Existence and regularity of solutions of nonlinear anisotropic elliptic problem with Hardy potential

Abstract

In this paper, we are interested in the existence and regularity of solutions for some anisotropic elliptic equations with Hardy potential and $L^{m} (Ω)$ data in appropriate anisotropic Sobolev spaces. The aim of this work is to get natural conditions to show the existence and regularity results for the solutions, that is related to an anisotropic Hardy inequality.

Keywords

Anisotropic elliptic problems existence and regularity Hardy potential irregular data Hardy inequality

1. Introduction

Let Ω be a bounded open set in $R^{N} (N > 2)$ with smooth boundary $\partial Ω$ and $\vec{p} = (p_{1}, \dots, p_{N})$ are restricted as follows $\begin{matrix} (1.1) & 2 ⩽ p^{-} = min_{1 ⩽ i ⩽ N} {p_{i}} < p_{i} < p^{+} = max_{1 ⩽ i ⩽ N} {p_{i}}, 2 ⩽ \overline{p} < N, \frac{1}{\overline{p}} = \frac{1}{N} \sum_{i = 1}^{N} \frac{1}{p_{i}} \end{matrix}$ The anisotropic Laplace operator $Δ_{\vec{p}} u$ is defined by $\begin{matrix} Δ_{\vec{p}} u = \sum_{i = 1}^{N} D_{i} (| D_{i} u |^{p_{i} - 2} D_{i} u), D_{i} u = \frac{\partial u}{\partial x_{i}} \forall i = \overline{1, N} . \end{matrix}$ This paper deals with existence and regularity of solutions for a class of nonlinear anisotropic elliptic problems $\begin{matrix} (1.2) & \{\begin{array}{ll} - Δ_{\vec{p}} u = \frac{| u |^{p^{-} - 1}}{| x |^{p^{-}}} + f & in Ω \\ u = 0 & on \partial Ω, \end{array} \end{matrix}$ where f belongs to $L^{m} (Ω)$ with $m ⩾ 1$ and $μ > 0$ , such that $\begin{aligned} (1.3) & 0 ⩽ μ & < \frac{(λ - 1) N^{\frac{p^{-} + 2}{2}}}{M_{p^{+}, p^{-}}^{- 1}} {(\frac{p^{-}}{p^{-} + λ - 2})}^{p^{-}}, \end{aligned}$ where $\begin{aligned} (1.4) & M_{p^{-}, p^{+}} = {(\frac{p^{-} - 1}{p^{+}})}^{p^{-}}, and λ = \frac{N (m - 1) (\overline{p} - 1) + N - m \overline{p}}{N - m \overline{p}} . \end{aligned}$

Nowadays, Anisotropic equations appear in various mathematical models. For example, they are required for the investigation of such physical fields as electro-rheological and thermo-rheological liquids [3]. They also appear in biology, see Bendahmane–Langlais–Saad [4], as a model describing the spread of an epidemic disease in heterogeneous environments.

In the case $p_{i} = 2$ for all $i = \overline{1, N}$ and $μ > 0$ , Boccardo, Orsina and Peral in their leading work [6], obtained existence and summability of solutions to the elliptic problem $\begin{matrix} (1.5) & \{\begin{array}{ll} - div (M (x) \nabla u) = a \frac{u}{| x |^{2}} + f & in Ω \\ u = 0 & on \partial Ω, \end{array} \end{matrix}$ where $M : Ω \times R \to R^{N^{2}}$ , is a bounded measurable matrix, such that $\begin{matrix} α | ξ |^{2} ⩽ ⟨ M (x) ξ, ξ ⟩ ⩽ β | ξ |^{2}, \end{matrix}$ $f \in L^{m} (Ω)$ , $1 < m < \frac{N}{2}$ , and $\begin{aligned} (1.6) & a \in] 0, α \frac{N (m - 1) (N - 2 m)}{m^{2}} [= I . \end{aligned}$ The authors in [7] proved that existence of solutions to the problem $- Δ u = \frac{u^{p}}{| x |^{2}}$ , $u > 0 in Ω$ , where $1 < p < \frac{N + 2}{N - 2}$ depends on the geometry of the domain.

Our main motive in this article is to investigate the results of [6] in framework of the non-homogeneous operator $Δ_{\vec{p}} u$ . To reach this goal, we will face the following difficulties. First, let us note that (1.2) can be singular at the origin on the right-hand side, the so-called Hardy potential. On the other hand, it is difficult to apply the anisotropic Hardy inequality, which plays a major role in showing the desired results. To overcome these difficulties we approximate the term $\frac{| u |^{p^{-} - 1}}{| x |^{p^{-}}}$ by $\frac{| T_{n} (u_{n}) |^{p^{-} - 1}}{{(| x | + \frac{1}{n})}^{p^{-}}}$ , Then, we prove a priori estimates of the approximate solution sequence, by using the anisotropic Hardy inequality (see Corollary 2.6 and Corollary 2.4).

Our inspiration in this paper is taken from [8] (also see [5]), where the authors considered the same right-hand side, for elliptic problems whose model is $\begin{matrix} \{\begin{array}{ll} - Δ_{\vec{p}} u = f & in Ω, \\ u = 0 & on \partial Ω . \end{array} \end{matrix}$ The author prove that

If ${({\overline{p}}^{*})}^{'} ⩽ m < \frac{N}{2}$ , then $u \in L^{s} (Ω)$ with $s = \frac{m N (\overline{p} - 1)}{N - m \overline{p}}$ .

If $1 < m < {({\overline{p}}^{*})}^{'}$ , then $u \in W_{0}^{1, (s_{i})} (Ω)$ with $s_{i} = \frac{m N (\overline{p} - 1)}{\overline{p} (N - m)} p_{i}$ .

