Anisotropic 1-Laplacian problems with unbounded weights

In this work we prove the existence of nontrivial bounded variation solutions to quasilinear elliptic problems involving a weighted 1-Laplacian operator. A key feature of these problems is that weights are unbounded. One of our main tools is the well-known Caffarelli-Kohn-Nirenberg’s inequality, which is established in the framework of weighted spaces of functions of bounded variation (and that provides us the necessary embeddings between weighted spaces). Additional tools are suitable variants of the Mountain Pass Theorem as well as an extension of the pairing theory by Anzellotti to this new setting.


Introduction
In the celebrated paper [13], Caffarelli, Kohn and Nirenberg established an interpolation inequality involving weighted Lebesgue norms of functions and their first derivatives. This inequality, in turn, allows one to show continuous and compact embeddings theorems dealing with weighted Sobolev spaces. Furthermore, this inequality and the connected embeddings have been applied to analyze several elliptic and parabolic problems involving weighted Laplacian and p-Laplacian operators (for elliptic problems, see for instance [1,2,9,12,14,36] and the references therein).
Regarding anisotropic problems involving the 1-Laplacian operator, we refer to [32] as the first paper which studies the existence and uniqueness of the anisotropic total variation flow. On the other hand, in [29], the author We could also cite [37], where the author studies questions about the existence and the regularity of minimizers of (1.1), where φ(x, Du) = a(x)|Du| and the weight function a(·) is a smooth bounded function. As a common hypothesis in all of these articles, we have the fact that the weight w satisfies 0 < α ≤ w(x) ≤ β < ∞. This assumption implies that the natural space to analyze the corresponding problem is BV , the space of functions of bounded variation (as in the isotropic case).
The aim of this paper is to consider some anisotropic problems with unbounded weights related to the Caffarelli-Kohn-Nirenberg inequality. More precisely, we study existence of positive solutions to the following problem where Ω is a bounded open set in R N (with N ≥ 2) containing the origin and having Lipschitz boundary ∂Ω, and the two parameters satisfy 0 < a < N − 1 and a < b < a + 1. Hypotheses on function f : R → R will be listed further below.
To the best of our knowledge, this work is the first attempt to deal with anisotropic problems having unbounded weights. In this situation, BV (Ω) is unsuitable and it cannot be the natural space to analyze this problem. Now, the energy space turns out to be a weighted BV -space. In the first step this weighted space, denoted by BV a (Ω), is introduced. Since our weights are related to the Caffarelli-Kohn-Nirenberg inequality, one of our main endeavors is to adapt this inequality to our setting. More specifically, we prove the following result. Theorem 1.1. Let 0 < a < N − 1, 0 < θ ≤ 1 and a < b < a + 1. Then there exists a constant C CKN > 0 such that holds for all u ∈ BV a (Ω), where α = θb and r θ = The concept of solution to problems involving the 1-Laplacian operator lies on the theory of L ∞ -divergence-measure vector fields (see [7,18]). It provides tools to handle bounded vector fields and gradients of BV -functions, including a Green's formula. Since in our context this theory can no longer be used, it follows that we must extend it to establish the necessary tools to NoDEA Anisotropic 1-Laplacian problems with unbounded weights Page 3 of 40 57 deal with it. This extension is far from being trivial, since the weight which is included in the vector field is unbounded. Using this tool, we may introduce the concept of solution to problem (1.2) (see Definition 4.9 below) and broach its study.
Before stating our main result in this paper, we list the assumptions on function f in problem (1. Our main result is the following. Theorem 1. 4. Suppose that f satisfies conditions (f 1 ) − (f 4 ). Then there exists a nontrivial nonnegative solution to problem (1.2). This solution is actually a ground-state solution (i.e., that solution which has the lowest energy among all nontrivial ones) if we further require condition (f 5 ).
Two different approaches will be used to prove this result. In each case a suitable variant of Mountain Pass Theorem (see [3]) is applied. In the first of them, we consider approximate solutions to problems involving the p-Laplacian operator and next we let p go to 1. Then we find a hindrance due to the assumptions on the function f which are needed to find solutions to p-problems. Indeed, in the literature on the p-Laplacian setting, our assumption (f 2 ) is too general to get a solution and a hypothesis as lim s→0 f (s) |s| p−1 = 0 is required. The difficulty is overcome by modifying the reaction term in the p-problems and then control the convergence process. In the second, we work by using variational methods applied to the problem itself defined in BV a (Ω). We apply a version of Mountain Pass Theorem suitable for functionals defined

