Nonstandard competing anisotropic $(p,q)$-Laplacians with convolution

Abstract

A competing anisotropic $(p,q)$-Laplacian

$$ -\overset{N}{\underset{i=1}{\sum}}\frac{\partial }{\partial x_{i}} \biggl( \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}-2}- \mu \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{q_{i}-2} \biggr) \frac{\partial u}{\partial x_{i}} =f \bigl(x, \phi \star u,\nabla (\phi \star u)\bigr) $$

as a nonstandard Dirichlet problem with convolutions on a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq 3$ is considered. Assume $f:\Omega \times \mathbb{R}\times \mathbb{R}^{N}\to \mathbb{R}$ is a Carathéodory function and $\phi \in L^{1}(\mathbb{R}^{N})$. If $\mu >0$, the existence of a generalized solution is proved. By the Galerkin basis for the space, a sequence that converges strongly to the solution is constructed. If $\mu \leq 0$, it is proved that any generalized solution is a weak solution.

1 Introduction

The $(p,q)$-Laplacian comes from a general reaction-diffusion system that has a wide spectrum of applications in physics and related sciences such as biophysics, plasma physics, solid-state physics, fractional quantum mechanics in the study of particles on stochastic fields, fractional superdiffusion and fractional white-noise limit, etc. (see [1, 5–7, 23–25, 31, 32] and the references therein).

Recently, Motreanu [20] proved the existence of solutions (generalized and weak) for

$$ \textstyle\begin{cases} -\operatorname{div} ( \vert \nabla u \vert ^{p-2}\nabla u-\mu \vert \nabla u \vert ^{q-2}\nabla u )=f(x,\rho \star u,\nabla (\rho \star u))&\text{in } \Omega , \\ u=0&\text{on } \partial \Omega , \end{cases} $$

under suitable condition of f and ρ, where he overcame the lack of ellipticity.

Here, with the inspiration of [20], the multiplicity of nontrivial solutions for the nonstandard Dirichlet problem with an anisotropic competing $(p, q)$-Laplacian

$$ \textstyle\begin{cases} -\overset{N}{\underset{i=1}{\sum}}\frac{\partial }{\partial x_{i}} ( \vert \frac{\partial u}{\partial x_{i}} \vert ^{p_{i}-2}- \mu \vert \frac{\partial u}{\partial x_{i}} \vert ^{q_{i}-2} ) \frac{\partial u}{\partial x_{i}} =f(x, \phi \star u,\nabla (\phi \star u))&\text{in } \Omega , \\ u=0&\text{on } \partial \Omega , \end{cases} $$

(1.1)

is proved, where Ω is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq 3$, with a Lipschitz boundary ∂Ω, $f:\Omega \times \mathbb{R}\times \mathbb{R}^{N}\to \mathbb{R}$ is a Carathéodory function, $\phi \in L^{1}(\mathbb{R}^{N})$, $u\in W_{0}^{1,\overrightarrow{p}}(\Omega )$, and the convolution $\phi \star u(x)$ is defined by

$$ \phi \star u(x):= \int _{\mathbb{R}^{N}}\phi (x-y)u(y)\,dy\quad \text{for a.e. } x\in \mathbb{R}^{N}. $$

We set $\overrightarrow{p}:=(p_{1},\ldots , p_{N})$ and $\overrightarrow{q}:=(q_{1},\ldots , q_{N})$ where

$$ \begin{aligned} &1< p_{1}, p_{2},\ldots ,p_{N}, \quad \sum^{N}_{i=1} \frac{1}{p_{i}}>1, \\ &1< q_{1}, q_{2}, \ldots , q_{N}, \quad \sum ^{N}_{i=1}\frac{1}{q_{i}}>1. \end{aligned} $$

Let p̅ and q̅ denote the harmonic means $\overline{p}= N/ (\sum^{N}_{i=1}\frac{1}{p_{i}} )$ and $\overline{q}= N/ (\sum^{N}_{i=1}\frac{1}{q_{i}} )$, respectively, and define

$$\begin{aligned}& p^{\star }:=\frac{N}{ (\sum^{N}_{i=1}\frac{1}{p_{i}} )-1}= \frac{N\overline{p}}{N-\overline{p}},\qquad q^{\star }:= \frac{N}{ (\sum^{N}_{i=1}\frac{1}{q_{i}} )-1}= \frac{N\overline{q}}{N-\overline{q}}, \\& p_{\infty}:=\max \bigl\{ p_{+},p^{\star }\bigr\} \quad \text{and}\quad p_{+}:=\max \{p_{i}: i=1,\ldots , N\}. \end{aligned}$$