Remark 1.1.
It should be noted that the constraints on the parameter μ are less stringent compared to the isotropic case, where $p_{i} = 2$ for all i ( $p^{-} = p^{+} = \overline{p} = 2$ ). Indeed, by using (1.3) and (1.4), we obtain $λ = \frac{N (m - 1) + N - 2 m}{N - 2 m}$ , $M_{p^{-}, p^{+}} = \frac{1}{4}$ , consequently, we have $\begin{aligned} (1.7) & μ \in] 0, \frac{N^{3} (m - 1) (N - 2 m)}{N^{2} m^{2} + 4 m^{2} (1 - N)} [= J . \end{aligned}$ Now, by setting $α = 1$ in (1.6) and considering that $N > 1$ , we can compare the intervals I and J to reach the conclusion $\in I \subset J$ .

2. Anisotropic Hardy inequality

Let Ω be a bounded, smooth domain of $R^{N}$ , $N ⩾ 3$ , $p_{i} > 1$ for any $i = \overline{1, N}$ . The anisotropic Sobolev spaces $W^{1, (p_{i})} (Ω)$ and $W_{0}^{1, (p_{i})} (Ω)$ , which are defined by $\begin{matrix} W^{1, (p_{i})} (Ω) = {v \in W^{1, 1} (Ω) : D_{i} v \in L^{p_{i}} (Ω)}, \end{matrix}$ and $\begin{matrix} W_{0}^{1, (p_{i})} (Ω) = W^{1, (p_{i})} (Ω) \cap W_{0}^{1, 1} (Ω) . \end{matrix}$ The space $W_{0}^{1, (p_{i})} (Ω)$ can also be defined as the closure of $C_{0}^{\infty} (Ω)$ in $W^{1, (p_{i})} (Ω)$ with respect to the norm $\begin{matrix} ‖ v ‖_{W_{0}^{1, (p_{i})} (Ω)} = \sum_{i = 1}^{N} ‖ D_{i} v ‖_{L^{p_{i}} (Ω)} . \end{matrix}$ Equipped with this norm, $W_{0}^{1, (p_{i})} (Ω)$ is a separable and reflexive Banach space. The following embedding result for the anisotropic Sobolev space is well-known [9,10,12].

Lemma 2.1.
There exists a positive constant C, depending only on Ω, such that for $v \in W_{0}^{1, (p_{i})} (Ω)$ , $\overline{p} < N$ we have $\begin{aligned} (2.1) & ‖ v ‖_{L^{r} (Ω)} ⩽ C \prod_{i = 1}^{N} ‖ D_{i} v ‖_{L^{p_{i}} (Ω)}^{\frac{1}{N}}, \forall r \in [1, {\overline{p}}^{}], \\ (2.2) & ‖ v ‖_{L^{{\overline{p}}^{}} (Ω)}^{p_{N}} ⩽ C \sum_{i = 1}^{N} ‖ D_{i} v ‖_{L^{p_{i}} (Ω)}^{p_{i}}, \end{aligned}$ where ${\overline{p}}^{} = \frac{N \overline{p}}{N - \overline{p}}$ .

The following technical Lemma, is a weighted anisotropic Sobolev type inequality, see [12].
Lemma 2.2.
Let* $\overline{p} < N$ and $v \in W_{0}^{1, (p_{i})} (Ω) \cap L^{\infty} (Ω)$ . Then there exists a positive constant C, depending only on $p_{i}, i = \overline{1, N}$ and N, such that $\begin{matrix} (2.3) & ‖ v ‖_{L^{r} (Ω)}^{r (\frac{N}{\overline{p}} - 1)} ⩽ C \prod_{i = 1}^{N} {(\int_{Ω} | D_{i} v |^{p_{i}} | v |^{t_{i} p_{i}} d x)}^{\frac{1}{p_{i}}}, \end{matrix}$ or every r and $t_{i} ⩾ 0$ satisfying $\begin{matrix} \{\begin{array}{ll} \frac{1}{r} = \frac{p_{i} (t_{i} + 1)}{p_{i} a_{i} (N - 1) - p_{i} + 1} > 0 & for all i = \overline{1, N}, \\ \sum_{i = 1}^{N} a_{i} = 1, a_{i} ⩾ 0 & for all i = \overline{1, N} . \end{array} \end{matrix}$

The problem (1.2) is related to the following Anisotropic Hardy type inequality, see [11].
Theorem 2.3 ([11]).

Let $v \in C_{0}^{1} (B)$ , $1 < p_{i} < N$ , $i = \overline{1, N}$ , $B = {x \in R^{N} such that x_{i} \neq 0, \forall i = \overline{1, N}}$ . Then we have $\begin{aligned} (2.4) & \sum_{i = 1}^{N} \int_{B} | D_{i} v |^{p_{i}} d x & ⩾ \sum_{i = 1}^{N} {(\frac{p_{i} - 1}{p_{i}})}^{p_{i}} \int_{B} \frac{| v |^{p_{i}}}{| x_{i} |^{p_{i}}} d x ⩾ M_{p^{-}, p^{+}} \sum_{i = 1}^{N} \int_{B} \frac{| v |^{p_{i}}}{| x_{i} |^{p_{i}}} d x, \end{aligned}$ where $M_{p^{-}, p^{+}} = {(\frac{p^{-} - 1}{p^{+}})}^{p^{-}}$ .

Corollary 2.4.
Let $v \in C_{0}^{1} (Ω)$ , $1 < p_{i} < N$ , $i = \overline{1, N}$ , then we have $\begin{aligned} (2.5) & \sum_{i = 1}^{N} \int_{Ω} | D_{i} v |^{p_{i}} d x ⩾ M_{p^{-}, p^{+}} \sum_{i = 1}^{N} \int_{Ω} \frac{| v |^{p_{i}}}{{(| x_{i} | + \frac{1}{n})}^{p_{i}}} d x, \end{aligned}$ where $M_{p^{-}, p^{+}} = {(\frac{p^{-} - 1}{p^{+}})}^{p^{-}}$ , for every $n \in N$ .

