Abstract
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.
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1 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 finds the Euler-Lagrange equation for the anisotropic least gradient problem
We could also cite [37], where the author studies questions about the existence and the regularity of minimizers of (1.1), where \(\phi (x,Du) = a(x)|Du|\) and the weight function \(a(\cdot )\) is a smooth bounded function.
As a common hypothesis in all of these articles, we have the fact that the weight w satisfies \(0<\alpha \le w(x)\le \beta <\infty \). 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 \(\Omega \) is a bounded open set in \({\mathbb {R}}^N\) (with \(N\ge 2\)) containing the origin and having Lipschitz boundary \(\partial \Omega \), and the two parameters satisfy \(0<a<N-1\) and \(a<b<a+1\). Hypotheses on function \(f\>:\>{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}\) 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, \(BV(\Omega )\) 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 \(BV_a(\Omega )\), 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 \(0<a<{N-1}\), \(0<\theta \le 1\) and \(a<b<a+1\). Then there exists a constant \({\mathfrak {C}}_{CKN}>0\) such that
holds for all \(u\in BV_a(\Omega )\), where \( \alpha =\theta b\) and \(r_{\theta }=\frac{N}{N -\theta (1+a-b)}\). Here \(BV_a(\Omega )\) denotes the appropriate weighted BV–space, which was introduced in [10] (see Sect. 2.3 below).
The concept of solution to problems involving the 1–Laplacian operator lies on the theory of \(L^\infty \)–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 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 (1.2):
- \((f_1)\):
-
\(f \in C^0([0,+\infty ),{\mathbb {R}})\);
- \((f_2)\):
-
\(f(0) = 0\);
- \((f_3)\):
-
There exist constants \(c_1, c_2 > 0\) and \(1< q < \frac{N}{N-(1+a-b)}\), such that
$$\begin{aligned}|f(s)| \le c_1 + c_2s^{q-1}, \quad s \in [0,+\infty );\end{aligned}$$ - \((f_4)\):
-
There exist \(\mu >1\) and \(s_0>0\) such that
$$\begin{aligned}0<\mu F(s)\le f(s)s,\quad \forall \, s\ge s_0,\end{aligned}$$where \(F(t) = \int _0^t f(s)ds\);
- \((f_5)\):
-
f is increasing on \([0,+\infty )\).
Remark 1.2
Some consequences of \((f_4)\) are in order. It is not difficult to deduce from \((f_4)\) that there exist two positive constants \(d_1\) and \(d_2\) satisfying
for all \(s>0\). Applying \((f_4)\) again, we get \(f(s)\ge \mu (d_1 s^{\mu -1}-d_2s^{-1})\) for all \(s\ge s_0\) and so, having in mind \(\mu >1\), it yields
Remark 1.3
Since we are looking for nonnegative solutions, we may (and will) extend f(s) as usual defining \(f(s) = 0\) if \(s<0\). As a consequence, we have \(F(s)=0\) for all \(s<0\).
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 \(\displaystyle \lim _{s\rightarrow 0}\frac{f(s)}{|s|^{p-1}}=0\) 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 \(BV_a(\Omega )\). We apply a version of Mountain Pass Theorem suitable for functionals defined on this sort of spaces. In addittion, by using this approach, we are able also to show that this mountain pass solution is in fact a ground-state solution of the problem, i. e., its energy level is the lowest one among all the nontrivial solutions.
We briefly explain the plan of this paper. In Sect. 2 we present some preliminary results and define the space \(BV_a(\Omega )\). In Sect. 3 we set the Caffarelli–Kohn–Nirenberg inequality in the space \(BV_a(\Omega )\). In Sect. 4 we extend the Anzellotti pairing theory to include unbounded vector fields and also define the sense of solution we deal with. Section 5 is devoted to prove Theorem 1.4 by using the approximation method by problems involving weighted p-Laplacian problems. Finally, in Sect. 6 we present the proof of Theorem 1.4 by using the purely variational approach.
2 Preliminaries
We denote by \({\mathcal {H}}^{N-1}(E)\) 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 \(\pm k\) defined by
In what follows, \(\Omega \subset {\mathbb {R}}^N\) (\(N\ge 1\)) is an open and bounded set such that \(0\in \Omega \). Moreover, its boundary \(\partial \Omega \) is Lipschitz–continuous. Thus, an outward normal unit vector \(\nu (x)\) is defined for \({\mathcal {H}}^{N-1}\)–almost every \(x\in \partial \Omega \).
From now on, we denote:
-
\(C_c^1(\Omega )\), stands for the space of functions with compact support which are continuously differentiable on \(\Omega \)
-
\(C_c^\infty (\Omega )\), 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 \(\mu \) will be written as \(L^q(\Omega , \mu )\). The measure will be deleted when it is Lebesgue measure.
Sometimes we will need to use convolution with mollifiers. We will denote by \(\rho \in C^\infty _c({\mathbb {R}}^N)\) a symmetric mollifier whose support is \(\overline{B(0,1)}\) and its associated approximation to the identity by \(\rho _\epsilon (x):=\frac{1}{\epsilon ^N}\rho \left( \frac{x}{\epsilon }\right) \), for \(\epsilon >0\). 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.
2.1 Weighted spaces
Our objective in this subsection is to study spaces having a weight of the form \(x\mapsto |x|^{-a}\), with \(a>0\). We refer to [25, 26, 28] as sources for a more extensive study on weights and weighted spaces. We begin by introducing some features of these weights.
Recall that w, a nonnegative locally integrable function on \({\mathbb {R}}^N\), belongs to Muckenhoupt’s class \(A_1\) if there exists a constant \(C_w>0\) such that
where 
.
It is well-known that the weight function \(w(x)=\frac{1}{|x|^a}\) 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 \(B(x,r)\subset {\mathbb {R}}^N\). We point out that this fact implies an inequality connecting mollifiers and this weight. Indeed,
and, as a consequence of belonging to \(A_1\),
holds for all \(x\in \Omega \).
Given \(a>0\) and \(s \ge 1\), let us denote by \(L^s_a(\Omega )\) the set of measurable functions u such that
Remark 2.1
Since \(\Omega \) is a bounded set, it follows that
is positive. We note that this implies that the embedding \(L_a^s(\Omega )\hookrightarrow L^s(\Omega )\) is continuous for all \(s\ge 1\).