We define an order as follows:

$$ \overrightarrow{q}\leq \overrightarrow{p} \quad \text{if and only if}\quad q_{i}\leq p_{i}\quad \text{for all } i=1,\ldots , N. $$

(1.2)

Throughout the paper, we assume that

$$ \overrightarrow{q}\leq \overrightarrow{p},\qquad q_{N} < q^{\star },\qquad p_{N}< p^{ \star }\quad \text{and}\quad q^{\star }< p^{\star }. $$

(1.3)

Also, we assume

$(H_{1})$:

$|f(x,t,\xi )|\leq \sigma (x)+c_{1}|t|^{p^{+}-1}+c_{2} \overset{N}{\underset{i=1}{\sum}}|\xi _{i}|^{p_{i}-1}$ for a.e. $x\in \Omega$ and for all $(t,\xi )\in \mathbb{R}\times \mathbb{R}^{N}$, where $\xi =(\xi _{1},\ldots ,\xi _{N})$, $\sigma \in L^{\gamma '}(\Omega )$ for $\gamma \in (1,p^{+})$, $\gamma '=\frac{\gamma}{\gamma -1}$ and constants $c_{1}\geq 0$, $c_{2}\geq 0$, satisfying

$$ \Vert \phi \Vert _{L^{1}(\mathbb{R}^{N})}^{p^{+}-1} c_{1}S_{p^{+}}+c_{2}\Pi < 1, $$

(1.4)

where $\Pi =\max_{1\leq i\leq N}\{S^{\prime}_{p_{i}}\|\phi \|^{p_{i}-1}_{L^{1}( \mathbb{R}^{N})}\}$ and $S^{\prime}_{p_{i}}$ is the Sobolev constant for the embedding $W_{0}^{1,p_{i}}(\Omega )\subset L^{p_{i}}(\Omega )$ for $i=1,\ldots , N$.

The differential operator in (1.1), i.e.,

$$ u\to \overset{N}{\underset{i=1}{\sum}} \frac{\partial }{\partial x_{i}} \biggl( \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}-2}-\mu \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{q_{i}-2} \biggr) \frac{\partial u}{\partial x_{i}} $$

is the difference of the anisotropic degenerated p-Laplacian and q-Laplacian. In fact, the negative anisotropic ϱ-Laplacian (for $\varrho =p,q$)

$$ -\Delta _{\overrightarrow{\varrho}}: W_{0}^{1, \overrightarrow{\varrho}}(\Omega )\to W^{-1,\overrightarrow{\varrho}^{ \prime}(\Omega )} $$

is expressed as

$$ \langle -\Delta _{\overrightarrow{\varrho}}u,v\rangle = \overset{N}{\underset{i=1}{\sum }} \int _{\Omega} \frac{\partial }{\partial x_{i}} \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{\varrho _{i}-2} \frac{\partial u}{\partial x_{i}}\cdot \frac{\partial v}{\partial x_{i}}\,dx $$

for all $u,v\in W_{0}^{1,\overrightarrow{\varrho}}(\Omega )$, where $\overrightarrow{\varrho}:=(\varrho _{1},\ldots ,\varrho _{N})$ and $\overrightarrow{\varrho}^{\prime}:=( \frac{\varrho _{1}}{\varrho _{1}-1},\ldots , \frac{\varrho _{N}}{\varrho _{N}-1})$.

Since $1< q_{1}$, $\overrightarrow{q}<\overrightarrow{p}$, $p_{N}<\infty$, the continuous embedding $W_{0}^{1,\overrightarrow{p}}(\Omega )\hookrightarrow W_{0}^{1, \overrightarrow{q}}(\Omega )$ holds and the operator $-\Delta _{\overrightarrow{p}}+\mu \Delta _{\overrightarrow{q}}$ is well defined on $W_{0}^{1,\overrightarrow{p}}(\Omega )$.