To prove Corollary 2.4, we need a lemma Lemma 2.5 ([11]).

If there exist a constant $k_{i} > 0$ and a function $h_{i} (x), i = \overline{1, N}$ , such that a differentiable function $v > 0$ in the set Ω satisfies $\begin{matrix} - Δ_{\vec{p}} v ⩾ k_{i} h_{i} (x) v^{p_{i} - 1}, \end{matrix}$ then, for any $0 ⩽ v \in C_{0}^{1} (Ω)$ , we have $\begin{matrix} \sum_{i = 1}^{N} \int_{Ω} | D_{i} v |^{p_{i}} d x ⩾ \sum_{i = 1}^{N} k_{i} \int_{Ω} h_{i} (x) v^{p_{i}} d x . \end{matrix}$

Proof of Corollary 2.4.
Let us consider $0 ⩽ u \in C_{0}^{\infty} (Ω)$ . To use Lemma 2.5, we introduce the auxiliary function $\begin{matrix} v = \prod_{j = 1}^{N} {(| x_{j} | + \frac{1}{n})}^{β_{j}} : = {(| x_{i} | + \frac{1}{n})}^{β_{i}} β_{i} {\overline{v}}_{i}, \end{matrix}$ where $β_{j} = \frac{p_{j} - 1}{p_{j}}$ and ${\overline{v}}_{i} = \prod_{j = 1, j \neq i}^{N} {(| x_{j} | + \frac{1}{n})}^{β_{j}}$ , hence $\begin{aligned} D_{i} v = β_{i} {\overline{v}}_{i} \frac{x_{i}}{| x_{i} |} {(| x_{i} | + \frac{1}{n})}^{β_{i} - 1}, \end{aligned}$ then $\begin{aligned} | D_{i} v |^{p_{i} - 2} D_{i} v = β_{i}^{p_{i} - 1} {\overline{v}}_{i}^{p_{i} - 1} {(| x_{i} | + \frac{1}{n})}^{β_{i} p_{i} - β_{i} - p_{i} + 1} \frac{x_{i}}{| x_{i} |}, \end{aligned}$ and $\begin{array}{r} - D_{i} (| D_{i} v |^{p_{i} - 2} D_{i} v) = {(\frac{p_{i} - 1}{p_{i}})}^{p_{i}} \frac{v^{p_{i} - 1}}{{(| x_{i} | + \frac{1}{n})}^{p_{i}}} . \end{array}$ Taking $k_{i} = {(\frac{p_{i} - 1}{p_{i}})}^{p_{i}}$ and $h_{i} (x) = \frac{1}{{(| x_{i} | + \frac{1}{n})}^{p_{i}}}$ and using Lemma 2.5 we obtain (2.5). □
Corollary 2.6.
Let $v \in C_{0}^{1} (Ω)$ , $1 < p_{i} < N$ , $i = \overline{1, N}$ , then we have $\begin{matrix} (2.6) & \int_{Ω} \frac{| v |^{p^{-}}}{{(| x | + \frac{1}{n})}^{p^{-}}} d x ⩽ M_{p^{-}, p^{+}}^{- 1} N^{- \frac{p^{-} + 2}{2}} \sum_{i = 1}^{N} \int_{Ω} | D_{i} v |^{p_{i}} d x + M_{p^{-}, p^{+}}^{- 1} N^{- \frac{p^{-}}{2}} | Ω | . \end{matrix}$
Proof.
In a similar way to the proof of Corollary 3.3 in [11] with $p = p^{-} ⩾ 2$ , we can write $\begin{matrix} (2.7) & \int_{Ω} \frac{| v |^{p^{-}}}{{(| x | + \frac{1}{n})}^{p^{-}}} d x ⩽ N^{- \frac{p^{-} + 2}{2}} \sum_{i = 1}^{N} \int_{Ω} \frac{| v |^{p^{-}}}{{(| x_{i} | + \frac{1}{n})}^{p^{-}}} d x . \end{matrix}$ Using (2.7), Corollary 2.4 with $p_{i} = p^{-}$ for all $i = \overline{1, N}$ and the fact that $| D_{i} v |^{p^{-}} ⩽ | D_{i} v |^{p_{i}} + 1$ for every $i = \overline{1, N}$ , we have $\begin{aligned} \int_{Ω} \frac{| v |^{p^{-}}}{{(| x | + \frac{1}{n})}^{p^{-}}} d x & ⩽ M_{p^{-}, p^{+}}^{- 1} N^{- \frac{p^{-} + 2}{2}} \sum_{i = 1}^{N} \int_{Ω} | D_{i} v |^{p^{-}} d x \\ (2.8) & ⩽ M_{p^{-}, p^{+}}^{- 1} N^{- \frac{p^{-} + 2}{2}} [\sum_{i = 1}^{N} \int_{Ω} | D_{i} v |^{p_{i}} d x + N | Ω |] . \end{aligned}$ By the previous inequality, we deduce (2.6). □

3. Statements of results and the approximated problem

Now, we give definition of weak solution for problem (1.2).

Definition 3.1.
A measurable function $u \in W_{0}^{1, (p_{i})} (Ω)$ is weak solution of (1.2) if $\begin{aligned} (3.1) & | D_{i} u |^{p_{i} - 1} \in L^{1} (Ω), \forall i = \overline{1, N}, μ \frac{| u |^{p^{-} - 1}}{| x |^{p^{-}}} + f \in L^{1} (Ω), \end{aligned}$ and one has the identity $\begin{matrix} (3.2) & \sum_{i = 1}^{N} \int_{Ω} | D_{i} u |^{p_{i} - 2} D_{i} u D_{i} φ d x = μ \int_{Ω} \frac{| u |^{p^{-} - 1}}{| x |^{p^{-}}} φ d x + \int_{Ω} f φ d x, \end{matrix}$ for all $φ \in C_{0}^{\infty} (Ω)$ .