Preliminaries
We denote by H N −1 (E) the (N − 1)-dimensional Hausdorff measure of a set E while |E| stands for its N -dimensional Lebesgue measure. We will usually handle an auxiliary function: the truncation function at level ±k defined by is an open and bounded set such that 0 ∈ Ω. Moreover, its boundary ∂Ω is Lipschitz-continuous. Thus, an outward normal unit vector ν(x) is defined for H N −1 -almost every x ∈ ∂Ω.
From now on, we denote: , stands for the space of functions with compact support which are continuously differentiable on Ω • C ∞ c (Ω), denotes the space of all functions with compact support having derivatives of all orders We will make use of the usual Lebesgue and Sobolev spaces. Lebesgue spaces with respect to a measure μ will be written as L q (Ω, μ). The measure will be deleted when it is Lebesgue measure.
Sometimes we will need to use convolution with mollifiers. We will denote by ρ ∈ C ∞ c (R N ) a symmetric mollifier whose support is B(0, 1) and its associated approximation to the identity by ρ (x) := 1 N ρ x , for > 0. The main properties of approximation to identity can be found, for instance, in [4,11].
We explicitly remark that, if not otherwise specified, we will denote by C several positive constants whose value may change from line to line. These values will only depend on the data but they will never depend on p or other indexes we will introduce.

Weighted spaces
Our objective in this subsection is to study spaces having a weight of the form x → |x| −a , with a > 0. We refer to [25,26,28] as sources for a more extensive NoDEA Anisotropic 1-Laplacian problems with unbounded weights Page 5 of 40 57 study on weights and weighted spaces. We begin by introducing some features of these weights.
Recall that w, a nonnegative locally integrable function on R N , belongs to Muckenhoupt's class A 1 if there exists a constant C w > 0 such that It is well-known that the weight function w(x) = 1 |x| a belongs to Muckenhoupt's class A 1 if and only if 0 < a < N, so that in this case there exists a constant C a > 0 such that for all B(x, r) ⊂ R N . We point out that this fact implies an inequality connecting mollifiers and this weight. Indeed, 1 |y| a dy and, as a consequence of belonging to A 1 , holds for all x ∈ Ω. Given a > 0 and s ≥ 1, let us denote by L s a (Ω) the set of measurable functions u such that  [27] is proved that the space W 1,p (Ω, |x| −ap ) is equal to the closure of {ϕ ∈ C ∞ (Ω); u p,a < ∞}.
The Sobolev space D 1,p 0,a (Ω) is defined as the completion of C ∞ c (Ω) with respect to the norm · p,a . Notice that there is a continuous embedding D 1,p 0,a (Ω) → W 1,p 0 (Ω). A Poincaré type inequality implies that this norm is equivalent in D 1,p 0,a (Ω) to the norm given by (2.8) This will be the norm we will use in what follows. For more information on weighted Sobolev spaces, we refer to [27] (see also [2,36]).

The space BV (Ω)
In this subsection, we just introduce some properties of the space of functions of bounded variation. As mentioned in the introduction, it is the natural space to study problems involving the 1-Laplacian operator. This space is defined as where Du : Ω → R N denotes the distributional gradient of u. Henceforth, we denote the distributional gradient by ∇u when it belongs to L 1 (Ω; R N ).
We recall that the space BV (Ω) endowed with the norm is a Banach space which is non reflexive and non separable. On the other hand, the notion of a trace on the boundary can be extended to functions u ∈ BV (Ω), so that we may write u ∂Ω . Indeed, there exists a continuous linear operator BV (Ω) → L 1 (∂Ω) extending the boundaries values of functions in C(Ω). As a consequence, an equivalent norm on BV (Ω) can be defined: We will often use this norm in what follows. In addition, the following continuous embeddings hold which are compact for 1 ≤ m < N N −1 . For further properties of functions of bounded variations, we refer to [4] and [21].

