ ∂ Ω g n u n u n + 1 n η d ℋ N − 1 ≤ c ‖ g ‖ L r ( ∂ Ω ) ‖ u n ‖ H 1 ( Ω ) 1 − η ,

where c does not depend on n . Therefore, gathering equations (2.13) and (2.14) in equation (2.12), one obtains that for η ≥ 1 , it holds

‖ u n ‖ λ , 2 2 ≤ c S 2 ‖ f ‖ L 2 N N + 2 ( Ω ) ‖ u n ‖ H 1 ( Ω ) + ‖ g ‖ L 1 ( ∂ Ω ) c ¯ η − 1 .

Otherwise, if η < 1 , one uses (2.15) in place of equation (2.14) in order to deduce that

‖ u n ‖ λ , 2 2 ≤ c S 2 ‖ f ‖ L 2 N N + 2 ( Ω ) ‖ u n ‖ H 1 ( Ω ) + c ‖ g ‖ L r ( ∂ Ω ) ‖ u n ‖ H 1 ( Ω ) 1 − η .

Recalling that ‖ ⋅ ‖ λ , 2 and ‖ ⋅ ‖ H 1 ( Ω ) are equivalent norms and applying the Young inequality, one simply deduces that u n is bounded in H 1 ( Ω ) with respect to n .□

We are ready to prove Theorem 2.2.

Proof of Theorem 2.2

Let u n be a solution to equation (2.5) whose existence is guaranteed from Lemma 2.4. Then, it follows from Lemma 2.7 that u n is bounded in H 1 ( Ω ) with respect to n . Moreover, classical embedding results give that u n (up to not relabelled subsequences) converges to a function u in L q ( Ω ) for any q < 2 N N − 2 and in L t ( ∂ Ω ) for any t < 2 ( N − 1 ) N − 2 as n → ∞ . This is sufficient to pass to the limit the first and the second term of equation (2.6). The third term simply passes to the limit in n . For the fourth term, we apply the Lebesgue theorem since

g n u n + 1 n η ≤ g c η ¯ ,

ℋ N − 1 almost everywhere on ∂ Ω . This concludes the proof.□

Remark 2.8

Let us stress once again that in the current section, we heavily used that λ ∈ L ∞ ( ∂ Ω ) and that is not identically null. Indeed, these facts allowed us to exploit the maximum principle deducing that the approximating solutions are bounded away from zero on the boundary of Ω . In some sense, if λ is bounded, Problem (2.1) seems to be non-singular. On the other hand, in the next section, under more general assumptions, this procedure cannot be carried over and we need to control the singularity through the use of suitable test functions.

3 L 1 -data and entropy solutions

In this section, let Ω be an open bounded set of R N ( N ≥ 2 ) with Lipschitz boundary. Here, we generalize the results obtained for (2.1) in the previous section to the following more general problem:

(3.1) − div ( a ( x , ∇ u ) ) = f , in Ω , u ≥ 0 , in Ω , a ( x , ∇ u ) ⋅ ν + λ σ ( u ) = h ( u ) g , on ∂ Ω ,

where a ( x , ξ ) : Ω × R N → R N is a Carathéodory function such that

(3.2) a ( x , ξ ) ⋅ ξ ≥ α ∣ ξ ∣ p , for some α > 0 ,

(3.3) ∣ a ( x , ξ ) ∣ ≤ β ( z ( x ) + ∣ ξ ∣ p − 1 ) , for some β > 0 and 0 ≤ z ∈ L p p − 1 ( Ω ) ,

(3.4) ( a ( x , ξ ) − a ( x , ξ ′ ) ) ⋅ ( ξ − ξ ′ ) > 0 ,

for 1 < p < N , for almost every x in Ω and for every ξ ≠ ξ ′ in R N . Regarding the data, 0 ≤ f ∈ L 1 ( Ω ) and 0 ≤ λ ∈ L 1 ( ∂ Ω ) is not identically null in Ω . Finally, 0 ≤ g ∈ L 1 ( ∂ Ω ) . Here, the function h is continuous on ( 0 , ∞ ) , it is finite outside the origin, and it can blow up at zero satisfying the following growth condition:

(3.5) ∃ η ≥ 0 , c 1 , s 1 > 0 : h ( s ) ≤ c 1 s η , if s ≤ s 1 .

In what follows, we denote as h ( 0 ) ≔ lim s → 0 h ( s ) , which exists. Moreover, we require that

(3.6) limsup s → ∞ h ( s ) < ∞ .

Finally, the function σ is continuous and such that:

(3.7) σ ( s ) ≥ s p − 1 , if s ≥ 0 and σ ( 0 ) = 0 .

Remark 3.1

Under the aforementioned assumptions, we are not in position to reason as in Section 2; in particular, we can not deduce the existence of subsolution as (2.11) for the approximating sequence that is bounded from below by a positive constant at the boundary of Ω .