The first result deals with a given f which yields unbounded solutions in energy space $W_{0}^{1, (p_{i})} (Ω)$ .
Theorem 3.2.
Assume that ( 1.1 ), ( 1.3 ) and ( 1.4 ) hold true. Let $f \in L^{m} (Ω)$ , such that $\begin{matrix} (3.3) & \frac{N \overline{p}}{N (\overline{p} - 1) + \overline{p}} ⩽ m < \frac{N}{\overline{p}} . \end{matrix}$ Then, there exists a weak solution $u \in L^{s} (Ω) \cap W_{0}^{1, (p_{i})} (Ω)$ to problem ( 1.2 ), where $\begin{matrix} (3.4) & s = \frac{N m (\overline{p} - 1)}{N - \overline{p} m} . \end{matrix}$
Remark 3.3.
Since $N > \overline{p}$ , the interval in (3.3) is not empty.

If $μ = 0$ the summability of the solution in Theorem 3.2 in the sense as in Theorem 3.1 in [8].

The next result deals with the case when the summability of f gives the existence of solution $u \in W_{0}^{1, (η_{i})}$ , with $1 < η_{i} < p_{i}$ for every $i = \overline{1, N}$ .
Theorem 3.4.
Assume that ( 1.1 ), ( 1.3 ) and ( 1.4 ) hold true. Let $f \in L^{m} (Ω)$ , such that $\begin{matrix} (3.5) & 1 < m < \frac{N \overline{p}}{N (\overline{p} - 1) + \overline{p}}, \end{matrix}$ and for all $i = \overline{1, N}$ $\begin{aligned} (3.6) & \frac{\overline{p} (N - m)}{m N (\overline{p} - 1)} < p_{i} < \frac{\overline{p} (N - m)}{\overline{p} (N - m) - m N (\overline{p} - 1)} . \end{aligned}$ Then, there exists a weak solution $u \in L^{{\overline{η}}^{}} (Ω) \cap W_{0}^{1, (η_{i})} (Ω)$ to problem (* 1.2 ), where $\begin{matrix} (3.7) & η_{i} = \frac{N m (\overline{p} - 1)}{\overline{p} (N - m)} p_{i} \forall i = \overline{1, N} . \end{matrix}$
Remark 3.5.
If $μ = 0$ the summability of the solution in Theorem 3.2 in the sense as in Theorem 3.3 in [8]. The hypothesis (3.5) implies $η_{i} < p_{i}$ for all $i = \overline{1, N}$ . By the assumption (3.6), we have $η_{i} > 1$ and $\frac{η_{i}}{p_{i} - 1} > 1$ for any $i = \overline{1, N}$ .
Remark 3.6 (see [6]).

Notice that if $f \in L^{m} (Ω)$ , $m ⩾ \frac{N}{\overline{p}}$ , even if $f \in L^{\infty} (Ω)$ , then any solution $u \in L^{1} (Ω)$ of (1.2) is unbounded. On the other hand Boccardo et al in [6] proved that the problem (1.2) is not well posed in $L^{1} (Ω)$ If $p_{i} = 2$ . for all $i = \overline{1, N}$ .

Remark 3.7.
In the extremal case $μ = M_{p^{+}, p^{-}} N^{\frac{p^{-} + 2}{2}}$ we not can prove the existence of a solution to problem (1.2). Although the existence of the solution in the isotropic case (i.e. $p_{i} = p$ for all $i = \overline{1, N}$ ) obtained in [1], by using the Improved Hardy–Sobolev inequality established in [2], this inequality has not proved in the case anisotropic.

Let us first consider the following approximating problems $\begin{aligned} (3.8) & \{\begin{array}{ll} - Δ_{\vec{p}} u_{n} = μ \frac{| T_{n} (u_{n}) |^{p^{-} - 1}}{{(| x | + \frac{1}{n})}^{p^{-}}} + f_{n} & in Ω, \\ u_{n} = 0 & on \partial Ω, \end{array} \end{aligned}$ where $f_{n} = T_{n} (f)$ . The existence of a weak solution $w \in W_{0}^{1, (p_{i})} (Ω) \cap L^{\infty} (Ω)$ to (3.8) is guaranteed by [8] for every $μ ⩾ 0$ .
4. A priori estimates

In this section, We shall denote by C various constants depending only on the structure of $p_{i}$ , μ, $| Ω |$ . Let $u_{n} \in W_{0}^{1, (p_{i})} (Ω) \cap L^{\infty} (Ω)$ be a solution to problem (3.8). In the following Lemma, we give the $L^{s}$ -estimates for the approximate solutions.