Definition 2.2
Let \(\, p\ge 1\) and fix \(\,0<a<\frac{N-p}{p}\). The weighted Sobolev space \({\mathcal {D}}_a^{1,p}(\Omega )\) is defined as the completion of restrictions of \(C_c^\infty ({{\mathbb {R}}}^N)\) with respect to the norm given by
Observe that functions in this space belong to
Reasoning as in Remark 2.1, we deduce that there is a continuous embedding \({\mathcal {D}}_a^{1,p}(\Omega )\hookrightarrow W^{1,p}(\Omega )\).
Remark 2.3
In [27] is proved that the space \(W^{1,p}(\Omega , |x|^{-ap})\) is equal to the closure of \(\{\varphi \in C^\infty (\Omega );\,\Vert u\Vert _{p,a}<\infty \}\).
The Sobolev space \({\mathcal {D}}_{0,a}^{1,p}(\Omega )\) is defined as the completion of \(C^\infty _c(\Omega )\) with respect to the norm \(\Vert \cdot \Vert _{p,a}\). Notice that there is a continuous embedding \({\mathcal {D}}_{0,a}^{1,p}(\Omega )\hookrightarrow W_0^{1,p}(\Omega )\). A Poincaré type inequality implies that this norm is equivalent in \({\mathcal {D}}_{0,a}^{1,p}(\Omega )\) to the norm given by
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]).
2.2 The space \(BV(\Omega )\)
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 \(Du:\Omega \rightarrow {{\mathbb {R}}}^N\) denotes the distributional gradient of u. Henceforth, we denote the distributional gradient by \(\nabla u\) when it belongs to \(L^1(\Omega ;{{\mathbb {R}}}^N)\).
We recall that the space \(BV(\Omega )\) 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 \(u\in BV(\Omega )\), so that we may write \(u\big |_{\partial \Omega }\). Indeed, there exists a continuous linear operator \(BV(\Omega )\hookrightarrow L^1(\partial \Omega )\) extending the boundaries values of functions in \(C({{\overline{\Omega }}})\). As a consequence, an equivalent norm on \(BV(\Omega )\) can be defined:
We will often use this norm in what follows.
In addition, the following continuous embeddings hold
which are compact for \(1\le m <\frac{N}{N-1}\).
For further properties of functions of bounded variations, we refer to [4] and [21].
2.3 The space \(BV_a(\Omega )\)
In this subsection, we study the definition and main properties of the space \(BV_a(\Omega )\), which is our energy space. We mainly follow [10] to where we refer for a wider analysis.
Let us define \(\text{ var}_au(\Omega )\) as
We remark that the Riesz representation Theorem implies that \(\text{ var}_au(\Omega )\) defines a Radon measure (see, for instance, [21, Section 1.8]).
We point out that the function
is continuous in \(\Omega \setminus \{0\}\), and hence it is lower semicontinuous. Then, appealing to [10, Theorem 4.1], we obtain the next result.
Theorem 2.4
The following statements are equivalent:
-
(a)
\(\text{ var}_au(\Omega )<\infty \);
-
(b)
\(u\in BV(\Omega )\) and \(\frac{1}{|x|^a}\in L^1(\Omega , |Du|)\).
Moreover,
Definition 2.5
Let \(BV_a(\Omega )\) be the space of functions \(u \in L^1(\Omega )\) such that \(|\cdot |^{-a} |Du|\) is a finite Radon measure, i.e.,
The space \(BV_a(\Omega )\) is a Banach space when endowed with the norm
Moreover, note that \(m_a\int _\Omega |Du|\le \int _\Omega \frac{1}{|x|^a}|Du|\) (\(m_a\) as in Remark 2.1), so that
Then
and so every \(u\in BV_a(\Omega )\) has a trace on \(\partial \Omega \).
We point out that the functional given by
is lower semicontinuous with respect to the \(L^1\)–convergence since each \(u\mapsto \int _{\Omega } u \, \hbox {div}\,\,\phi dx\) 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 \(\varphi \in C_c^\infty (\Omega )\), 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 \(BV(\Omega )\), we can have an equivalent norm in \(BV_a(\Omega )\) which involves an integral over \(\partial \Omega \). Its proof is a consequence of being equivalent \(\Vert \cdot \Vert _{BV(\Omega )}\) and \(\Vert \cdot \Vert _{BV(\Omega ),1}\), and using that the positive quantities
are finite.
Proposition 2.6
The norm \(\Vert \cdot \Vert _{BV_a}\) is equivalent to the norm given by
Proposition 2.7
Let \(u,v\in BV_a(\Omega )\), then \(\max \{u,v\},\min \{u,v\}\in BV_a(\Omega )\) and the following inequality is valid
In particular, choosing \(v=0\), we have that \(u^+:=\max \{u,0\},\,\,u^-=\min \{u,0\}\in BV_a(\Omega )\), with \(u=u^++u^-\), and it holds
3 The Caffarelli–Kohn–Nirenberg inequality in \(BV_a(\Omega )\)
In this section we are going to present a version of the Caffarelli–Kohn–Nirenberg inequality [13] in the space \(BV_a(\Omega )\). 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 \(BV_a(\Omega )\).
First of all we state the particular cases of the Caffarelli–Kohn–Nirenberg inequality we are interested in.
Lemma 3.1
Let \(p\ge 1\) and consider parameters satisfying \(0<a<\frac{N-p}{p}\), \(0<\theta \le 1\) and \(a<b<a+1\). Then there exists a constant \({\mathfrak {C}}_{CKN}>0\) such that the following inequality holds for all \(u\in C^\infty _c({\mathbb {R}}^N)\):
where \( \alpha =\theta b\) and \(r_{\theta }=\frac{Np}{\theta N-p[\theta (1+a-b)-N(1-\theta )]}.\)
Now we present the proof of Theorem 1.1.
Proof of Theorem 1.1
Let \(u\in BV_a(\Omega )\) and consider its extension to \({{\mathbb {R}}}^N\) defined by
We remark that 
(see [4, Theorem 3.87]).