The space BV a (Ω)
In this subsection, we study the definition and main properties of the space BV a (Ω), which is our energy space. We mainly follow [10] to where we refer for a wider analysis.
Let us define var a u(Ω) as We remark that the Riesz representation Theorem implies that var a u(Ω) defines a Radon measure (see, for instance, [21, Section 1.8]).
We point out that the function Definition 2.5. Let BV a (Ω) be the space of functions u ∈ L 1 (Ω) such that | · | −a |Du| is a finite Radon measure, i.e., BV a (Ω) = u ∈ L 1 (Ω) : The space BV a (Ω) is a Banach space when endowed with the norm Moreover, note that m a Ω |Du| ≤ Ω 1 |x| a |Du| (m a as in Remark 2.1), so that Then BV a (Ω) → L 1 (∂Ω) and so every u ∈ BV a (Ω) has a trace on ∂Ω.
We point out that the functional given by NoDEA is lower semicontinuous with respect to the L 1 -convergence since each u → Ω u div φdx is so. Furthermore, similar arguments lead to the lower semicontinuity of the functional We also need to use the lower semicontinuity of another functional. For a fixed nonnegative ϕ ∈ C ∞ c (Ω), consider As a consequence of [10, Theorem 3.3], we may write from where the desired lower semicontinuity follows. We end this subsection by showing that just like in the space BV (Ω), we can have an equivalent norm in BV a (Ω) which involves an integral over ∂Ω. Its proof is a consequence of being equivalent · BV (Ω) and · BV (Ω),1 , and using that the positive quantities Proposition 2.7. Let u, v ∈ BV a (Ω), then max{u, v}, min{u, v} ∈ BV a (Ω) and the following inequality is valid (2.13) In particular, choosing v = 0, we have that u + := max{u, 0}, u − = min{u, 0} ∈ BV a (Ω), with u = u + + u − , and it holds (2.14)

The Caffarelli-Kohn-Nirenberg inequality in BV a (Ω)
In this section we are going to present a version of the Caffarelli-Kohn-Nirenberg inequality [13] in the space BV a (Ω). We do not prove it in its full generality, but just introduce those cases to be applied. In particular, we employ them to prove embeddings involving BV a (Ω). First of all we state the particular cases of the Caffarelli-Kohn-Nirenberg inequality we are interested in.

NoDEA
Anisotropic 1-Laplacian problems with unbounded weights Page 9 of 40 57 Lemma 3.1. Let p ≥ 1 and consider parameters satisfying 0 < a < N −p p , 0 < θ ≤ 1 and a < b < a + 1. Then there exists a constant C CKN > 0 such that the following inequality holds for all u ∈ C ∞ c (R N ): Now we present the proof of Theorem 1.1.
Proof of Theorem 1.1. Let u ∈ BV a (Ω) and consider its extension to R N defined byũ We remark that Dũ = Du + u| ∂Ω · H N −1 ∂Ω (see [4,Theorem 3.87]). Note also thatũ * ρ ∈ C ∞ c (R N ) and so we may apply Lemma 3.1 for p = 1 (so that r θ = N N −θ(1+a−b) ). Thus, for every > 0, we get We will separately take the limit as → 0 in each integral. We begin by analyzing the gradient term. Thanks to [4, Proposition 3.2(c)], we write Moreover, by the continuity of our weight, 1 and this fact, jointly with (2.7), allows us to apply the Dominated Convergence Theorem and obtain Furthermore, we deduce from Therefore, using (3.17), (3.19) and (3.20), we may pass to the limit in (3.15) and obtain the desired result.
In the following results, we denote C Ω = sup{|x| : x ∈ Ω}, which is finite since Ω is bounded.
Proof. In this proof, we consider several cases. All of them are consequence of some manipulations involving Hölder's inequality and the version of Caffarelli-Kohn-Nirenberg's inequality given in Theorem 1.1.
First of all, let us consider the case q = 1. We apply the mentioned inequalities to get Now consider 1 < q < r. In this case, arguing as above, we obtain Finally, the case q = r follows from a similar argument. Therefore, in any case, there exists C > 0 such that holds for every u ∈ BV a (Ω) and we are done. Proof. Let (u n ) be a bounded sequence in BV a (Ω) and note that, since BV a (Ω) → BV (Ω), (u n ) is also bounded in BV (Ω). Then, by the compact embedding in BV (Ω), there exist a subsequence (not relabeled) and u ∈ BV (Ω) such that Then, using first Hölder's inequality and then (1.3) we get which tends to 0 as n → ∞. Here we have used that (u n ) n is bounded in BV a (Ω) and (3.21). It remains to consider q = 1. Note that there exists 0 <θ < 1 such that θ(1 + a) > b. Performing similar manipulations, we get Hence, applying (1.3), it yields which tends to 0 as above.