Lemma 4.1.
Let m, $p_{i}$ and μ be as in Theorem 3.2 . Then, there exists a positive constant C (not depending on n), such that $\begin{aligned} (4.1) & ‖ u_{n} ‖_{L^{s} (Ω)} ⩽ C, \\ (4.2) & ‖ u_{n} ‖_{W_{0}^{1, (p_{i})} (Ω)} ⩽ C . \end{aligned}$
Proof.
Let us use $φ = | u_{n} |^{λ - 2} u_{n}$ as test function in (3.8), $λ ⩾ 2$ to be defined later, such that $\begin{matrix} (4.3) & M_{1} = (λ - 1) - μ \frac{M_{p^{+}, p^{-}}^{- 1}}{N^{\frac{p^{-} + 2}{2}}} {(\frac{p^{-} + λ - 2}{p^{-}})}^{p^{-}} > 0 . \end{matrix}$ Using the fact that $| T_{n} (u_{n}) | ⩽ | u_{n} |$ , we have $\begin{array}{r} (λ - 1) \sum_{i = 1}^{N} \int_{Ω} | D_{i} u_{n} |^{p_{i}} | u_{n} |^{λ - 2} d x ⩽ μ \int_{Ω} \frac{| u_{n} |^{\frac{p^{-} + λ - 2}{p^{-}} p^{-}}}{{(| x | + \frac{1}{n})}^{p^{-}}} d x + \int_{Ω} | f_{n} | | u_{n} |^{λ - 1} d x . \end{array}$ Applying Hölder inequality and (2.8) on the right-hand side and that $\begin{aligned} {| D_{i} u_{n}^{\frac{p^{-} + λ - 2}{p^{-}}} |}^{p^{-}} & = {(\frac{p^{-} + λ - 2}{p^{-}})}^{p^{-}} | D_{i} u_{n} |^{p^{-}} | u_{n} |^{λ - 2} \\ ⩽ {(\frac{p^{-} + λ - 2}{p^{-}})}^{p^{-}} [| D_{i} u_{n} |^{p_{i}} | u_{n} |^{λ - 2} + | u_{n} |^{λ - 2}] \forall i = \overline{1, N}, \end{aligned}$ we get $\begin{aligned} (λ - 1) \sum_{i = 1}^{N} \int_{Ω} | D_{i} u_{n} |^{p_{i}} | u_{n} |^{λ - 2} d x \\ ⩽ μ \frac{M_{p^{+}, p^{-}}^{- 1}}{N^{\frac{p^{-} + 2}{2}}} \sum_{i = 1}^{N} \int_{Ω} {| D_{i} u_{n}^{\frac{p^{-} + λ - 2}{p^{-}}} |}^{p^{-}} d x + ‖ f ‖_{L^{m} (Ω)} {(\int_{Ω} | u_{n} |^{(λ - 1) m^{'}} d x)}^{\frac{1}{m^{'}}} \\ ⩽ μ \frac{M_{p^{+}, p^{-}}^{- 1}}{N^{\frac{p^{-} + 2}{2}}} {(\frac{p^{-} + λ - 2}{p^{-}})}^{p^{-}} [\sum_{i = 1}^{N} \int_{Ω} | D_{i} u_{n} |^{p_{i}} | u_{n} |^{λ - 2} d x + N \int_{Ω} | u_{n} |^{λ - 2} d x] \\ (4.4) & + ‖ f ‖_{L^{m} (Ω)} {(\int_{Ω} | u_{n} |^{(λ - 1) m^{'}} d x)}^{\frac{1}{m^{'}}} . \end{aligned}$ By (4.4), (4.3) and that $| u_{n} |^{λ - 2} ⩽ | u_{n} |^{λ - 1} + 1$ , we obtain for all $i = \overline{1, N}$ $\begin{array}{r} \int_{Ω} | D_{i} u_{n} |^{p_{i}} | u_{n} |^{λ - 1} d x ⩽ C \int_{Ω} | u_{n} |^{λ - 1} d x + C {(\int_{Ω} | u_{n} |^{(λ - 1) m^{'}} d x)}^{\frac{1}{m^{'}}} + C . \end{array}$ Using Hölder’s inequality, we have $\begin{array}{r} \int_{Ω} | D_{i} u_{n} |^{p_{i}} | u_{n} |^{λ - 1} d x ⩽ C {(\int_{Ω} | u_{n} |^{(λ - 1) m^{'}} d x)}^{\frac{1}{m^{'}}} + C, i = \overline{1, N} . \end{array}$ Now taking the product on i in the last inequality, we obtain $\begin{aligned} \prod_{i = 1}^{N} {(\int_{Ω} | D_{i} u_{n} |^{p_{i}} | u_{n} |^{λ - 2} d x)}^{\frac{1}{p_{i}}} & ⩽ \prod_{i = 1}^{N} {[C {(\int_{Ω} | u_{n} |^{(λ - 1) m^{'}} d x)}^{\frac{1}{m^{'}}} + C]}^{\frac{1}{p_{i}}} \\ = {[C {(\int_{Ω} | u_{n} |^{(λ - 1) m^{'}} d x)}^{\frac{1}{m^{'}}} + C]}^{\frac{N}{\overline{p}}} \\ (4.5) & ⩽ C {(\int_{Ω} | u_{n} |^{(λ - 1) m^{'}} d x)}^{\frac{N}{\overline{p} m^{'}}} + C . \end{aligned}$ Since we also want $s = (λ - 1) m^{'}$ , in (4.5), for any $i = \overline{1, N}$ , we have to solve the following system $\begin{matrix} \{\begin{array}{ll} t_{i} p_{i} = λ - 2, \\ s = \frac{(1 + t_{i}) p_{i}}{p_{i} a_{i} (N - 1) - p_{i} + 1}, & t_{i} ⩾ 0, \\ \sum_{i = 1}^{N} a_{i} = 1, & a_{i} ⩾ 0 . \end{array} \end{matrix}$ After a trivial computation, we get $\begin{aligned} (4.6) & λ = \frac{N (m - 1) (\overline{p} - 1) + N - m \overline{p}}{N - m \overline{p}}, and s = \frac{m N (\overline{p} - 1)}{N - m \overline{p}} . \end{aligned}$ By (2.3) of Lemma 2.2 with $v = | u_{n} |$ , $r = s$ and (4.5), we have $\begin{matrix} (4.7) & {(\int_{Ω} | u_{n} |^{s} d x)}^{\frac{N}{\overline{p}} - 1} ⩽ C_{7} {(\int_{Ω} | u_{n} |^{s} d x)}^{\frac{N}{\overline{p} m^{'}}} + C_{8} . \end{matrix}$ Since $m < \frac{N}{\overline{p}}$ , then $\frac{N}{\overline{p}} - 1 > \frac{N}{m^{'} \overline{p}}$ . Therefore, from (4.6) and (4.7), it follows (4.1), which implies that $\begin{matrix} (4.8) & \int_{Ω} | D_{i} u_{n} |^{p_{i}} | u_{n} |^{λ - 2} d x ⩽ C_{9} \forall i = \overline{1, N} . \end{matrix}$ Choosing $λ = 2$ in (4.8), we obtain (4.2). □
Lemma 4.2.
Let m, μ, $p_{i}$ be restricted as in Theorem 3.4 . Then, there exists a positive constant C (not depending on n), such that $\begin{aligned} (4.9) & ‖ u_{n} ‖_{L^{{\overline{η}}^{}} (Ω)} ⩽ C, \\ (4.10) & ‖ u_{n} ‖_{W_{0}^{1, (η_{i})} (Ω)} ⩽ C . \end{aligned}$
Proof.