Note also that \({\tilde{u}}*\rho _\epsilon \in C^\infty _c({\mathbb {R}}^N)\) and so we may apply Lemma 3.1 for \(p=1\) (so that \(r_{\theta }=\frac{N}{N-\theta (1+a-b)}\)). Thus, for every \(\epsilon >0\), we get
We will separately take the limit as \(\epsilon \rightarrow 0\) 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,
and this fact, jointly with (2.7), allows us to apply the Dominated Convergence Theorem and obtain
On the other hand, since
it follows that
Furthermore, we deduce from
and Fatou’s Lemma that
Therefore, using (3.17), (3.19) and (3.20), we may pass to the limit in (3.15) and obtain the desired result. \(\square \)
In the following results, we denote \(C_\Omega =\sup \{|x|\>:\> x\in \Omega \}\), which is finite since \(\Omega \) is bounded.
Theorem 3.2
Let \(a<b< a+1\) and \(r=\frac{N}{N-(1+a-b)}\). Then for all \(q \in {\mathbb {R}}\), \(1\le q\le r\), the embedding
is continuous.
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 \(u\in BV_a(\Omega )\) and we are done. \(\square \)
Theorem 3.3
Let \(a<b< a+1\) and \(r=\frac{N}{N-(1+a-b)}\). Then for all q, \(1\le q<r\) the embedding
is compact.
Proof
Let \((u_n)\) be a bounded sequence in \(BV_a(\Omega )\) and note that, since \(BV_a(\Omega ) \hookrightarrow BV(\Omega )\), \((u_n)\) is also bounded in \(BV(\Omega )\). Then, by the compact embedding in \(BV(\Omega )\), there exist a subsequence (not relabeled) and \(u\in BV(\Omega )\) such that
Let \(1<q<r\). Note that there exists \(\theta \in (0,1)\) such that
Then, using first Hölder’s inequality and then (1.3) we get
which tends to 0 as \(n\rightarrow \infty \). Here we have used that \((u_n)_n\) is bounded in \(BV_a(\Omega )\) and (3.21).
It remains to consider \(q=1\). Note that there exists \(0<{\bar{\theta }}<1\) such that \({\bar{\theta }}(1+a)>b\). Performing similar manipulations, we get
Observe that, since \({\bar{\theta }}(a+1)>b\), it follows that \(b(1-{\bar{\theta }})\frac{r_{{\bar{\theta }}}}{r_{{\bar{\theta }}}-1}<N\), so that
Hence, applying (1.3), it yields
which tends to 0 as above. \(\square \)
4 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.
4.1 Remainder of Anzellotti’s theory
We recall the notion of weak trace on \(\partial \Omega \) of the normal component defined in [7] for every \(z\in L^\infty (\Omega ; {{\mathbb {R}}}^N)\) such that its distributional divergence \(\hbox {div}\,z\) is a Radon measure having finite total variation. This trace is a function \(\left[ z, \nu \right] :\partial \Omega \rightarrow {{\mathbb {R}}}\) satisfying \(\left[ z, \nu \right] \in L^\infty (\partial \Omega )\) and \(\Vert \left[ z, \nu \right] \Vert _{L^\infty (\partial \Omega )} \le \Vert z \Vert _{L^\infty (\Omega ;{{\mathbb {R}}}^N)}\), being \(\nu (\cdot )\) the outer normal unitary vector on \(\partial \Omega \).
In [7], it was also introduced a distribution \( (z,Du):{\mathcal {C}}_c^\infty (\Omega )\rightarrow {{\mathbb {R}}}\) defined by
where
among other possible pairings. It is then proved
for all open sets \(U\subset \Omega \) such that \(\hbox {supp}\,\varphi \subset U\). As a consequence, (z, Du) is a Radon measure whose total variation satisfies
Finally, a Green formula involving the measure \(\left( z, Du \right) \) and the weak trace \(\left[ z, \nu \right] \) is established in [7], namely:
being z and u as in (4.23).
4.2 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
Note that, if \(z\in \mathcal {DM}_a(\Omega )\), then div \(z\in L^1 (\Omega )\). Indeed, choose \(\varphi \in C_c^{\infty }(\Omega )\), mollify the function \(\varphi {|x|^a}\) and have in mind [1, Lemma 1.5] to get
Thus,
and this fact means that
On the other hand, the following equalities are valid in the sense of distributions
Hence, letting \(k\rightarrow \infty \), it also holds
in the sense of distributions. This last identity implies that
Then Anzellotti’s theory supplies us with the weak trace \([z,\nu ]\) on \(\partial \Omega \) and the Radon measure (z, Du) for every \(u\in BV(\Omega )\cap L^\infty (\Omega )\) (and so for every \(u\in BV_a(\Omega )\cap L^\infty (\Omega )\)).
It is easy to compare \(\left[ \frac{1}{|x|^a}z,\nu \right] \) and \(\frac{1}{|x|^a}[z,\nu ]\). To see that they are equal, we just employ the inequality
for certain finite constant \(M_a\).
Lemma 4.1
For every \(z\in \mathcal {DM}_a(\Omega )\) we have that
Proof
For each \(k>0\), by the Proposition 2 of [15], we obtain
Now it is enough to take \(k\ge M_a\) to get our result. \(\square \)
4.3 Measures \(\left( \frac{1}{|\cdot |^a}z, Du\right) \) and \(\frac{1}{|\cdot |^a}\left( z, Du\right) \)
In this subsection, we take \(z\in \mathcal {DM}_a(\Omega )\) and \(u\in BV_a(\Omega )\cap L^\infty (\Omega )\), and introduce two distributions \(\left( \frac{1}{|x|^a}z, Du\right) \) and \(\frac{1}{|x|^a}\left( z, Du\right) \), which turn out to be equal. Finally, we will prove a Green’s formula that connects them to traces \(\left[ \frac{1}{|x|^a}z,\nu \right] =\frac{1}{|x|^a}[z,\nu ]\).