The sign of $-\Delta _{\overrightarrow{p}}+\mu \Delta _{\overrightarrow{q}}$ for $\mu >0$ and sufficiently large is different from $\mu >0$ and sufficiently small. This makes it difficult to study (1.1). We owe essential ideas to [20] to overcome the lack of ellipticity, monotonicity, and variational structure in problem (1.1) (see [18–20, 22]). Therefore, for problem (1.1), the existence of a solution is proved by Theorem 1.1.

Theorem 1.1

Suppose that $(H_{1})$ holds. Then, there exists a generalized solution to problem (1.1). In particular, if $\mu \leq 0$, there exists a weak solution to problem (1.1).

The rest of the paper is organized as follows: In Sect. 2, the suitable function spaces and some lemmas are recalled. In Sect. 3, the associated Nemytskij operator is introduced and then we show the anisotropic competing $(p,q)$-Laplacian (1.1) has a solution, i.e., the proof of Theorem 1.1 is presented.

2 Function space

Consider the anisotropic Sobolev spaces $W^{1,\overrightarrow{p}}(\Omega )$, with the norm

$$ \Vert u \Vert _{W^{1,\overrightarrow{p}}(\Omega )}:= \int _{\Omega } \bigl\vert u(x) \bigr\vert \,dx+ \sum _{i=1}^{N} \biggl( \int _{\Omega } \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{\frac{1}{p_{i}}}, $$

and $W^{1,\overrightarrow{p}}_{0}(\Omega )$ with the norm

$$ \begin{aligned} \Vert u \Vert _{W^{1,\overrightarrow{p}}_{0}(\Omega )}&:=\sum _{i=1}^{N} \biggl( \int _{\Omega } \biggl\vert \frac{\partial u}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{\frac{1}{p_{i}}} \\ &=\sum_{i=1}^{N} \Vert u \Vert _{W^{1,p_{i}}_{0}(\Omega )}. \end{aligned} $$

Note that $W^{1,\overrightarrow{p}}_{0}(\Omega )$ is a reflexive and uniformly convex Banach space (see [26–28] and references therein for more details or more literature in [2, 4, 8–14, 30]). Here, is an embedding theorem [15, Theorem 1].

Theorem 2.1

Let $\Omega \subset \mathbb{R}^{N}$ be an open bounded domain with Lipschitz boundary. If

$$ p_{i}>1, \quad \textit{for all } i=1,\ldots ,N, \qquad \sum _{i=1}^{N} \frac{1}{p_{i}}>1, $$

then for all $r\in [1,p_{\infty}]$, there is a continuous embedding $W_{0}^{1,\overrightarrow{p}}(\Omega )\subset L^{r}(\Omega )$. For $r< p_{\infty}$, the embedding is compact.

Note that the Sobolev space $W^{1,\overrightarrow{p}}_{0}(\Omega )$ is embedded in $W^{1,\overrightarrow{p}}(\mathbb{R}^{N})$ by identifying every $u\in W^{1,\overrightarrow{p}}_{0}(\Omega )$ with its extension equal to zero outside Ω. Thus, one can define the convolution $\phi \star u$ of $\phi \in L^{1}(\mathbb{R}^{N})$ with $u\in W^{1,\overrightarrow{p}}_{0} (\Omega )$ (see [3, Sect. 4.4 and Sect. 9.1]) by

$$ \phi \star u(x)= \int _{\mathbb{R}^{N}}\phi (x-y)u(y)\,dy\quad \text{for a.e. } x\in \mathbb{R}^{N}. $$

Also,

$$ \frac{\partial}{\partial x_{i}}(\phi \star u) =\phi \star \frac{\partial u}{\partial x_{i}}\in L^{p_{i}}\bigl(\mathbb{R}^{N}\bigr), \text{for all} i=1,2,\ldots , N. $$

Remark 2.2

Assume $\phi \in L^{1}(\mathbb{R}^{N})$ with $u\in W^{1,\overrightarrow{p}}_{0} (\Omega )$, then

• (i)
  $$ \Vert \phi \star u \Vert _{L^{r}(\mathbb{R}^{N})}\leq \Vert \phi \Vert _{L^{1}( \mathbb{R}^{N})} \Vert u \Vert _{L^{r}(\Omega )} $$

  (2.1)

  whenever $r\in [1,p^{\star }]$;