Let $φ = [{(| u_{n} | + ε)}^{λ - 1} - ε^{λ - 1}] sgn (u_{n})$ , where $1 < λ < 2$ is defined in (4.6), such that (4.3) holds true. Taking φ as test function in (3.8), we get $\begin{aligned} (λ - 1) \sum_{i = 1}^{N} \int_{Ω} {(| u_{n} | + ε)}^{λ - 2} | D_{i} u_{n} |^{p_{i}} d x \\ ⩽ μ \int_{Ω} \frac{| u_{n} |^{p^{-} - 1} {(| u_{n} | + ε)}^{λ - 1}}{{(| x | + \frac{1}{n})}^{p^{-}}} d x + \int_{Ω} | f_{n} | | {(u_{n} + ε)}^{λ - 1} - ε^{λ - 1} | d x . \end{aligned}$ Using the fact that $u_{n}^{p^{-} - 1} {(u_{n} + ε)}^{λ - 1} ⩽ {(u_{n} + ε)}^{λ + p^{-} - 2}$ and Hölder’s inequality, we obtain $\begin{aligned} (λ - 1) \sum_{i = 1}^{N} \int_{Ω} \frac{| D_{i} u_{n} |^{p_{i}}}{{(| u_{n} | + ε)}^{2 - λ}} d x \\ ⩽ μ \int_{Ω} \frac{{(| u_{n} | + ε)}^{\frac{λ + p^{-} - 2}{p^{-}} p^{-}}}{{(| x | + \frac{1}{n})}^{p^{-}}} d x + ‖ f ‖_{L^{m} (Ω)} {(\int_{Ω} {(| u_{n} | + ε)}^{(λ - 1) m^{'}} d x)}^{\frac{1}{m^{'}}} . \end{aligned}$ Applying the anisotropic Hardy inequality on the right-hand side and using the fact that $\begin{matrix} {| D_{i} {(| u_{n} | + ε)}^{\frac{p^{-} + λ - 2}{p^{-}}} |}^{p^{-}} ⩽ {(\frac{p^{-} + λ - 2}{p^{-}})}^{p^{-}} [\frac{| D_{i} u_{n} |^{p_{i}}}{{(| u_{n} | + ε)}^{2 - λ}} + \frac{1}{{(| u_{n} | + ε)}^{2 - λ}}] \forall i = \overline{1, N}, \end{matrix}$ we find that $\begin{aligned} (λ - 1) \sum_{i = 1}^{N} \int_{Ω} \frac{| D_{i} u_{n} |^{p_{i}}}{{(| u_{n} | + ε)}^{2 - λ}} d x \\ ⩽ μ \frac{M_{p^{+}, p^{-}}^{- 1}}{N^{\frac{p^{-} + 2}{2}}} {(\frac{p^{-} + λ - 2}{p^{-}})}^{p^{-}} [\sum_{i = 1}^{N} \int_{Ω} \frac{| D_{i} u_{n} |^{p_{i}}}{{(| u_{n} | + ε)}^{2 - λ}} d x + N \int_{Ω} \frac{1}{{(| u_{n} | + ε)}^{2 - λ}} d x] \\ + ‖ f ‖_{L^{m} (Ω)} {(\int_{Ω} {(| u_{n} | + ε)}^{(λ - 1) m^{'}} d x)}^{\frac{1}{m^{'}}} . \end{aligned}$ By (4.3) and the inequality ${(| u_{n} | + ε)}^{2 - λ} > ε^{2 - λ}$ , we get $\begin{aligned} (4.11) & \sum_{i = 1}^{N} \int_{Ω} \frac{| D_{i} u_{n} |^{p_{i}}}{{(| u_{n} | + ε)}^{2 - λ}} d x ⩽ C {(\int_{Ω} {(| u_{n} | + ε)}^{(λ - 1) m^{'}} d x)}^{\frac{1}{m^{'}}} + \frac{C}{ε^{2 - λ}} . \end{aligned}$ From (4.11) and by Hölder’s inequality, we obtain $\begin{aligned} \int_{Ω} | D_{i} u_{n} |^{η_{i}} d x & = \int_{Ω} \frac{| D_{i} u_{n} |^{η_{i}}}{{(| u_{n} | + ε)}^{(2 - λ) \frac{η_{i}}{p_{i}}}} {(| u_{n} | + ε)}^{(2 - λ) \frac{η_{i}}{p_{i}}} d x \\ ⩽ C_{10} {(\int_{Ω} \frac{| D_{i} u_{n} |^{p_{i}}}{{(| u_{n} | + ε)}^{(2 - λ)}} d x)}^{\frac{η_{i}}{p_{i}}} \\ \times {(\int_{Ω} {(| u_{n} | + ε)}^{(2 - λ) \frac{η_{i}}{p_{i} - η_{i}}} d x + 1)}^{\frac{p_{i} - η_{i}}{p_{i}}} \\ ⩽ C_{11} {(\int_{Ω} {(| u_{n} | + ε)}^{(λ - 1) m^{'}} d x + 1)}^{\frac{η_{i}}{p_{i} m^{'}}} \\ (4.12) & \times {(\int_{Ω} {(| u_{n} | + ε)}^{(2 - λ) \frac{η_{i}}{p_{i} - η_{i}}} d x + 1)}^{\frac{p_{i} - η_{i}}{p_{i}}} \forall i = \overline{1, N} . \end{aligned}$ Now we take $η_{i} = θ p_{i}$ , with $θ \in [0, 1)$ such that $\begin{matrix} (λ - 1) m^{'} = \frac{(2 - λ) η_{i}}{p_{i} - η_{i}} = \frac{(2 - λ) θ}{1 - θ} \forall i = \overline{1, N}, \end{matrix}$ the previous equality is equivalent to $\begin{matrix} (4.13) & (λ - 1) m^{'} = \frac{m θ}{m - θ} . \end{matrix}$ Thus, we choose $\begin{matrix} 1 < 1 + \frac{θ (m - 1)}{m - θ} = λ = 2 - \frac{m (1 - θ)}{m - θ} < 2 . \end{matrix}$ Therefore, by (4.12) and (4.13) we can write $\begin{matrix} {(\int_{Ω} | D_{i} u_{n} |^{η_{i}} d x)}^{\frac{1}{N η_{i}}} ⩽ C {(\int_{Ω} {(| u_{n} | + ε)}^{\frac{m θ}{m - θ}} d x + 1)}^{(\frac{η_{i}}{p_{i} m^{'}} + \frac{p_{i} - η_{i}}{p_{i}}) \frac{1}{N η_{i}}} \forall i = \overline{1, N} . \end{matrix}$ By anisotropic Sobolev inequality (2.1) with $r = {\overline{η}}^{}$ , we have $\begin{aligned} (4.14) & {(\int_{Ω} | u_{n} |^{{\overline{η}}^{}} d x)}^{\frac{N - θ \overline{p}}{N θ \overline{p}}} ⩽ C_{12} {(\int_{Ω} | u_{n} |^{\frac{m θ}{m - θ}} d x)}^{(\frac{θ (m - 1)}{m} + 1 - θ) \frac{1}{θ \overline{p}}} + C_{13} . \end{aligned}$ Since $m < \frac{N}{\overline{p}}$ , we obtain $\begin{matrix} (4.15) & \frac{N - θ \overline{p}}{N θ \overline{p}} > (θ \frac{m - 1}{m} + 1 - θ) \frac{1}{θ \overline{p}}, \end{matrix}$ that, implies $\begin{matrix} (4.16) & \frac{m θ}{m - θ} = {\overline{η}}^{} \Leftrightarrow θ = \frac{m N (\overline{p} - 1)}{\overline{p} (N - m)} < 1 . \end{matrix}$ Hence, it follows from (4.14), (4.15) and (4.16) that (4.9) is proved. Finally, since $η_{i} = θ p_{i}$ for all $i = \overline{1, N}$ we obtain (4.10) from (4.12). This proves Lemma 4.2. □