We begin by observing that \(\left( T_k\left( \frac{1}{|x|^a}\right) z, Du\right) =T_k\left( \frac{1}{|x|^a}\right) \left( z, Du\right) \) as measures for all \(k>0\). In order to do so, first notice that \(\hbox {div}\,\left( \frac{1}{|x|^a}z\right) \in L^1(\Omega )\). Then \(\left( T_k\left( \frac{1}{|x|^a}\right) z, Du\right) \) is defined as in (4.22) by
On the other hand, \(T_k\left( \frac{1}{|x|^a}\right) \left( z, Du\right) \) is such that
It is not difficult to connect both distributions. To this end, denote \(w(x)=T_k\left( \frac{1}{|x|^a}\right) \) and consider the mollification of \(\varphi w\). Then
and so
We stress that (4.27) implies that both distributions are equal. So, we have proved the following lemma.
Lemma 4.2
For every \(z\in \mathcal {DM}_a(\Omega )\) and \(u\in BV_a(\Omega )\cap L^\infty (\Omega )\), we have that
We define the weighted pairings as the limit of the above functionals.
Definition 4.3
Let \(z\in \mathcal {DM}_a(\Omega )\) and \(u\in BV_a(\Omega )\cap L^\infty (\Omega )\). Then we define the functional \(\left( \frac{1}{|x|^a}z, Du\right) :C_c^\infty (\Omega )\rightarrow {\mathbb {R}}\) as
Lemma 4.4
For every \(z\in \mathcal {DM}_a(\Omega )\) and \(u\in BV_a(\Omega )\cap L^\infty (\Omega )\), we have that
As a consequence, since \(\frac{1}{|x|^a}(z,Du)\) is a Radon measure in \(\Omega \), so is \(\left( \frac{1}{|x|^a}z,Du\right) \).
Proof
We point out that \(\frac{1}{|x|^a}\in L^1(\Omega , (z,Du))\), since \(|(z,Du)|\le \Vert z\Vert _\infty |Du|\) and \(u\in BV_a(\Omega )\cap L^\infty (\Omega )\). Moreover, we have
Thus, having in mind (4.28), both distributions are equal. \(\square \)
Theorem 4.5
Let \(z\in \mathcal {DM}_a(\Omega )\) and \(u\in BV_a(\Omega )\cap L^\infty (\Omega )\). For all open sets \(U\subset \Omega \) and for all functions \(\varphi \in C_c^\infty (U) \), it yields
Proof
Note that, from (4.24) we have that
what proves the result. \(\square \)
Corollary 4.6
The measures \(\left( \frac{1}{|x|^a}z, Du\right) \) and \(\left| \left( \frac{1}{|x|^a}z, Du\right) \right| \) are absolutely continuous with respect to the measure \(\frac{1}{|x|^a}|Du|\) and the inequality
holds for all Borel sets B and for all open sets U such that \(B\subset U\subset \Omega .\)
Theorem 4.7
Let \(z\in \mathcal {DM}_a(\Omega )\) and \(u\in BV_a(\Omega )\cap L^\infty (\Omega )\). Then we have
Proof
It follows from (4.26), jointly with Lemmas 4.1 and 4.2, that
for all \(k>0\). Since \(x \mapsto \frac{1}{|x|^a}\) is a bounded function on \(\partial \Omega \), then for k large enough, \(T_k\left( \frac{1}{|x|^a}\right) =\frac{1}{|x|^a}\). Hence,
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 \(\nabla \left( \frac{1}{|x|^a}\right) \in L^1(\Omega )\). On the other hand, we denote by \(\theta (z, Du)\) the Radon–Nikodým derivative of (z, Du) with respect to |Du|, so that \(|\theta (z, Du)|\le \Vert z\Vert _\infty \). Then
and
Owing to \(\frac{1}{|x|^a}\in L^1(\Omega , |Du|)\), we are allowed to use the Dominated Convergence Theorem. Therefore, when \(k\rightarrow \infty \), identity (4.34) becomes
as desired. \(\square \)
Remark 4.8
Note that, by Lemmas 4.1 and 4.4, the last identity can also be written as
4.4 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).
Definition 4.9
We say that \(u \in BV_a(\Omega )\cap L^\infty (\Omega )\) is a solution of problem (1.2) if there exists a vector field \(z\in L^\infty (\Omega ,{\mathbb {R}}^N)\) with \(\Vert z\Vert _\infty \le 1\) and such that
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].
Proposition 4.10
For \(u \in BV_a(\Omega )\cap L^\infty (\Omega )\), the following assertions are equivalent.
-
(a)
u is a solution to problem (1.2).
-
(b)
there exists a vector field \(z\in L^\infty (\Omega ,{\mathbb {R}}^N)\) satisfying \(\Vert z\Vert _\infty \le 1\),
$$\begin{aligned} -\mathrm{div} \left( \frac{1}{|x|^a}z\right) = \frac{1}{|x|^b}f(u), \text{ in } {\mathcal {D}}'(\Omega ), \end{aligned}$$and
$$\begin{aligned} \int _\Omega \frac{1}{|x|^b}f(u)(v-u) dx =\int _\Omega \frac{1}{|x|^a}(z,Dv)-\int _{\partial \Omega }\frac{1}{|x|^a}v[z,\nu ]\, d{\mathcal {H}}^{N-1} -\Vert u\Vert _{BV(\Omega ),1} \end{aligned}$$(4.36)for all \(v \in BV_a(\Omega )\cap L^\infty (\Omega )\).
Proof
To see that \((a)\Rightarrow (b)\), just take \(v \in BV_a(\Omega )\cap L^\infty (\Omega )\), multiply the equality (1) of Definition 4.9 by \(v-u\) and apply Green’s formula and conditions (2) and (3).
The reverse implication \((b)\Rightarrow (a)\) is deduced by taken \(v=u\) in (4.36). Indeed, we obtain
and conditions (2) and (3) follow since \(\Vert z\Vert _\infty \le 1\). \(\square \)
Corollary 4.11
If u is a solution to problem (1.2), then
holds for every \(v \in BV_a(\Omega )\).
Proof
When \(v \in BV_a(\Omega )\cap L^\infty (\Omega )\), it is an easy consequence of Proposition 4.10 and the condition \(\Vert z\Vert _\infty \le 1\). For a general \(v \in BV_a(\Omega )\), apply this inequality to \(T_k(v)\) to get
Now, on account of Theorem 3.2, \(v\in L_b^1(\Omega )\) and so we may let k go to \(\infty \) on the left hand side of (4.38). \(\square \)
Corollary 4.12
Every solution to problem (1.2) is nonnegative.