Extension of the Anzellotti theory
In this Section, we extend the Anzellotti theory to a setting which involves unbounded vector fields. To begin with, we recall this theory. Not only these results will be applied, but they will also serve us as a guide for its broadening.

Remainder of Anzellotti's theory
We recall the notion of weak trace on ∂Ω of the normal component defined in [7] for every z ∈ L ∞ (Ω; R N ) such that its distributional divergence div z is a Radon measure having finite total variation. This trace is a function [z, ν] : , being ν(·) the outer normal unitary vector on ∂Ω. In [7], it was also introduced a distribution (z, among other possible pairings. It is then proved for all open sets U ⊂ Ω such that supp ϕ ⊂ U . As a consequence, (z, Du) is a Radon measure whose total variation satisfies Finally, a Green formula involving the measure (z, Du) and the weak trace [z, ν] is established in [7], namely: being z and u as in (4.23).

Weighted theory
In this subsection, we consider weights w(x) = |x| −a , with 0 < a < N − 1.
Nevertheless, we point out that most of the results holds for more general weights. We define the space Thus, and this fact means that On the other hand, the following equalities are valid in the sense of distributions (4.29) in the sense of distributions. This last identity implies that Then Anzellotti's theory supplies us with the weak trace [z, ν] on ∂Ω and the Radon measure (z, Du) for every u ∈ BV (Ω) ∩ L ∞ (Ω) (and so for every It is easy to compare 1 |x| a z, ν and 1 |x| a [z, ν]. To see that they are equal, we just employ the inequality for certain finite constant M a .
Proof. For each k > 0, by the Proposition 2 of [15], we obtain Now it is enough to take k ≥ M a to get our result.

Measures 1 |·| a z, Du and 1 |·| a (z, Du)
In this subsection, we take z ∈ DM a (Ω) and u ∈ BV a (Ω) ∩ L ∞ (Ω), and introduce two distributions 1 |x| a z, Du and 1 |x| a (z, Du), which turn out to be equal. Finally, we will prove a Green's formula that connects them to traces Du) as measures for all k > 0. In order to do so, first notice that div It is not difficult to connect both distributions. To this end, denote w(x) = T k 1 |x| a and consider the mollification of ϕw. Then We stress that (4.27) implies that both distributions are equal. So, we have proved the following lemma.
We define the weighted pairings as the limit of the above functionals. Let z ∈ DM a (Ω) and u ∈ BV a (Ω) ∩ L ∞ (Ω). Then we define the functional Proof. We point out that 1 Thus, having in mind (4.28), both distributions are equal.
Proof. Note that, from (4.24) we have that what proves the result.
Proof. It follows from (4.26), jointly with Lemmas 4.1 and 4.2, that for all k > 0. Since x → 1 |x| a is a bounded function on ∂Ω, then for k large enough, T k On the left hand side of (4.34), we will apply the Dominated Convergence Theorem. In the first term, we may pass to the limit as in the proof of the Theorem 4.5, taking into account (4.27), (4.29) and ∇ 1 |x| a ∈ L 1 (Ω). On the other hand, we denote by θ(z, Du) the Radon-Nikodým derivative of (z, Du) with respect to |Du|, so that |θ(z, Du)| ≤ z ∞ . Then Owing to 1 |x| a ∈ L 1 (Ω, |Du|), we are allowed to use the Dominated Convergence Theorem. Therefore, when k → ∞, identity (4.34) becomes

Concept of solution to problem (1.2)
Once we have the weighted theory available, we may introduce the definition of solution to problem (1.2).
We will need a variational formulation of our concept of solution. We begin with the following equivalence, whose proof in the non weighted setting can be found in [6, Proposition 2]. (a) u is a solution to problem (1.2).
holds for every v ∈ BV a (Ω).