5. Proof of the main results

Because the proofs of Theorem 3.2, are similar to that of Theorem 3.4, we restrict to the proof of Theorem 3.4. By Lemma 4.1, there exist a subsequence of ${u_{n}}_{n}$ (still denoted by ${u_{n}}_{n}$ ), and $u \in W_{0}^{1, (η_{i})} (Ω)$ , such that $\begin{aligned} (5.1) & u_{n} ⇀ u weakly in W_{0}^{1, (η_{i})} (Ω), a.e. in Ω, \\ (5.2) & D_{i} u_{n} ⇀ D_{i} u weakly in L^{η_{i}} (Ω) and a.e. in Ω, \\ (5.3) & u_{n} \to u strongly in L^{\overline{η}} (Ω) . \end{aligned}$ Now, adapting the approach of the proof of Theorem 2.3 in [8], there exists a subsequence (still denoted ${u_{n}}_{n}$ ) such that for all $i = \overline{1, N}$ $\begin{aligned} (5.4) & D_{i} u_{n} \to D_{i} u strongly in L^{r_{i}} (Ω) and a.e. in Ω, \forall r_{i} < η_{i} . \end{aligned}$ Now we fix $φ \in C_{0}^{\infty} (Ω)$ , by (5.1), (5.2), (5.4) and Lebesgue’s Theorem, we obtain $\begin{aligned} (5.5) & lim_{n \to \infty} \sum_{i = 1}^{N} \int_{Ω} | D_{i} u_{n} |^{p_{i} - 2} D_{i} u_{n} D_{i} φ d x = \sum_{i = 1}^{N} \int_{Ω} | D_{i} u |^{p_{i} - 2} D_{i} u D_{i} φ d x . \end{aligned}$ Let $1 < γ < \frac{η^{-}}{p^{-} - 1}$ ( $p^{-} - 1 < η^{-}$ ). Thanks Young’s inequality with exponent $\frac{η^{-}}{γ (p^{-} - 1)}$ , Hölder’s inequality with exponent $\frac{{\overline{η}}^{*}}{η^{-}}$ and (4.9), we have $\begin{aligned} \int_{Ω} {| \frac{| T_{n} (u_{n}) |^{p^{-} - 1}}{{(| x | + \frac{1}{n})}^{p^{-}}} |}^{γ} d x & ⩽ \int_{Ω} | u_{n} |^{γ (p^{-} - 1)} \frac{1}{| x |^{γ p^{-}}} \\ ⩽ \frac{γ (p^{-} - 1)}{η^{-}} \int_{Ω} | u_{n} |^{η^{-}} d x + \int_{Ω} \frac{1}{| x |^{\frac{γ η^{-} p^{-}}{η^{-} - γ (p^{-} - 1)}}} d x \\ ⩽ \frac{γ (p^{-} - 1)}{η^{-}} | Ω |^{1 - \frac{η^{-}}{{\overline{η}}^{*}}} {(\int_{Ω} | u_{n} |^{{\overline{η}}^{*}} d x)}^{\frac{η^{-}}{{\overline{η}}^{*}}} \\ + \frac{η^{-} - γ (p^{-} - 1)}{η^{-}} \int_{Ω} \frac{1}{| x |^{\frac{γ η^{-} p^{-}}{η^{-} - γ (p^{-} - 1)}}} d x \\ (5.6) & ⩽ c_{1} + c_{2} \int_{Ω} \frac{1}{| x |^{\frac{γ η^{-} p^{-}}{η^{-} - γ (p^{-} - 1)}}} d x . \end{aligned}$ Using (5.6), and the fact that $\begin{matrix} \frac{γ η^{-} p^{-}}{η^{-} - γ (p^{-} - 1)} < N \Leftrightarrow 1 < γ < \frac{N η^{-}}{p^{-} η^{-} + (p^{-} - 1) N} < \frac{η^{-}}{p^{-} - 1}, \end{matrix}$ we obtain for every $1 < γ < \frac{N η^{-}}{p^{-} η^{-} + (p^{-} - 1) N}$ $\begin{aligned} (5.7) & \int_{Ω} {| \frac{| T_{n} (u_{n}) |^{p^{-} - 1}}{{(| x | + \frac{1}{n})}^{p^{-}}} |}^{γ} d x ⩽ c_{3}, \forall n \in N . \end{aligned}$ Therefore, using (5.7), (5.1) and applying Lebesgue’s dominated convergence theorem with respect to the index n, we can conclude that $\begin{matrix} (5.8) & \frac{| T_{n} (u_{n}) |^{p^{-} - 1}}{{(| x | + \frac{1}{n})}^{p^{-}}} \to \frac{| u |^{p^{-} - 1}}{| x |^{p^{-}}} strongly in L^{γ} (Ω) . \end{matrix}$ Choosing now $φ \in W_{0}^{1, (p_{i})} (Ω) \cap L^{\infty} (Ω)$ as a test function in (3.8), by the convergences results (5.5), (5.8), $f_{n} \to f$ in $L^{1} (Ω)$ and letting $n \to + \infty$ , we get $\begin{matrix} \sum_{i = 1}^{N} \int_{Ω} | D_{i} u |^{p_{i} - 2} D_{i} u D_{i} φ d x = μ \sum_{i = 1}^{N} \int_{Ω} \frac{| u |^{p_{i} - 1}}{| x_{i} |^{p_{i}}} d x + \int_{Ω} f φ d x . \end{matrix}$