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 \(s\le 0\)) and we get
Therefore, \(\Vert u^-\Vert _{BV(\Omega ),1}\le 0\) and so \(u=u^+\ge 0\). \(\square \)
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.
Proposition 4.13
Let \(h\in L^1(\Omega )\) and assume that problem
has a bounded solution w. (By a solution to problem (4.39) we mean that w satisfies Definition 4.9 with the obvious replacement of \(\frac{1}{|x|^b}f(w)\) by h.)
If \(u \in BV_a(\Omega )\cap L^\infty (\Omega )\) and \(h\in \partial \Vert u\Vert _{BV(\Omega ),1}\), then u is also a solution to problem (4.39).
Proof
Let \(w \in BV_a(\Omega )\cap L^\infty (\Omega )\) be a solution to problem (4.39). Then there exists a vector field \(z\in L^\infty (\Omega ,{\mathbb {R}}^N)\) such that \(\Vert z\Vert _\infty \le 1\) and
jointly with conditions (2) and (3). Taken \(w-u\) as test function, it yields
On the other hand, assumption \(h\in \partial \Vert u\Vert _{BV(\Omega ),1}\) implies
Hence, gathering (4.40) and (4.41), it follows that
and the result is a consequence of being \(\Vert z\Vert _\infty \le 1\). \(\square \)
5 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 \(p\rightarrow 1^+\).
5.1 Approximating problems involving p-Laplacian operators
First of all, we consider \(1<{{\bar{p}}}<2\) and so \({{\bar{p}}}<N<\frac{N}{1+a-b}\). Since \(0<a<N-1\), \(\mu >1\) and \(1<q<\frac{N}{N-(1+a-b)}\), we may assume that \({{\bar{p}}}\) also satisfies
This implies that, denoting \({{\bar{q}}}=q+{{\bar{p}}}-1\),
for every \(1<p\le {{\bar{p}}}\). Now, for each \(1<p\le {\bar{p}}\), 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:
- \((f_{1p})\):
-
\(f_p \in C^0([0,+\infty ), {\mathbb {R}})\);
- \((f_{2p})\):
-
\(\displaystyle \lim _{s\rightarrow 0^+} \frac{f_p(s)}{|s|^{p-1}} = 0\);
- \((f_{3p})\):
-
There exist constants \(c_1, c_2 > 0\) and \(p< {{\bar{q}}} < \frac{Np}{N-p(1+a-b)}\), such that
$$\begin{aligned} |f_p(s)|\le c_1+c_2 s^{{{\bar{q}}}-1}\quad \text{ for } \text{ all } s\in [0,+\infty ). \end{aligned}$$ - \((f_{4p})\):
-
There exists \(\mu > p\) such that
$$\begin{aligned}0<\mu F_p(s)\le f_p(s)s,\quad \forall \, s\ge s_0,\end{aligned}$$where \(F_p(t) = \int _0^t f_p(s)ds\).
Remark 5.1
The conditions \((f_{1p})-(f_{3p})\) are straightforward to check. To prove the condition \((f_{4p})\), just integrate by parts to obtain
when \(s>0\). Hence,
for s large enough.
Problem (5.42) has been studied in [12] using the lower and upper–solutions 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 \(p\rightarrow 1\).
In order to get a nontrivial solution to (5.42), we work in the space \({\mathcal {D}}_{0,a}^{1,p}(\Omega )\) that is defined in Sect. 2.1. Moreover, the functions of this space satisfy the following Caffarelli–Kohn–Nirenberg inequality.
Theorem 5.2
Let \(0<a<\frac{N-{\bar{p}}}{{\bar{p}}}\), \(0<\theta \le 1\) and \(a<b<a+1\). Then there exists a constant \({\mathfrak {C}}_{CKN}>0\) such that the following inequality holds for all \(u\in {\mathcal {D}}_{0,a}^{1,p}(\Omega )\)
where \( \alpha =\theta b\) and \(r_{\theta p}=\frac{Np}{\theta N-p[\theta (1+a-b)-N(1-\theta )]}.\)
Proof
The proof follows as that one of Theorem 3.1, with the difference that
as showed in Theorem 2.5 of [27]. \(\square \)
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.
Theorem 5.3
Let \(0<a<\frac{N-p}{p}\), \(a<b<a+1\) and \(r_p=r_{1p}=\frac{Np}{N-p(1+a-b)}\). Then the embedding
is continuous for all \(q\in [1,r_p]\) and compact for all \(q\in [1,r_p).\)
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 \((PS)_c\) sequence \((w_n)_{n\in {\mathbb {N}}}\) in \({\mathcal {D}}_{0,a}^{1,p}(\Omega )\), i.e.,
where
and
Well–known arguments can be used to show that \((w_n)_{n \in {\mathbb {N}}}\) is a bounded sequence in \(\mathcal D_{0,a}^{1,p}(\Omega )\) and consequently, that there exists \(w_p \in {\mathcal {D}}_{0,a}^{1,p}(\Omega )\) in such a way that
Since \(J_p\in C^1({\mathcal {D}}_{0,a}^{1,p}(\Omega ))\) the previous convergence implies that
and consequently \(w_p\) is a nontrivial solution in \(\mathcal D_{0,a}^{1,p}(\Omega )\) to problem (5.42).
Once we have got the family of approximate solutions \((w_p)_{1<p\le {{\bar{p}}}}\), our main concern is to get bounds of this family which do not depend on p. To this end, let us consider the functional \(I_p\>:\>{\mathcal {D}}_{0,a}^{1,p}(\Omega )\rightarrow {\mathbb {R}}\) defined by
It is straightforward to see that \(p\mapsto I_p(u)\) is a nondecreasing function, for every \(u \in W^{1,{\overline{p}}}_0(\Omega , |x|^{-a})\). Indeed, let \(1< p_1< p_2 < {\overline{p}}\) and note that, by Young’s inequality,
Moreover, the critical points of \(J_p\) are the same of those of \(u\mapsto I_p(u)-\int _{\Omega }\frac{1}{|x|^b}F_p(u)dx\).