Proof.
When v ∈ BV a (Ω) ∩ L ∞ (Ω), it is an easy consequence of Proposition 4.10 and the condition z ∞ ≤ 1. For a general v ∈ BV a (Ω), apply this inequality to T k (v) to get (4.38) Now, on account of Theorem 3.2, v ∈ L 1 b (Ω) and so we may let k go to ∞ on the left hand side of (4.38). Proof. Let u be a solution to problem (1.2). By Proposition 2.7, we may take v = u + in Corollary 4.11 obtaining On the left hand side, the integrand vanishes (recall that f (s) = 0 for all s ≤ 0) and we get To characterize the sub-differential of the norm, we could try to adapt the proof of [6, Section 5] to our weighted framework. Nevertheless, for our purposes, the following result will be enough. Proof. Let w ∈ BV a (Ω) ∩ L ∞ (Ω) be a solution to problem (4.39). Then there exists a vector field z ∈ L ∞ (Ω, R N ) such that z ∞ ≤ 1 and jointly with conditions (2) and (3). Taken w − u as test function, it yields On the other hand, assumption h ∈ ∂ u BV (Ω),1 implies Hence, gathering (4.40) and (4.41), it follows that and the result is a consequence of being z ∞ ≤ 1.

Proof of Theorem 1.4 through p-Laplacian problems
This section is devoted to prove Theorem 1.4 assuming conditions (f 1 ) − (f 4 ) by an approximating approach. We first consider problems involving the p-Laplacian operator and, following the arguments of [31], we prove a priori estimates which allow us to find the solution w of Problem (1.2) as p → 1 + .
for every 1 < p ≤p. Now, for each 1 < p ≤p, we consider the problem where f p (s) = f (s)|s| p−1 . Observe that, as a consequence of (f 1 ) − (f 4 ), the function f p satisfies: for s large enough.
Problem (5.42) has been studied in [12] using the lower and uppersolutions method. Nevertheless, we need to obtain a solution applying the Mountain Pass Theorem to get estimates independent of p and thus be able to pass to the limit as p → 1.
In order to get a nontrivial solution to (5.42), we work in the space D 1,p 0,a (Ω) that is defined in Sect. 2.1. Moreover, the functions of this space satisfy the following Caffarelli-Kohn-Nirenberg inequality.
Thanks to this version of the Caffarelli-Kohn-Nirenberg inequality and using the arguments of the proofs of Theorems 3.2 and 3.3, we can show the following embedding result. Probably this result already has been proved in the literature (for a related result, see [36, Theorem 2.1]). However, we state it here for the sake of completeness. The functional associated to problem (5.42) is given by By the conditions (f 2p ), (f 3p ), (f 4p ) and the Theorem 5.3, the functional J p satisfies the geometric conditions of the Mountain Pass Theorem (see [35]), which imply that there exists a (P S) c sequence (w n ) n∈N in D 1,p 0,a (Ω), i.e., J p (w n ) → c p and J p (w n ) → 0, as n → ∞, NoDEA Anisotropic 1-Laplacian problems with unbounded weights J p (γ(t)) and Γ = {γ ∈ C([0, 1], D 1,p 0,a (Ω)); γ(0) = 0, J p (γ(1)) < 0}. Well-known arguments can be used to show that (w n ) n∈N is a bounded sequence in D 1,p 0,a (Ω) and consequently, that there exists w p ∈ D 1,p 0,a (Ω) in such a way that w n → w p in D 1,p 0,a (Ω), as n → ∞. Since J p ∈ C 1 (D 1,p 0,a (Ω)) the previous convergence implies that J p (w p ) = c p and J p (w p ) = 0 and consequently w p is a nontrivial solution in D 1,p 0,a (Ω) to problem (5.42). Once we have got the family of approximate solutions (w p ) 1<p≤p , our main concern is to get bounds of this family which do not depend on p. To this end, let us consider the functional I p : D 1,p 0,a (Ω) → R defined by It is straightforward to see that p → I p (u) is a nondecreasing function, for every u ∈ W 1,p 0 (Ω, |x| −a ). Indeed, let 1 < p 1 < p 2 < p and note that, by Young's inequality, Moreover, the critical points of J p are the same of those of u → I p (u) − Next, we show that there exists e ∈ C ∞ c (Ω) such that J p (e) < 0, for all 1 < p ≤p.
Fix a nontrivial φ ∈ C ∞ c (Ω) such that φ ≥ 0 and φ ∞ ≤ 1. This fact leads to where have also used (5.47). Therefore, from (5.46) Since e does not depend on p, thanks to the Mountain Pass Theorem, we know that w p satisfies