Moreover, besides the unboundedness of λ , we also require σ and h to be functions where no monotonicity is assumed. Finally, f and g are merely integrable functions. All the aforementioned arguments force us to employ a different technique to pass to the limit in the approximation sequence.

As we will see, since we also deal with uniqueness of solutions under some restrictive hypotheses, the entropy setting better adapts with L 1 -data. Firstly, we precisely set what we mean by entropy solution to Problem (3.1) and, after that, we make some comments on the notion of solution.

Definition 3.2

A measurable function u that is almost everywhere finite in Ω and such that T k ( u ) ∈ W 1 , p ( Ω ) for all k > 0 is an entropy solution to equation (3.1) if λ σ ( u ) , h ( u ) g ∈ L 1 ( ∂ Ω ) and it holds

(3.8) ∫ Ω a ( x , ∇ u ) ⋅ ∇ T k ( u − v ) + ∫ ∂ Ω λ σ ( u ) T k ( u − v ) d ℋ N − 1 = ∫ Ω f T k ( u − v ) + ∫ ∂ Ω h ( u ) g T k ( u − v ) d ℋ N − 1 ,

for all v ∈ W 1 , p ( Ω ) ∩ L ∞ ( Ω ) and all k > 0 .

Remark 3.3

Let us clarify the meaning of ∇ u since we do not necessarily deal with functions in W 1 , 1 ( Ω ) .

It is classical nowadays that from Lemma 2.1 of [4], there exists a unique measurable function v such that

∇ T k ( u ) = v χ { ∣ u n ∣ ≤ k } ,

for almost every x ∈ Ω and for every k > 0 . Moreover, it is shown that u ∈ W 1 , 1 ( Ω ) if and only if v ∈ L 1 ( Ω ) and v = ∇ u in the usual distributional sense.

This motivates the choice of referring to the above-cited function v when dealing with the gradient of a function u having only its truncations in a Sobolev space.

Remark 3.4

Let us stress that the first term on the left-hand of equation (3.8) is finite. Indeed, ∇ T k ( u − v ) is different from zero only on { ∣ u − v ∣ < k } , where ∣ u ∣ < ‖ v ‖ L ∞ ( Ω ) + k ≕ M . Hence, since T M ( u ) ∈ W 1 , p ( Ω ) , we deduce a ( x , ∇ T M ( u ) ) ∈ L p p − 1 ( Ω ) N and ∇ T k ( T M ( u ) − v ) ∈ L p ( Ω ) N . Clearly, it is simple to convince that also, all the other terms are well defined. Moreover, let us explicitly underline that it is easy to see that a solution u ∈ W 1 , p ( Ω ) satisfying a formulation analogous to equation (2.3) is also an entropy solution. Conversely, any entropy solution u belonging to W 1 , p ( Ω ) is also a solution in the sense of equation (2.3) if f ∈ L N p N p − N + p ( Ω ) .

Hence, we state the existence result to equation (3.1).

Theorem 3.5

Let a satisfy equations (3.2), (3.3), and (3.4). Let 0 ≤ f ∈ L 1 ( Ω ) , let 0 ≤ λ ∈ L 1 ( ∂ Ω ) not identically null, and let 0 ≤ g ∈ L 1 ( ∂ Ω ) . Finally, let h satisfy (3.5), (3.6) and let σ satisfy (3.7). Then, there exists a nonnegative entropy solution u to equation (3.1) such that u ∈ M N ( p − 1 ) N − p ( Ω ) , u ∈ M ( N − 1 ) ( p − 1 ) N − p ( ∂ Ω ) , and ∣ ∇ u ∣ ∈ M N ( p − 1 ) N − 1 ( Ω ) .

Under some restrictive assumptions, we show that there is at most one entropy solution to equation (3.1).

Theorem 3.6

Let a satisfy equations (3.2), (3.3), and (3.4), and let λ , g ≥ 0 ℋ N − 1 almost everywhere on ∂ Ω . Finally, assume that σ ( s ) is increasing and h ( s ) is nonincreasing with respect to s. Then, there is at most one entropy solution to equation (3.1).

3.1 Approximation scheme and a priori estimates

Once again, we work by approximation through the following scheme:

(3.9) − div ( a ( x , ∇ u n ) ) = f n , in Ω , a ( x , ∇ u n ) ⋅ ν + λ n σ n ( u n ) = h n ( u n ) g n , on ∂ Ω ,

where f n ≔ T n ( f ) , λ n ≔ T n ( λ ) , σ n ( s ) ≔ T n ( σ ( s ) ) , h n ( s ) ≔ T n ( h ( s ) ) , and g n ≔ T n ( g ) . We start proving the existence of a solution u n to equation (3.9).

Lemma 3.7

Let a satisfy (3.2), (3.3), and (3.4). Let 0 ≤ f ∈ L 1 ( Ω ) , 0 ≤ λ ∈ L 1 ( ∂ Ω ) not identically null, and let 0 ≤ g ∈ L 1 ( ∂ Ω ) . Let h satisfy (3.5) and (3.6), and finally, let σ satisfy (3.7). Then, there exists a nonnegative weak solution u n ∈ H 1 ( Ω ) ∩ L ∞ ( Ω ) to equation (3.9).