Next, we show that there exists \(e\in C^\infty _c(\Omega )\) such that
Fix a nontrivial \(\phi \in C^\infty _c(\Omega )\) such that \(\phi \ge 0\) and \(\Vert \phi \Vert _\infty \le 1\). This fact leads to
for every \(1<p\le {{\bar{p}}}\). Moreover, the Lebesgue Dominated Convergence Theorem implies
and, as a consequence, we may assume that
Analogously, there is no loss of generality in assuming that
for every \(1<p\le {{\bar{p}}}\).
Now let \(t>1\). Then, owing to \(\displaystyle \lim _{s\rightarrow +\infty }f(s)=+\infty \), given
we can find \(M>0\) such that \(f(s)>K\), and consequently \(f_p(s)>Ks^{p-1}\), for all \(s>M\). Hence, if \(s>M\), then
Denoting \(K_1=K(1+M)^{{{\bar{p}}}}\int _\Omega \frac{1}{|x|^b}\, dx\) and taking t large enough such that
we deduce
where have also used (5.47). Therefore, from (5.46)
since \(p<2\) and \(t>1\). Thus, choosing t large enough, we find \(e=t\phi \) satisfying
Since e does not depend on p, thanks to the Mountain Pass Theorem, we know that \(w_p\) satisfies
where
5.2 Estimate of the family \(\{w_p\}\)
We claim that the sequence \(\left( I_p(w_p)-\int _{\Omega }\frac{1}{|x|^b}F_p(w_p)dx \right) _{1<p<\bar{p}}\) is bounded by a constant which does not depend on p. Indeed, let \(1<p_1<p_2<{{\bar{p}}}\) and let us apply the monotonicity of \(I_p\) and the fact that \(\Gamma _{p_2}\subset \Gamma _{p_1}\) (because \({\mathcal {D}}_{0,a}^{1,p_2}(\Omega )\subset \mathcal D_{0,a}^{1,p_1}(\Omega )\)). Then
It yields
where \(\gamma _0(t)=te\). Now, for \(1<p<{{\bar{p}}}\), it is straightforward to see that
and so
It follows that if \(1<p<{{\bar{p}}}\), then
and the claim is proved. Thus, there exists \(C>0\) such that
where the constant C is independent of p.
Let \(\Omega _p=\{x\in \Omega \,:\, |w_p(x)|\le s_0 \}\), for any \(p\in (1,{\bar{p}})\). Then, by \((f_{3p})\), we have
and so
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
Gathering together (5.50), (5.51) and (5.52), we have
Moreover, since \(1<p\le {\bar{p}}<\mu \) by the last inequality we have that there exists \({\tilde{C}}>0\) independent of p such that
Now, using the previous estimate, Young and Hölder’s inequalities we have
where \({\hat{C}}\) is a constant independent of p.
5.3 Convergence of \((w_p)_p\)
Recalling that \(w_p\big |_{\partial \Omega }=0\), it follows from (5.56) that the sequence \(\{w_p\}_{1<p<{\bar{p}}}\) is bounded in \(BV_a(\Omega )\). Then, up to a subsequence, there exists w such that, by Theorem 3.3,
for all \(q\in \left[ 1, \frac{N}{N-(1+a-b)}\right) \) as well as, by (2.9),
for all \(s\in \left[ 1, \frac{N}{N-1}\right) \). Up to a further subsequence, by [11, Theorem 4.9], we may also assume
and that there exists \(g\in L^q_b(\Omega )\), \(1\le q< \frac{N}{N-(1+a-b)}\), such that
holds for all \(p\in (1,{\bar{p}}]\). Finally, the lower semicontinuity of the functional \(u\mapsto \int _\Omega \frac{1}{|x|^a}|Du|\) guarantees that \(w\in BV_a(\Omega )\).
5.4 Boundedness of the limit
Let \(k\ge 0\) and let \(w_p\in {\mathcal {D}}_{0,a}^{1,p}(\Omega )\) be a solution of problem (5.42). Define
Lemma 5.4
Let \(p>1\) be small enough. For each \(\epsilon >0\) there exists \(k_0>0\) (which does not depend on p) such that
where \({{\bar{q}}}\) is as in \((f_{3p})\).
Proof
Note that
Now we denote \(\alpha =\frac{({{\bar{q}}}-1)[N-(1+a-b)]}{1+a-b}\) and \(l=\frac{N}{1+a-b}\), which satisfy \(0<\alpha <1\) and \(l>1\). Using (5.61) and Hölder’s inequality, we obtain
Hence, we have got
where \(\omega (k)\) stands for a quantity independent on p that tends to 0 as \(k\rightarrow +\infty \). 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 \(k\rightarrow \infty \). \(\square \)
Now, let us deduce from Lemma 5.4 that \(w\in L^\infty (\Omega )\). To this end, given \(k>0\), we define the auxiliary function \(G_k:{\mathbb {R}}\rightarrow {\mathbb {R}}\) as
Choosing \(G_k(w_p)\) as a test function in problem (5.42), we get
Set \(1^*_a=\frac{N}{N-(1+a-b)}\). 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 \(k_0\in {\mathbb {N}}\) such that
Since \(w_p(x)\rightarrow w(x)\) a. e. in \(\Omega \) when \(p\rightarrow 1^+\), Fatou’s Lemma implies
Therefore \(\Vert w\Vert _\infty \le k_0.\)
5.5 Existence of the vector field
We begin by using the notation of Remark 2.1 and observing that (5.55) yields
and then
So, we may apply the same argument than that in [30, Theorem 3.5.] and obtain a subsequence (not relabeled) and \(z\in L^\infty (\Omega ;{{\mathbb {R}}}^N)\) satisfying \(\Vert z\Vert _\infty \le 1\) and
In order to pass to the limit in the following stages, these weak convergences must slightly be improved. Fix \(1<s<\infty \) such that \(1<s'<\frac{N}{a}\), and take \({{\bar{p}}}\) small enough to have \(1<s'<\frac{N}{a{{\bar{p}}}}\), so that \(\int _\Omega \frac{1}{|x|^{a\bar{p}s'}}dx<\infty \). Since
for all \(1<p<{{\bar{p}}}\), Lebesgue Convergence Dominated Theorem implies
Thus, the convergences \(\frac{1}{|x|^{ap}}\rightarrow \frac{1}{|x|^{a}}\) strongly in \(L^{s'}(\Omega )\) and \( |\nabla w_p|^{p-2}\nabla w_p \rightharpoonup z\) weakly in \(L^s(\Omega ; {{\mathbb {R}}}^N)\) lead to
5.6 w satisfies condition (1) of Definition 4.9
Let \(\varphi \in C^\infty _c(\Omega )\) and take it as test function in (5.42) to obtain
Our aim is to let \(p\rightarrow 1^+\) in (5.71). On the left hand side it is enough to apply (5.70), while in the right hand side, just observe that
due to (5.59). Moreover, by \((f_{3p})\) and Young’s inequality, we get
and \(g\in L_b^{{{\bar{q}}}}(\Omega )\). Hence, the Lebesgue Dominated Convergence Theorem implies
Therefore, letting \(p\rightarrow 1^+\) in (5.71), we obtain
and thus item (1) of Definition 4.9 is verified.