Estimate of the family {w p }
We claim that the sequence is bounded by a constant which does not depend on p. Indeed, let 1 < p 1 < p 2 <p and let us apply the monotonicity of I p and the fact that Γ p2 ⊂ Γ p1 (because D 1,p2 0,a (Ω) ⊂ D 1,p1 0,a (Ω)). Then

It yields
where γ 0 (t) = te. Now, for 1 < p <p, it is straightforward to see that and the claim is proved. Thus, there exists C > 0 such that Let Ω p = {x ∈ Ω : |w p (x)| ≤ s 0 }, for any p ∈ (1,p). Then, by (f 3p ), we have By the condition (f 4p ) and since w p is a solution of (5.42), it holds On the other hand, note that condition (f 3p ) also implies Thus, by (5.52) and (5.53), we get (5.54) Gathering together (5.50), (5.51) and (5.52), we have Moreover, since 1 < p ≤p < μ by the last inequality we have that there exists C > 0 independent of p such that Now, using the previous estimate, Young and Hölder's inequalities we have whereĈ is a constant independent of p.

Convergence of (w p ) p
Recalling that w p ∂Ω = 0, it follows from (5.56) that the sequence {w p } 1<p<p is bounded in BV a (Ω). Then, up to a subsequence, there exists w such that, by Theorem 3.3, as well as, by (2.9), for all s ∈ 1, N N −1 . Up to a further subsequence, by [11,Theorem 4.9], we may also assume w p (x) → w(x) a. e. x ∈ Ω . (5.59) and that there exists holds for all p ∈ (1,p]. Finally, the lower semicontinuity of the functional u → Ω 1 |x| a |Du| guarantees that w ∈ BV a (Ω).

Boundedness of the limit
Let k ≥ 0 and let w p ∈ D 1,p 0,a (Ω) be a solution of problem (5.42). Define A k,p = {x ∈ Ω; |w p (x)| ≥ k a. e. in Ω}.

Lemma 5.4.
Let p > 1 be small enough. For each > 0 there exists k 0 > 0 (which does not depend on p) such that whereq is as in (f 3p ). and l = N 1+a−b , which satisfy 0 < α < 1 and l > 1. Using (5.61) and Hölder's inequality, we obtain

Hence, we have got
where ω(k) stands for a quantity independent on p that tends to 0 as k → +∞.
On the other hand, by the Caffarelli-Kohn-Nirenberg inequality, the Hölder inequality and the estimate (5.55) we obtain due to (5.56).
Therefore using (5.63) in (5.62) we get which tends to 0 as k → ∞. Now, let us deduce from Lemma 5.4 that w ∈ L ∞ (Ω). To this end, given k > 0, we define the auxiliary function G k : R → R as Choosing G k (w p ) as a test function in problem (5.42), we get . Then the previous identity, Caffarelli-Kohn-Nirenberg's, Young's and Hölder's inequalities and the condition (f 3p ) lead to On the other hand, by Lemma 5.4 there exists k 0 ∈ N such that (5.66) Using (5.66) in (5.65) we get Since w p (x) → w(x) a. e. in Ω when p → 1 + , Fatou's Lemma implies Therefore w ∞ ≤ k 0 .