• (ii)
  $$ \biggl\Vert \phi \star \frac{\partial u}{\partial x_{i}} \biggr\Vert _{L^{p_{i}}( \mathbb{R}^{N})}\leq \Vert \phi \Vert _{L^{1}(\mathbb{R}^{N})} \biggl\Vert \frac{\partial u}{\partial x_{i}} \biggr\Vert _{L^{p_{i}}(\Omega )} $$

  (2.2)

  for all $i=1,\ldots , N$;

• (iii) By (2.2), we have

  $$ \begin{aligned} \Vert \phi \star u \Vert _{W_{0}^{1\overrightarrow{p}}(\mathbb{R}^{N})}={}& \sum_{i=1}^{N} \biggl( \int _{\mathbb{R}^{N}} \biggl\vert \frac{\partial (\phi \star u)}{\partial x_{i}} \biggr\vert ^{p_{i}}\,dx \biggr)^{ \frac{1}{p_{i}}} \\ ={}& \sum_{i=1}^{N} \biggl\Vert \frac{\partial (\phi \star u)}{\partial x_{i}} \biggr\Vert _{L^{p_{i}}( \mathbb{R}^{N})} \\ \leq {}& \sum_{i=1}^{N} \Vert \phi \Vert _{L^{1}(\mathbb{R}^{N})} \biggl\Vert \frac{\partial u}{\partial x_{i}} \biggr\Vert _{L^{p_{i}}(\mathbb{R}^{N})} \\ = {}& \Vert \phi \Vert _{L^{1}(\mathbb{R}^{N})}\sum_{i=1}^{N} \biggl\Vert \frac{\partial u}{\partial x_{i}} \biggr\Vert _{L^{p_{i}}(\mathbb{R}^{N})} \\ ={}& \Vert \phi \Vert _{L^{1}(\mathbb{R}^{N})} \Vert u \Vert _{W_{0}^{1\overrightarrow{p}}( \mathbb{R}^{N})}. \end{aligned} $$

  (2.3)

Before ending this section we require a generalized solution for (1.1).

Definition 2.3

A function $u\in W^{1,\overrightarrow{p}}_{0}(\Omega )$ is called a generalized solution to problem (1.1) if there exists a sequence $\{u_{n}\}_{n\geq 1}$ in $W^{1,\overrightarrow{p}}_{0}(\Omega )$ such that

• (I) $u_{n}\rightharpoonup u$ in $W^{1,\overrightarrow{p}}_{0}(\Omega )$ as $n\to \infty$;

• (II) $-\Delta _{\overrightarrow{p}}u_{n}+\mu \Delta _{\overrightarrow{q}}u_{n}-f (\cdot , \phi \star u_{n}(\cdot ),\nabla (\phi \star \nabla u)(\cdot )) \rightharpoonup 0$ in $W^{-1,\overrightarrow{p}^{\prime}}(\Omega )$ as $n\to \infty$;

• (III) $\lim_{n\to \infty}\langle -\Delta _{\overrightarrow{p}}u_{n}+ \mu \Delta _{\overrightarrow{q}}u_{n}, u_{n}-u\rangle =0$.

Remark 2.4

Assume u is a weak solution of (1.1), i.e., u satisfies

$$ \bigl\langle (-\Delta _{\overrightarrow{p}}+\mu \Delta _{ \overrightarrow{q}} ) (u),v\bigr\rangle _{W_{0}^{1,\overrightarrow{p}}( \Omega )}= \int _{\Omega }f\bigl(x,\phi \star u(x),\nabla \bigl(\phi \star u(x) \bigr)\bigr)v(x)\,dx $$

for all $v\in W^{1,\overrightarrow{p}}_{0}(\Omega )$. Set $u_{n} = u$ for all n, then any weak solution is a generalized solution to problem (1.1).

3 Weak and generalized solutions

Here, we study the behavior of the Nemytskij operator and construct a sequence (by the Galerkin basis of the space) that converges strongly to the generalized (weak) solution of (1.1) when $\mu \geq 0$ ($\mu <0$). First, we recall an embedding result.