Remark 5.1.

All the results in this work also hold if our problem is exchanged by a more general one, $\begin{matrix} (5.9) & \{\begin{array}{ll} - \sum_{i = 1}^{N} D_{i} [a_{i} (x, u, \nabla u)] = \frac{| u |^{p^{-} - 1}}{| x |^{p^{-}}} + f & in Ω, \\ u = 0 & on \partial Ω, \end{array} \end{matrix}$ where $a_{i} : Ω \times R \times R^{N} \to R$ , ( $\forall i = \overline{1, N}$ ) are Carathéodory functions satisfying for almost every x in Ω, for every $σ \in R$ , for all $ξ, η \in R^{N}$ and for any $i = \overline{1, N}$ the following $\begin{aligned} a_{i} (x, σ, ξ) \cdot ξ_{i} ⩾ α | ξ_{i} |^{p_{i}}, \\ | a_{i} (x, σ, ξ_{i}) | ⩽ β (k (x) + | σ |^{p_{i} - 1} + | ξ_{i} |^{p_{i} - 1}), k \in L^{p_{i}^{'}} (Ω), \\ [a_{i} (x, σ, ξ) - a_{i} (x, σ, η)] \cdot (ξ - η) > 0, ξ \neq η, \end{aligned}$ where $α, β > 0$ .

References

Abdellaoui and

Attar, Quasilinear elliptic problem with Hardy potential and singular term, Commun. Pure Appl. Anal 12(3) (2013), 1363–1380. doi:10.3934/cpaa.2013.12.1363.

Abdellaoui,

Colorado and

Peral, Some improved Caffarelli–Kohn–Nirenberg inequalities, Calc. Var 23 (2005), 327–345. doi:10.1007/s00526-004-0303-8.

Bear, Dynamics of Fluids in Porous Media, American Elsevier, New York, 1972.

Bendahmane,

Langlais and

Saad, On some anisotropic reaction-diffusion systems with

L^{1}

-data modeling the propagation of an epidemic disease, Nonlinear Anal 54(4) (2003), 617–636. doi:10.1016/S0362-546X(03)00090-7.

Boccardo,

Gallouët and

Marcellini, Anisotropic equations in

L^{1}

, Differential Integral Equations 9 (1996), 209–212.

Boccardo,

Orsina and

Peral, A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential, DCDS 16(3) (2006), 513–523. doi:10.3934/dcds.2006.16.513.

Dávila and

Peral, Nonlinear elliptic problems with a singular weight on the boundary, Calc. Var 41 (2011), 576–586.

Di Castro, Existence and regularity results for anisotropic elliptic problems, Adv. Nonlin. Stud 9 (2009), 367–393. doi:10.1515/ans-2009-0207.

S.N.

Kruzhkov and

I.M.

Kolodii, On the theory of embedding of anisotropic Sobolev spaces, Russ. Math. Surv 38 (1983), 188–189. doi:10.1070/RM1983v038n02ABEH003476.

10.

S.M.

Nikolskii, Imbedding theorems for functions with partial derivatives considered in various metrics, Izd. Akad. Nauk SSSR 22 (1958), 321–336.

11.

Tingfu and

Xuewei, Anisotropic Picone identities and anisotropic Hardy inequalities, J. Inequalities Appl 2017 (2017), 1–9. doi:10.1186/s13660-016-1272-0.

12.

Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat 18(3) (1969), 3–24.