5.7 w satisfies condition (2) of Definition 4.9
In this subsection, we show that the identity
holds as Radon measures.
Firstly note that we may apply Corollary 4.6 (since \(\Vert z\Vert _\infty \le 1\)) getting
Now let us check the opposite inequality, i. e.,
for all \(\varphi \in C^1_c(\Omega )\) such that \(\varphi \ge 0\).
Fix \(0\le \varphi \in C^1_c(\Omega )\) and choose \(k>\Vert w\Vert _\infty \). Taking \(T_k(w_p)\varphi \in {\mathcal {D}}_{0,a}^{1,p}(\Omega )\) as test function in (5.42), we get
Moreover, applying Young’s inequality, one deduces
Our next objective is to let \(p\rightarrow 1^+\). On the left hand side, since \(T_k(w_p)\rightarrow T_k(w)\) in \(L^1(\Omega )\), 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.
Therefore, from (5.77), (5.78) and (5.79), inequality (5.76) becomes
Our choice of k leads to
so that (5.73) implies
Thus (5.74) holds.
5.8 w satisfies condition (3) of Definition 4.9
It only remains to check
It is equivalent to show that
Indeed, \(\Vert z\Vert _\infty \le 1\) yields
and so the integrand is nonnegative. Then (5.675.81) implies \(\frac{1}{|x|^a}|w|+w\frac{1}{|x|^a}\left[ z,\nu \right] =0\) 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 \(w_p\big |_{\partial \Omega }=0\), we get
Our aim is to let \(p \rightarrow 1^+\) 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
and the following consequence of condition \((f_{3p})\):
Thus,
and the remainder term tends to 0.
Consequently, using (5.85) and (5.86) in (5.84) we get
Applying (5.73) and Green’s formula (Theorem 4.7), we arrive at
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).
5.9 w is a nontrivial solution of (1.2)
Now, what is left to do is to show that \(w \ne 0\). In order to do so, we should introduce the energy functional \(\Phi : BV_a(\Omega ) \rightarrow {\mathbb {R}}\) given by
First of all, let us prove that
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 \(p \rightarrow 1^+\)
Moreover, again by \((f_{3p})\), (5.57) and the Lebesgue Dominated Convergence Theorem, as \(p \rightarrow 1^+\), we have that
Then, (5.91) and (5.92) imply in (5.90).
We remark that, by \((f_1)\) and \((f_2)\), given \(\epsilon > 0\), we may find \(\delta >0\) satisfying
so that \((f_3)\) implies that there exists a positive constant \({{\tilde{C}}}_{\epsilon } > 0\) such that
Integrating this inequality, we deduce
for certain constant \(C_{\epsilon }>0\). Thus, by Theorem 3.2,
Let us consider \(\epsilon > 0\) small enough such that \(1-\epsilon C_1 > 1/2\). So, if \(\Vert u\Vert _{BV_a(\Omega ),1} \le \rho \), where \(\displaystyle 0< \rho < \left( \frac{(1-\epsilon C_1)-1/2}{C_\epsilon C_q}\right) ^\frac{1}{q-1}\), then
On the other hand, for all \(1< p < {\bar{p}}\), Young’s inequality implies that \(I_p(u) - \int _\Omega \frac{1}{|x|^b}F_p(u) dx \ge \Phi (u) + o_p(1)\). Then, for all \(\gamma \in \Gamma _p\), from the continuity of \(t \mapsto I_p(\gamma (t)) - \int _\Omega \frac{1}{|x|^b}F_p(\gamma (t)) dx\) and from the fact that \(I_p(e) - \int _\Omega \frac{1}{|x|^b}F_p(e) dx < 0\), it follows that there exists \(t_0 \in [0,1]\) such that \(\Vert \gamma (t_0)\Vert _{BV_a(\Omega ),1} = \rho \). Then,
Hence, from the last inequality and (5.90), it follows that
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.
Corollary 5.5
If \(u \in BV_a(\Omega )\cap L^\infty (\Omega )\) and \(\frac{1}{|x|^b}f(w)\in \partial \Vert u\Vert _{BV(\Omega ),1}\), then u is a solution to problem (1.2).
6 Existence by variational methods
First of all, let us consider the energy functional \(\Phi : BV_a(\Omega ) \rightarrow {\mathbb {R}}\), given by
where
and
It is straightforward to see that \({\mathcal {F}}_b\) is a smooth functional. Moreover, by the same arguments of [8], it is possible to show that the functional \({\mathcal {J}}_a\) admits some directional derivatives. More specifically, given \(u \in BV_a(\Omega )\), for all \(v \in BV_a(\Omega )\) such that \((Dv)^s\) is absolutely continuous with respect to \((Du)^s\), \((Dv)^a\) vanishes a.e. on the set \(\{x \in \Omega :\, (Du)^a(x) = 0\}\) and \(v \equiv 0\), \({\mathcal {H}}^{N-1}-\)a.e. on \(\{x \in \partial \Omega :\, u(x) = 0\}\), it follows that
In particular, note that, for all \(u \in BV_a(\Omega )\),
Then, the directional derivatives \(\Phi '(u)u\) exist and
Note that \(\Phi \) can we written as the difference between a Lipschitz and a smooth functional in \(BV_a(\Omega )\). Taking into account the theory of subdifferentials of Clarke (see [16, 17]) , we say that \(w \in BV_a(\Omega )\) is a critical point of \(\Phi \) if \(0 \in \partial \Phi (w)\), where \(\partial \Phi (w)\) denotes the generalized gradient of \(\Phi \) in w. It follows that this is equivalent to \({\mathcal {F}}'(w) \in \partial {\mathcal {J}}_a(w)\) and, since \({\mathcal {J}}_a\) is convex, this can be written as
Henceforth, every \(w \in BV_a(\Omega )\) such that (6.98) holds is going to be called a critical point of \(\Phi \).