Existence of the vector field
We begin by using the notation of Remark 2.1 and observing that (5.55) yields So, we may apply the same argument than that in [30,Theorem 3.5.] and obtain a subsequence (not relabeled) and z ∈ L ∞ (Ω; R N ) satisfying z ∞ ≤ 1 and In order to pass to the limit in the following stages, these weak convergences must slightly be improved. Fix 1 < s < ∞ such that 1 < s < N a , and takep small enough to have 1 < s < N ap , so that Ω 1 |x| aps dx < ∞. Since

w satisfies condition (1) of Definition 4.9
Let ϕ ∈ C ∞ c (Ω) and take it as test function in (5.42) to obtain Our aim is to let p → 1 + in (5.71). On the left hand side it is enough to apply (5.70), while in the right hand side, just observe that f p (w p (x)) → f (w(x)) a. e. x ∈ Ω due to (5.59). Moreover, by (f 3p ) and Young's inequality, we get and g ∈ Lq b (Ω). Hence, the Lebesgue Dominated Convergence Theorem implies (5.72) Therefore, letting p → 1 + in (5.71), we obtain Moreover, applying Young's inequality, one deduces Our next objective is to let p → 1 + . On the left hand side, since T k (w p ) → T k (w) in L 1 (Ω), the lower semicontinuity of (2.11) may be applied: We turn to analyze the right hand side of (5.76). The convergence of the first integral is a consequence of (5.59) and (5.70). Thus, We deal with the second integral applying the Lebesgue Dominated Convergence Theorem as in the previous subsection. So, we obtain The last term on the right hand side, obviously, tends to 0.
Our choice of k leads to Thus (5.74) holds.

w satisfies condition (3) of Definition 4.9
It only remains to check It is equivalent to show that (5.82) and so the integrand is nonnegative. Then (5.81) implies 1 |x| a |w|+w 1 |x| a [z, ν] = 0 and it follows from (5.82) that (5.80) holds. Actually, due to the nonnegativeness of the integrand, it is enough to check In order to do so, we take w p as a test function in (5.42) obtaining Using Young's inequality and the boundary condition w p ∂Ω = 0, we get Our aim is to let p → 1 + again. The lower semicontinuity of the functional in (2.10) gives On the other hand, we may apply the Lebesgue Dominated Convergence Theorem on the right hand side of (5.84), owing to x ∈ Ω and the following consequence of condition (f 3p ): Thus, (5.86) and the remainder term tends to 0. Consequently, using (5.85) and (5.86) in (5.84) we get (5.87) Applying (5.73) and Green's formula (Theorem 4.7), we arrive at (5.88) Gathering together (5.87) and (5.88), we obtain and we are done. Therefore, since w satisfies conditions (1), (2) and (3) of Definition 4.9, we conclude that w is a solution to problem (1.2).

w is a nontrivial solution of (1.2)
Now, what is left to do is to show that w = 0. In order to do so, we should introduce the energy functional Φ : BV a (Ω) → R given by Indeed, since w satisfies (1), (2) and (3) in Definition 4.9 and w p satisfies (5.42), it follows from Remark 4.8, (5.57), (f 3p ) and the Lebesgue Dominated Convergence Theorem that, as Moreover, again by (f 3p ), (5.57) and the Lebesgue Dominated Convergence Theorem, as p → 1 + , we have that Then, (5.91) and (5.92) imply in (5.90). We remark that, by (f 1 ) and (f 2 ), given > 0, we may find δ > 0 satisfying |f (s)| < ∀|s| < δ so that (f 3 ) implies that there exists a positive constantC > 0 such that Integrating this inequality, we deduce NoDEA Anisotropic 1-Laplacian problems with unbounded weights Let us consider > 0 small enough such that 1− On the other hand, for all 1 < p <p, Young's inequality implies that (1). Then, for all γ ∈ Γ p , from the continuity of t → I p (γ(t)) − Ω 1 |x| b F p (γ(t))dx and from the fact that I p (e) − Ω 1 |x| b F p (e)dx < 0, it follows that there exists t 0 ∈ [0, 1] such that γ(t 0 ) BVa(Ω),1 = ρ. Then, Hence, from the last inequality and (5.90), it follows that Φ(w) > 0 and then w is a nontrivial solution of (1.2). It remains to prove that w is a nonnegative solution of (1.2), but Corollary 4.12 does the job. This finishes the proof of Theorem 1.4. As a consequence of Proposition 4.13, we deduce the following result.