Proof

Let us provide a very brief idea of the proof.

Let us consider the function w ∈ W 1 , p ( Ω ) nonnegative solution to:

− div ( a ( x , ∇ w ) ) = f n , in Ω , a ( x , ∇ w ) ⋅ ν + λ n σ n ( ∣ w ∣ ) = h n ( ∣ v ∣ ) g n , on ∂ Ω ,

where v ∈ L p ( ∂ Ω ) , constructed as in [17]. This allows us to define an application T : L p ( ∂ Ω ) ↦ L p ( ∂ Ω ) such that T ( v ) = w ∣ ∂ Ω . Then, a very similar reasoning to the one of Lemma 2.4 gives that T has a fixed point. The main difference, apart from the estimates in which one heavily uses (3.2), lies in the continuity request; if w k ∣ ∂ Ω = T ( v k ) and v k converges to v in L p ( ∂ Ω ) , in this case one has also to show that ∇ w k converges almost everywhere to some ∇ w in Ω to pass to the limit the principal operator in order to have that w ∣ ∂ Ω = T ( v ) . Since f n is independent of k and h n ( ∣ v ∣ ) g n ≤ n 2 , one can reason as in Lemma 3.9 in order to deduce the desired convergence. Then, the continuity part is analogous to the one proven in Lemma 2.4.□

Let us now show some a priori estimates for u n in n .

Lemma 3.8

Under the assumptions of Lemma 3.7, let u n be a solution to equation (3.9). Then, it holds:

(3.10) ∫ { u n > t } λ n σ n ( u n ) d ℋ N − 1 ≤ ∫ { u n > t } f n + ∫ { u n > t } h n ( u n ) g n d ℋ N − 1 , ∀ t > 0 .

It holds that λ n σ n ( u n ) is bounded in L 1 ( ∂ Ω ) , u n is bounded in M N ( p − 1 ) N − p ( Ω ) , and M ( N − 1 ) ( p − 1 ) N − p ( ∂ Ω ) , and ∣ ∇ u n ∣ is bounded in M N ( p − 1 ) N − 1 ( Ω ) with respect to n. Moreover, h n ( u n ) g n is bounded in L 1 ( ∂ Ω ) with respect to n. In particular, it holds

(3.11) ‖ T k ( u n ) ‖ W 1 , p ( Ω ) ≤ C k 1 p ,

for some positive constant C, which does not depend on n.

Proof

Let t , ε > 0 and let us take ϕ t , ε ( u n ) ( ϕ t , ε is defined in equation (1.4)) as a test function in the weak formulation of equation (3.9) yielding (recall (3.2))

(3.12) α ∫ Ω ∣ ∇ u n ∣ p ϕ t , ε ′ ( u n ) + ∫ ∂ Ω λ n σ n ( u n ) ϕ t , ε ( u n ) d ℋ N − 1 ≤ ∫ Ω f n ϕ t , ε ( u n ) + ∫ ∂ Ω h n ( u n ) g n ϕ t , ε ( u n ) d ℋ N − 1 .

Since ϕ t , ε is nondecreasing and it is different from zero only on { u n > t } , one obtains

∫ ∂ Ω λ n σ n ( u n ) ϕ t , ε ( u n ) d ℋ N − 1 ≤ ∫ { u n > t } f n + ∫ { u n > t } h n ( u n ) g n d ℋ N − 1 ,

and (3.10) is obtained by an application of the Fatou lemma as ε → 0 + . Let also highlight that the right-hand side of equation (3.12) is bounded by a constant, which is independent of n :

∫ { u n > t } f n + ∫ { u n > t } h n ( u n ) g n d ℋ N − 1 ≤ ∫ Ω f + sup s ∈ ( t , ∞ ) h ( s ) ∫ ∂ Ω g d ℋ N − 1 = c ( t ) .

This implies

∫ ∂ Ω λ n σ n ( u n ) d ℋ N − 1 = ∫ { u n ≤ 1 } λ n σ n ( u n ) d ℋ N − 1 + ∫ { u n > 1 } λ n σ n ( u n ) d ℋ N − 1 ≤ max s ∈ [ 0 , 1 ] σ ( s ) ∫ ∂ Ω λ d ℋ N − 1 + c ( 1 )

so that λ n σ ( u n ) is bounded in L 1 ( ∂ Ω ) with respect to n .

Now, we focus on the Sobolev estimate for u n . Let us take T k ( u n ) − k as a test function in the weak formulation of equation (3.9) yielding

α ∫ Ω ∣ ∇ T k ( u n ) ∣ p + ∫ ∂ Ω λ n σ n ( u n ) ( T k ( u n ) − k ) d ℋ N − 1 ≤ 0 ,

which means that

∫ Ω ∣ ∇ T k ( u n ) ∣ p ≤ k α ∫ ∂ Ω λ n σ n ( u n ) d ℋ N − 1 ≤ C k ,

where C does not depend on n since λ n σ n ( u n ) is bounded in L 1 ( ∂ Ω ) with respect to n . Then, we have shown that

∫ Ω ∣ ∇ T k ( u n ) ∣ p + ∫ ∂ Ω λ n σ n ( u n ) T k ( u n ) ≤ C k , ∀ k > 0 .