Let us prove that \(\Phi \) 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 \(\epsilon > 0\), there exists \(A_\epsilon > 0\) such that
Note also that, by (6.99) and the embeddings of \(BV_a(\Omega )\) (see Theorem 3.2), it follows that
for all \(u \in BV_a(\Omega )\), such that \(\Vert u\Vert _{BV_a(\Omega ),1} = \rho \), where \(0< \epsilon < 1\) is fixed, \(\displaystyle 0< \rho < \left( \frac{1-\epsilon C}{c_3}\right) ^\frac{1}{q-1}\) and \(\displaystyle \alpha = \rho (1 - \epsilon C - c_3\rho ^{q-1})\).
Now let us check that \(\Phi \) satisfies the second geometric condition of the Mountain Pass Theorem. Recall (see Remark 1.2) that condition \((f_4)\) implies that there exists constants \(d_1,d_2 > 0\) such that
Let \(\phi \in C^\infty _c(\Omega )\) be nontrivial and nonnegative and let \(t > 0\). Since \(\mu > 1\), it follows that
as \(t \rightarrow +\infty \), and so we can choose \(e \in BV_a(\Omega )\) such that \(\Phi (e) < 0\).
Then, the Mountain Pass Theorem (see [22, Theorem 4.1]) implies that there exist sequences \(\tau _n \rightarrow 0\) and \((w_n)\subset BV_a(\Omega )\) satisfying the following conditions
-
(1)
$$\begin{aligned} \lim _{n \rightarrow \infty }\Phi (w_n) = c \end{aligned}$$(6.101)
where c is given by
$$\begin{aligned} c = \inf _{\gamma \in \Gamma } \sup _{t \in [0,1]}\Phi (\gamma (t)) \end{aligned}$$and \(\Gamma = \{\gamma \in C^0([0,1],BV_a(\Omega )); \, \gamma (0) = 0 \, \, \text{ and } \, \, \gamma (1) = \phi \}\).
-
(2)
$$\begin{aligned}&\Vert v\Vert _{BV_a(\Omega ),1} - \Vert w_n\Vert _{BV_a(\Omega ),1}\nonumber \\&\quad \ge \int _\Omega \frac{1}{|x|^b}f(w_n)(v - w_n)dx - \tau _n\Vert v - w_n\Vert _{BV_a(\Omega ),1}, \end{aligned}$$(6.102)
for all \(v \in BV_a(\Omega )\).
Let us prove that the sequence \((w_n)\) is bounded in \(BV_a(\Omega )\). First of all, note that by taking \(v = w_n +t w_n\) in (6.102), dividing by t and letting \(t \rightarrow 0^\pm \), we have that
Then, by \((f_4)\) and (6.103), note that
for some \(C > 0\) uniform in \(n \in {\mathbb {N}}\). Then it follows that \((w_n)\) is bounded in \(BV_a(\Omega )\).
By the boundedness of \((w_n) \subset BV_a(\Omega )\) and Theorem 3.3, we find \(w \in BV_a(\Omega )\) such that
Then, by (6.104) and the lower semicontinuity of \({\mathcal {J}}_a\) with respect to the \(L^1(\Omega )\) convergence, calculating the \(\limsup \) 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
On the other hand, taking the limit as \(n \rightarrow +\infty \) in (6.103), it follows that
Hence, from (6.104), (6.105), (6.106) and the Lebesgue Dominated Convergence Theorem, it follows that
and then w is a nontrivial critical point of \(\Phi \).
Our next concern is to check that \(w\in L^\infty (\Omega )\). To this end, consider \(k>0\) and the function \(G_k(s)\) defined in (5.64). Taking \(v=w\pm G_k(w)\) in (6.98), it yields
and we infer that
Setting \(1_a^*=\frac{N}{N-(1+a-b)}\) again and reasoning as in Sect. 5.4, we obtain
Since
we may find \(k_0>0\) such that
and then
holds. Therefore, \(G_{k_0}(w)=0\) and so \(|w|\le k_0\).
As a consequence of Corollary 5.5, since \(w\in BV_a(\Omega )\cap L^\infty (\Omega )\) 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 \(\Phi \), given by
It can be proven as in [23] that \({\mathcal {N}}\) 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 \(\Phi (w) = \inf _{{\mathcal {N}}}\Phi \), 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 \(t \mapsto \Phi (tv)\) and verifying that it has a unique maximum point \(t_v > 0\), which is such that \(t_v v \in {\mathcal {N}}\) (\((f_5)\) is mandatory to prove the uniqueness); in the light of \((f_1) - (f_5)\), one can see that \({\mathcal {N}}\) is radially homeomorphic to the unit sphere in \(BV_a(\Omega )\) and also that the minimax level c satisfies
Since w is such that \(\Phi (w) = c\), it follows that w is a solution which has the lowest energy among all the nontrivial ones.
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Acknowledgements
Juan C. Ortiz Chata is supported by FAPESP 2017/06119-0 and 2019/13503-7. Marcos T. O. Pimenta is supported by FAPESP 2019/14330-9 and CNPq 303788/2018-6. Sergio Segura de León is supported by the Spanish Ministerio de Ciencia, Innovación y Universidades and FEDER, under project PGC2018–094775–B–I00.
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Ortiz Chata, J.C., Pimenta, M.T.O. & Segura de León, S. Anisotropic 1-Laplacian problems with unbounded weights. Nonlinear Differ. Equ. Appl. 28, 57 (2021). https://doi.org/10.1007/s00030-021-00717-4
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DOI: https://doi.org/10.1007/s00030-021-00717-4
Keywords
- 1-Laplacian operator
- Caffarelli–Kohn–Nirenberg inequality
- Weighted \(L^\infty \)–divergence–measure vector fields
- Weighted quasilinear elliptic problems