Existence by variational methods
First of all, let us consider the energy functional Φ : BV a (Ω) → R, given by It is straightforward to see that F b is a smooth functional. Moreover, by the same arguments of [8], it is possible to show that the functional J a admits some directional derivatives. More specifically, given u ∈ BV a (Ω), for all v ∈ BV a (Ω) such that (Dv) s is absolutely continuous with respect to (Du) s , (6.95) In particular, note that, for all u ∈ BV a (Ω), Then, the directional derivatives Φ (u)u exist and Note that Φ can we written as the difference between a Lipschitz and a smooth functional in BV a (Ω). Taking into account the theory of subdifferentials of Clarke (see [16,17]) , we say that w ∈ BV a (Ω) is a critical point of Φ if 0 ∈ ∂Φ(w), where ∂Φ(w) denotes the generalized gradient of Φ in w. It follows that this is equivalent to F (w) ∈ ∂J a (w) and, since J a is convex, this can be written as Henceforth, every w ∈ BV a (Ω) such that (6.98) holds is going to be called a critical point of Φ. Let us prove that Φ satisfies the first geometric condition of the Mountain Pass Theorem (see [22]). Note again (see inequality (5.93)) that, by (f 1 ), (f 2 ) and (f 3 ), it follows that for all > 0, there exists A > 0 such that |F (s)| ≤ |s| + A |s| q , ∀s ∈ R.
Then, the Mountain Pass Theorem (see [22,Theorem 4.1]) implies that there exist sequences τ n → 0 and (w n ) ⊂ BV a (Ω) satisfying the following conditions for all v ∈ BV a (Ω).
Let us prove that the sequence (w n ) is bounded in BV a (Ω). First of all, note that by taking v = w n + tw n in (6.102), dividing by t and letting t → 0 ± , we have that  By the boundedness of (w n ) ⊂ BV a (Ω) and Theorem 3.3, we find w ∈ BV a (Ω) such that w n → w in L r (Ω) for all r ∈ 1, N N − (1 + a − b) . (6.104) Then, by (6.104) and the lower semicontinuity of J a with respect to the L 1 (Ω) convergence, calculating the lim sup on both sides of (6.102), it yields that w satisfies (6.98). Moreover, by taking v = w+tw in (6.98) and considering the sign of t, we obtain Hence, from (6.104), (6.105), (6.106) and the Lebesgue Dominated Convergence Theorem, it follows that c = Φ(w) and then w is a nontrivial critical point of Φ. Our next concern is to check that w ∈ L ∞ (Ω). To this end, consider k > 0 and the function G k (s) defined in (5.64). Taking v = w ± G k (w) in (6.98), it yields ± Ω 1 |x| b f (w)G k (w) dx ≤ w ± G k (w) BV a (Ω),1 − w BV a (Ω),1 ≤ G k (w) BV a (Ω),1 and we infer that Since lim k→∞ {|w|≥k} 1 |x| b (1 + |w| q−1 ) dx = 0, we may find k 0 > 0 such that holds. Therefore, G k0 (w) = 0 and so |w| ≤ k 0 . As a consequence of Corollary 5.5, since w ∈ BV a (Ω) ∩ L ∞ (Ω) satisfies (6.98), it also satisfies all the conditions of Definition 4.9 and, moreover, it is nonnegative thanks to Corollary 4.12.
It just remains to justify that w is a ground-state solution, i.e., that w has the lowest energy level among all nontrivial bounded variation solutions. In order to prove it, we have to recall [23], where it is proved that we can define the Nehari set associated to Φ, given by It can be proven as in [23] that N is a set which contains all nontrivial bounded variation solutions of (1.2). Then, if we manage to prove that the solution w is such that Φ(w) = inf N Φ, then w would have the lowest energy level among the nontrivial solutions.
By using the same kind of arguments that Rabinowitz in [33], which consists in studying the map t → Φ(tv) and verifying that it has a unique maximum point t v > 0, which is such that t v v ∈ N ((f 5 ) is mandatory to prove the uniqueness); in the light of (f 1 ) − (f 5 ), one can see that N is radially homeomorphic to the unit sphere in BV a (Ω) and also that the minimax level c satisfies Since w is such that Φ(w) = c, it follows that w is a solution which has the lowest energy among all the nontrivial ones.