Thus, recalling (3.7) and that λ n ≥ λ 1 , the previous integral implies that for any k > 0 equation (3.11) holds since ‖ ⋅ ‖ λ 1 , p and ‖ ⋅ ‖ W 1 , p ( Ω ) are equivalent.

It follows from classical arguments that u n is bounded in M N ( p − 1 ) N − p ( Ω ) and ∣ ∇ u n ∣ is bounded in M N ( p − 1 ) N − 1 ( Ω ) with respect to n (see, for instance, [4]).

Here, we briefly sketch the boundedness in M ( N − 1 ) ( p − 1 ) N − p ( ∂ Ω ) . It follows from equations (3.11) and (1.5) that

k ℋ N − 1 ( { x ∈ ∂ Ω : u n ≥ k } ) N − p ( N − 1 ) p ≤ ‖ T k ( u n ) ‖ L ( N − 1 ) p N − p ( ∂ Ω ) ≤ C k 1 p ,

which implies that

ℋ N − 1 ( { x ∈ ∂ Ω : u n ≥ k } ) ≤ C k ( N − 1 ) ( p − 1 ) N − p ,

namely u n is bounded in M ( N − 1 ) ( p − 1 ) N − p ( ∂ Ω ) with respect to n .

Now, we show that h n ( u n ) g n is bounded in L 1 ( ∂ Ω ) with respect to n . Let us take V δ ( u n ) ( V δ is defined in equation (1.3)) as a test function in the weak formulation of equation (3.9). This takes to:

∫ Ω a ( x , ∇ u n ) ⋅ ∇ u n V δ ′ ( u n ) + ∫ ∂ Ω λ n σ n ( u n ) V δ ( u n ) d ℋ N − 1 = ∫ Ω f n V δ ( u n ) + ∫ ∂ Ω h n ( u n ) g n V δ ( u n ) d ℋ N − 1 .

Now, recalling that f n ≥ 0 , V δ ′ ( s ) ≤ 0 and that (3.2) holds, the previous integral implies

∫ { u n ≤ δ } h n ( u n ) g n d ℋ N − 1 ≤ ∫ ∂ Ω h n ( u n ) g n V δ ( u n ) d ℋ N − 1 ≤ ∫ ∂ Ω λ n σ n ( u n ) V δ ( u n ) d ℋ N − 1 ≤ C ,

since λ n σ n ( u n ) is bounded in L 1 ( ∂ Ω ) . This concludes the proof.□

3.2 Convergence results

This subsection is devoted to the proof of the convergence results concerning u n , which are needed to prove Theorem 3.5.

Lemma 3.9

Under the assumptions of Lemma 3.7, let u n be a solution to equation (3.9). Then, u n converges (up to a subsequence) almost everywhere in Ω and ℋ N − 1 almost everywhere in ∂ Ω as n → ∞ to a function u, which is almost everywhere finite in Ω and on ∂ Ω .

Moreover, λ n σ n ( u n ) and h n ( u n ) g n converge in L 1 ( ∂ Ω ) , respectively, to λ σ ( u ) and h ( u ) g as n → ∞ .

Finally, T k ( u n ) converges to T k ( u ) strongly in W 1 , p ( Ω ) as n → ∞ and for every k > 0 .

Proof

It follows from Lemma 3.8 that u n is bounded in M N ( p − 1 ) N − p ( Ω ) with respect to n and T k ( u n ) is bounded in W 1 , p ( Ω ) for any k > 0 , then u n converges (up to not relabelled subsequences) almost everywhere to a function u such that T k ( u ) ∈ W 1 , p ( Ω ) , and u is almost everywhere finite in Ω . Moreover, by a suitable compactness argument (in n ) for T k ( u n ) on ∂ Ω , u n converges (up to a subsequence) ℋ N − 1 almost everywhere to u in ∂ Ω . The function u is ℋ N − 1 -a.e. finite on ∂ Ω since u n is bounded in M ( N − 1 ) ( p − 1 ) N − p ( ∂ Ω ) as proven in Lemma 3.8.

Now, observe that (3.10) implies that λ n σ n ( u n ) is equiintegrable and it converges to λ σ ( u ) in L 1 ( ∂ Ω ) as n → ∞ .

Now, let us show that h n ( u n ) g n converges in L 1 ( ∂ Ω ) to h ( u ) g as n → ∞ . If h ( 0 ) < ∞ , then this is obvious; hence, without loss of generality, we assume that h ( 0 ) = ∞ . Firstly observe that h ( u ) g ∈ L 1 ( ∂ Ω ) ; indeed, it follows from the weak formulation of equation (3.9) that

∫ ∂ Ω h n ( u n ) g n d ℋ N − 1 ≤ ∫ ∂ Ω λ n σ n ( u n ) d ℋ N − 1 ≤ C ,

thanks to Lemma 3.8. Then, an application of the Fatou lemma gives h ( u ) g ∈ L 1 ( ∂ Ω ) , which also means

(3.13) { u = 0 } ⊂ { g = 0 } , if h ( 0 ) = ∞ ,

up to a set of zero ℋ N − 1 measure set.

Now, let us take V δ ( u n ) as a test function in the weak formulation of equation (3.9) yielding to

∫ { u n ≤ δ } h n ( u n ) g n d ℋ N − 1 ≤ ∫ ∂ Ω h n ( u n ) g n V δ ( u n ) d ℋ N − 1 ≤ ∫ ∂ Ω λ n σ n ( u n ) V δ ( u n ) d ℋ N − 1 ,

where we dropped a non-positive term. Now, one can simply take n → ∞ and δ → 0 + , obtaining that

(3.14) lim δ → 0 + limsup n → ∞ ∫ { u n ≤ δ } h n ( u n ) g n d ℋ N − 1 ≤ ∫ { u = 0 } λ σ ( u ) V δ ( u ) d ℋ N − 1 = 0 ,

since σ ( 0 ) = 0 .

Now, consider δ ∉ { t : ∣ { u = t } ∣ > 0 } , which is admissible since it is a countable set, and split the singular term as:

(3.15) ∫ ∂ Ω h n ( u n ) g n d ℋ N − 1 = ∫ { u n ≤ δ } h n ( u n ) g n d ℋ N − 1 + ∫ { u n > δ } h n ( u n ) g n d ℋ N − 1 .

For the first term of equation (3.15), it holds (3.14) as n → ∞ and δ → 0 + .

For the second term in the right-hand of the previous integral, one can apply the Lebesgue theorem since

h n ( u n ) g n χ { u n ≥ δ } ≤ sup s ∈ ( δ , ∞ ) h ( s ) g ∈ L 1 ( ∂ Ω ) ,

yielding

lim n → ∞ ∫ { u n > δ } h n ( u n ) g n d ℋ N − 1 = ∫ { u > δ } h ( u ) g d ℋ N − 1 .

Then, since h ( u ) g ∈ L 1 ( ∂ Ω ) , one can apply once again the Lebesgue theorem in order to obtain

lim δ → 0 + lim n → ∞ ∫ { u n > δ } h n ( u n ) g n d ℋ N − 1 = ∫ ∂ Ω h ( u ) g d ℋ N − 1 ,

thanks to equation (3.13). Since h n ( u n ) g n is nonnegative, this is sufficient to deduce that it converges to h ( u ) g in L 1 ( ∂ Ω ) as n → ∞ .

Now, we prove that T k ( u n ) converges to T k ( u ) strongly in W 1 , p ( Ω ) as n → ∞ and for every k > 0 . Let us take ( T k ( u n ) − T k ( u ) ) V l ( u n ) ( l > k ) as a test function in the weak formulation of equation (3.9) yielding

∫ Ω ( a ( x , ∇ T k ( u n ) ) − a ( x , ∇ T k ( u ) ) ) ⋅ ∇ ( T k ( u n ) − T k ( u ) ) = − ∫ { k < u n < 2 l } a ( x , ∇ u n ) ⋅ ∇ ( T k ( u n ) − T k ( u ) ) V l ( u n ) + 1 l ∫ { l < u n < 2 l } a ( x , ∇ u n ) ⋅ ∇ u n ( T k ( u n ) − T k ( u ) ) + ∫ Ω f n ( T k ( u n ) − T k ( u ) ) V l ( u n ) + ∫ ∂ Ω h n ( u n ) g n ( T k ( u n ) − T k ( u ) ) V l ( u n ) d ℋ N − 1 − ∫ ∂ Ω λ n σ n ( u n ) ( T k ( u n ) − T k ( u ) ) V l ( u n ) d ℋ N − 1 − ∫ Ω a ( x , ∇ T k ( u ) ) ⋅ ∇ ( T k ( u n ) − T k ( u ) ) ≕ ( A ) + ( B ) + ( C ) + ( D ) + ( E ) + ( F ) .

For ( A ) , one has

( A ) ≤ ∫ Ω ∣ a ( x , ∇ u n ) ∣ V l ( u n ) ∣ ∇ T k ( u ) ∣ χ { u n > k } .

Now, let us underline that ∣ a ( x , ∇ u n ) ∣ V l ( u n ) is bounded in L p p − 1 ( Ω ) with respect to n and that ∣ ∇ T k ( u ) ∣ χ { u n > k } converges to zero in L p ( Ω ) as n → ∞ , then one has

limsup n → ∞ ( A ) ≤ 0 .

In order to estimate ( B ) , we take ϕ l , l ( u n ) as a test function in the weak formulation of equation (3.9), yielding to

1 l ∫ Ω a ( x , ∇ u n ) ⋅ ∇ u n ≤ ∫ Ω f n ϕ l , l ( u n ) + ∫ ∂ Ω h n ( u n ) g n ϕ l , l ( u n ) d ℋ N − 1 ,

which simply goes to zero as n → ∞ and l → ∞ since both f n and h n ( u n ) g n converge in L 1 ( Ω ) and in L 1 ( ∂ Ω ) , respectively, as n → ∞ . Hence, one obtains that

lim l → ∞ limsup n → ∞ ( B ) = 0 .

Moreover, one simply has that

lim n → ∞ ( C ) = lim n → ∞ ( D ) = lim n → ∞ ( E ) = 0 ,

since f n converges in L 1 ( Ω ) , and both h n ( u n ) g n and λ n σ n ( u n ) converge in L 1 ( ∂ Ω ) with respect to n . Finally, it follows from the weak convergence of T k ( u n ) to T k ( u ) as n → ∞ in W 1 , p ( Ω ) that

lim n → ∞ ( F ) = 0 .

Therefore, we have proven that

limsup n → ∞ ∫ Ω ( a ( x , ∇ T k ( u n ) ) − a ( x , ∇ T k ( u ) ) ) ⋅ ∇ ( T k ( u n ) − T k ( u ) ) = 0 ,

which allows us to reason as in the proof of Lemma 5 of [5] in order to conclude the proof.□

3.3 Existence of an entropy solution

This section is devoted to the passage to the limit in weak formulation of the approximation scheme (3.9).

Proof of Theorem 3.5

Let u n be a solution to equation (3.9). Then, it follows from Lemma 3.9 that u n converges (up to a subsequence) almost everywhere in Ω and ℋ N − 1 almost everywhere on ∂ Ω to u as n → ∞ . Moreover, u is almost everywhere finite and T k ( u ) ∈ W 1 , p ( Ω ) .

Let us firstly observe that a ( x , ∇ T k ( u ) ) ∈ L p p − 1 ( Ω ) N since T k ( u ) ∈ W 1 , p ( Ω ) and thanks to equation (3.3). Let us also note that it follows from Lemma 3.9 that λ σ ( u ) , h ( u ) g ∈ L 1 ( ∂ Ω ) .

Let us prove (3.8). We take T k ( u n − v ) as a test function in the weak formulation of equation (3.9) where v ∈ W 1 , p ( Ω ) ∩ L ∞ ( Ω ) . Then, one obtains

(3.16) ∫ Ω a ( x , ∇ u n ) ⋅ ∇ T k ( u n − v ) + ∫ ∂ Ω λ n σ n ( u n ) T k ( u n − v ) d ℋ N − 1 = ∫ Ω f n T k ( u n − v ) + ∫ ∂ Ω h n ( u n ) g n T k ( u n − v ) d ℋ N − 1 ,

and we want to pass to the limit (3.16) as n → ∞ . For the first term on the left-hand side, one can write

∫ Ω a ( x , ∇ u n ) ⋅ ∇ T k ( u n − v ) = ∫ { ∣ u n − v ∣ ≤ k } a ( x , ∇ u n ) ⋅ ∇ u n − ∫ { ∣ u n − v ∣ ≤ k } a ( x , ∇ u n ) ⋅ ∇ v .

Let firstly observe that in the previous integrals, one has that u n ≤ ‖ v ‖ L ∞ ( Ω ) + k ≕ M . Then, since it follows from Lemma 3.9 that T k ( u n ) converges strongly to T k ( u ) in W 1 , p ( Ω ) as n → ∞ for any k > 0 , one has that a ( x , ∇ T M ( u n ) ) converges strongly to a ( x , ∇ T M ( u ) ) in L p p − 1 ( Ω ) N as n → ∞ . This is sufficient to deduce that

lim n → ∞ ∫ Ω a ( x , ∇ u n ) ⋅ ∇ T k ( u n − v ) = ∫ { ∣ u − v ∣ ≤ k } a ( x , ∇ u ) ⋅ ∇ u − ∫ { ∣ u − v ∣ ≤ k } a ( x , ∇ u ) ⋅ ∇ v = ∫ Ω a ( x , ∇ u ) ⋅ ∇ T k ( u − v ) .

Moreover, Lemma 3.9 also gives that λ n σ n ( u n ) and h n ( u n ) g n converge in L 1 ( ∂ Ω ) to λ σ ( u ) and h ( u ) g as n → ∞ . Hence, we may take to take n → ∞ in the second and in the fourth term of equation (3.16). The first term on the right-hand side simply passes to the limit as n → ∞ . This concludes the proof.□

3.4 Proof of the uniqueness result

In this section, we prove Theorem 3.6.

Proof of Theorem 3.6

Let u 1 and u 2 be entropy solutions to Problem (3.1) and let us take v = T m ( u 2 ) in the entropy formulation corresponding to u 1 and v = T m ( u 1 ) in that of u 2 . Adding up both identities, it leads to:

(3.17) ∫ { ∣ u 1 − T m ( u 2 ) ∣ < k } a ( x , ∇ u 1 ) ⋅ ∇ ( u 1 − T m ( u 2 ) ) + ∫ { ∣ u 2 − T m ( u 1 ) ∣ < k } a ( x , ∇ u 2 ) ⋅ ∇ ( u 2 − T m ( u 1 ) ) + ∫ ∂ Ω λ σ ( u 1 ) T k ( u 1 − T m ( u 2 ) ) d ℋ N − 1 + ∫ ∂ Ω λ σ ( u 2 ) T k ( u 2 − T m ( u 1 ) ) d ℋ N − 1 = ∫ Ω f ( T k ( u 1 − T m ( u 2 ) ) + T k ( u 2 − T m ( u 1 ) ) ) + ∫ ∂ Ω h ( u 1 ) g T k ( u 1 − T m ( u 2 ) ) d ℋ N − 1 + ∫ ∂ Ω h ( u 2 ) g T k ( u 2 − T m ( u 1 ) ) d ℋ N − 1 .

We let m → ∞ in equation (3.17).

For the first two terms of equation (3.17), one can reason as in Theorem 5.1 of [4], deducing that its liminf as m → ∞ is bigger than:

∫ { ∣ u 1 − u 2 ∣ < k } ( a ( x , ∇ u 1 ) − a ( x , ∇ u 2 ) ) ⋅ ∇ ( u 1 − u 2 ) .

It is easy to handle the other terms thanks to Lebesgue’s theorem. Indeed,

(3.18) lim m → ∞ ∫ ∂ Ω λ σ ( u 1 ) T k ( u 1 − T m ( u 2 ) ) d ℋ N − 1 + ∫ ∂ Ω λ σ ( u 2 ) T k ( u 2 − T m ( u 1 ) ) d ℋ N − 1 = ∫ ∂ Ω λ ( σ ( u 1 ) − σ ( u 2 ) ) T k ( u 1 − u 2 ) d ℋ N − 1 ≥ 0

since σ is an increasing function. Moreover,

lim m → ∞ ∫ Ω f ( T k ( u 1 − T m ( u 2 ) ) + T k ( u 2 − T m ( u 1 ) ) ) = 0

and

lim m → ∞ ∫ ∂ Ω h ( u 1 ) g T k ( u 1 − T m ( u 2 ) ) d ℋ N − 1 + ∫ ∂ Ω h ( u 2 ) g T k ( u 2 − T m ( u 1 ) ) d ℋ N − 1 = ∫ ∂ Ω g ( h ( u 1 ) − h ( u 2 ) ) T k ( u 1 − u 2 ) d ℋ N − 1 ≤ 0

since h is nonincreasing. Therefore, Identity (3.17) becomes

∫ { ∣ u 1 − u 2 ∣ < k } ( a ( x , ∇ u 1 ) − a ( x , ∇ u 2 ) ) ⋅ ∇ ( u 1 − u 2 ) + ∫ ∂ Ω λ ( σ ( u 1 ) − σ ( u 2 ) ) T k ( u 1 − u 2 ) d ℋ N − 1 ≤ 0 .

Now, the proof concludes by observing that it follows from (3.4) and (3.18) that both terms of the previous integrals are zero. This means that ∇ u 1 = ∇ u 2 almost everywhere in Ω . Moreover, the previous integral implies that λ u 1 = λ u 2 ℋ N − 1 almost everywhere on ∂ Ω since σ ( s ) is increasing in s .

Then, one has that, for all k > 0 , ‖ T k ( u 1 ) − T k ( u 2 ) ‖ λ , p = 0 , and we conclude that u 1 = u 2 almost everywhere in Ω and ℋ N − 1 almost everywhere on ∂ Ω .□

Funding information: The first author was partially supported by the MIUR-PRIN 2017 grant “Qualitative and quantitative aspects of nonlinear PDE’s,” by GNAMPA of INdAM, by the FRA Project (Compagnia di San Paolo and Università degli studi di Napoli Federico II) 000022–ALTRI_CDA_75_2021_FRA_PASSARELLI. The second author was partially supported by GNAMPA of INdAM. The third author was partially supported by CIUCSD (Generalitat Valenciana) under Project AICO/2021/223 and by Grant RED2022-134784-T funded by MCIN/AEI/10.13039/501100011033.
Conflict of interest: The authors state no conflict of interest.

References

[1] A. Alvino, C. Nitsch, and C. Trombetti, A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions. Comm. Pure Appl. Math. 76 (2023), no. 3, 586–603. 10.1002/cpa.22090Search in Google Scholar

[2] F. Andreu, N. Igbida, J. M. Mazón, and J. Toledo, L1 existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 24 (2007), no. 1, 61–89. 10.1016/j.anihpc.2005.09.009Search in Google Scholar

[3] F. Andreu, J. M. Mazón, S. Segura de León, and J. Toledo, Quasi-linear elliptic and parabolic equations in L1 with nonlinear boundary conditions, Adv. Math. Sci. Appl. 7 (1997), no. 1, 183–213. Search in Google Scholar

[4] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, and J. L. Vázquez, An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 2, 241–273. Search in Google Scholar

[5] L. Boccardo, F. Murat, and J. P. Puel, Existence of bounded solutions for nonlinear unilateral problems, Ann. Mat. Pura Appl. 152 (1988), 183–196. 10.1007/BF01766148Search in Google Scholar

[6] L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. and PDEs 37 (2010), no. 3–4, 363–380. 10.1007/s00526-009-0266-xSearch in Google Scholar

[7] M. G. Crandall, P. H. Rabinowitz, and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Part. Diff. Eq. 2 (1977), no. 2, 193–222. 10.1080/03605307708820029Search in Google Scholar

[8] L. M. De Cave, R. Durastanti, and F. Oliva, Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data, NoDEA Nonlinear Differential Equations Appl. 25 (2018), 18. 10.1007/s00030-018-0509-7Search in Google Scholar

[9] F. Della Pietra, F. Oliva, and S. Segura de León, Behaviour of solutions to p-Laplacian with Robin boundary conditions as p goes to 1, Proc. R. Soc. Edinburgh Sec. A Math. 154 (2024), 105–130, https://doi.org/10.1017/prm.2022.92. Search in Google Scholar

[10] S. El Manouni, G. Marino, and P. Winkert, Existence results for double phase problems depending on Robin and Steklov eigenvalues for the p-Laplacian, Adv. Nonlinear Anal. 11 (2022), no. 1, 304–320. 10.1515/anona-2020-0193Search in Google Scholar

[11] U. Guarnotta, S. Marano, and D. Motreanu, On a singular Robin problem with convection terms, Adv. Nonlinear Stud. 20 (2020), no. 4, 895–909. 10.1515/ans-2020-2093Search in Google Scholar

[12] U. Guarnotta, S. Marano, and A. Moussaoui, Singular quasilinear convective elliptic systems in RN, Adv. Nonlinear Anal. 11 (2022), no. 1, 741–756. 10.1515/anona-2021-0208Search in Google Scholar

[13] U. Guarnotta, S. Marano, and N. S. Papageorgiou, Multiple nodal solutions to a Robin problem with sign-changing potential and locally defined reaction, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30 (2019), no. 2, 269–294. 10.4171/rlm/847Search in Google Scholar

[14] O. Guibé and A. Oropeza, Renormalized solutions of elliptic equations with Robin boundary conditions, Acta Math. Sci. Ser. B (Engl. Ed.) 37 (2017), no. 4, 889–910. 10.1016/S0252-9602(17)30046-2Search in Google Scholar

[15] R. Hunt, On L(p,q) spaces, Enseign. Math. 12 (1966), no. 2, 249–276. Search in Google Scholar

[16] A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc. 111 (1991), no. 3, 721–730. 10.2307/2048410Search in Google Scholar

[17] J. Leray and J. L. Lions, Quelques résulatats de Vissssik sur les problémes elliptiques nonlinéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France 93 (1965), 97–107. 10.24033/bsmf.1617Search in Google Scholar

[18] A. Masiello and G. Paoli, A rigidity result for the Robin torsion problem, J. Geom. Anal. 33 (2023), no. 5, Paper No. 149, 14 pp. 10.1007/s12220-023-01202-3Search in Google Scholar

[19] M. Montenegro and J. A. L. Tordecilla, Existence of positive solution for elliptic equations with singular terms and combined nonlinearities, J. Math. Anal. Appl. 503 (2021), no. 2, Paper No. 125316, 21 pp. 10.1016/j.jmaa.2021.125316Search in Google Scholar

[20] J. Nečas, Direct Methods in the Theory of Elliptic Equations, Transl. from the French, Springer Monographs in Mathematics, Springer, Berlin, 2012. 10.1007/978-3-642-10455-8Search in Google Scholar

[21] F. Oliva and F. Petitta, Finite and Infinite energy solutions of singular elliptic problems: Existence and uniqueness, J. Differ. Equ. 264 (2018), 311–340. 10.1016/j.jde.2017.09.008Search in Google Scholar

[22] A. Prignet, Non-homogeneous boundary value conditions for elliptic problems with measure valued right-hand side, (Conditions aux limites non homogènes pour des problèmes elliptiques avec second membre mesure.) Ann. Fac. Sci. Toulouse, VI. Sér. Math. 6 (1997), no. 2, 297–318. 10.5802/afst.867Search in Google Scholar

[23] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 3, 191–202. 10.1007/BF01449041Search in Google Scholar

Received: 2023-03-31

Revised: 2023-10-20

Accepted: 2023-11-29

Published Online: 2024-02-21

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/anona-2023-0118

Keywords for this article

nonlinear elliptic equations; Robin boundary conditions; singular lower-order term; entropy solutions; uniqueness

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