Since $\overrightarrow{q}<\overrightarrow{p}$ and Ω is bounded then

$$ \begin{aligned} & W_{0}^{1\overrightarrow{p}}(\Omega )\quad \text{is continuously embedded in } W_{0}^{1\overrightarrow{q}}( \Omega ) \quad \text{and } \\ &W^{-1,\overrightarrow{q}^{\prime}}(\Omega )\quad \text{is continuously embedded in } W^{-1,\overrightarrow{p}^{\prime}}( \Omega ). \end{aligned} $$

(3.1)

Assume the operator $A: W_{0}^{1,\overrightarrow{p}}(\Omega )\to W^{-1,\overrightarrow{p}^{ \prime}}(\Omega )$ (see (1.1)) is defined by

$$ \bigl\langle A(u),v\bigr\rangle =\langle -\Delta _{\overrightarrow{p}}u+\mu \Delta _{\overrightarrow{q}}u,v\rangle - \int _{\Omega}f\bigl(x, \phi \star u(x), \nabla (\phi \star u) (x) \bigr)v(x)\,dx. $$

(3.2)

Lemma 3.1

The operator A defined by (3.2) is continuous, when ($H_{1}$) holds.

Proof

Define the operator

$$ T: W_{0}^{1,\overrightarrow{p}}(\Omega )\to L^{p^{+}}(\Omega )\times L^{p_{1}}( \Omega )\times \cdots \times L^{p_{N}}(\Omega ) $$

by $T(u)=(\phi \star u |_{\Omega}, \nabla (\phi \star u) |_{ \Omega})$. Relations (2.1) and (2.3) imply that T is linear and continuous. By ($H_{1}$) and Krasnoselskii’s theorem [16], the Nemytskii operator

$$ \begin{aligned} \mathcal{N}:{}&L^{p^{+}}(\Omega )\times \bigl(L^{{p_{1}}}( \Omega )\times \cdots \times L^{{p_{N}}}(\Omega ) \bigr)\to L^{{p^{+}}^{ \prime}}(\Omega ) \\ &{}(v,,w_{1},\ldots ,w_{N})\mapsto f\bigl(\cdot ,v(\cdot ),w_{1}(\cdot ), \ldots ,w_{N}(\cdot )\bigr) \end{aligned} $$

is well defined and continuous and so the composition operator

$$ W_{0}^{1,\overrightarrow{p}}(\Omega )\to L^{{p^{+}}^{\prime}}( \Omega ), \qquad u\mapsto f\bigl(\cdot ,\phi \star u(\cdot ),\nabla (\phi \star u) ( \cdot )\bigr) $$

(3.3)

is continuous. Note that $L^{{p^{+}}^{\prime}}(\Omega )$ is continuously embedded in $W^{-1,{p^{+}}^{\prime}}(\Omega )$.

The operator $-\Delta _{\overrightarrow{\varrho}}: W_{0}^{1, \overrightarrow{\varrho}}(\Omega )\to W^{-1,\overrightarrow{\varrho}^{ \prime}}(\Omega )$ (for $\varrho =p,q$) is continuous. Therefore, embedding (3.1) implies $-\Delta _{\overrightarrow{p}}+\mu \Delta _{\overrightarrow{q}}: W_{0}^{1 \overrightarrow{p}}(\Omega )\to W^{-1,\overrightarrow{p}^{\prime}}( \Omega )$ is continuous and finally the operator A is continuous. □

Assume $\{X_{n}\}$ (vector subspaces of $W_{0}^{1,\overrightarrow{p}}(\Omega )$) is a Galerkin basis for the separable Banach space $W_{0}^{1,\overrightarrow{p}}(\Omega )$, i.e.,

• (i) $\dim (X_{n})<\infty$, for all n;

• (ii) $X_{n}\subset X_{n+1}$, for all n;

• (iii) $\overline{\underset{n}{\cup}X_{n}}=W_{0}^{1,\overrightarrow{p}}( \Omega )$.

A consequence of Brouwer’s fixed-point theorem will resolve each approximate problem on $X_{n}$. Due to this, we construct a sequence $\{u_{n}\}$ by the next Proposition.

Proposition 3.2

Assume ($H_{1}$) holds. Then, for each $n\geq 1$ there exists $u_{n}\in X_{n}$ such that

$$ \begin{aligned} \bigl\langle (-\Delta _{\overrightarrow{p}}+ \mu \Delta _{ \overrightarrow{q}} ) (u_{n}),v\bigr\rangle _{W_{0}^{1, \overrightarrow{p}}(\Omega )} = \int _{\Omega }f\bigl(x,\phi \star u_{n}(x), \nabla \bigl(\phi \star u_{n}(x)\bigr)\bigr)v(x)\,dx \end{aligned} $$

(3.4)

for all $v\in X_{n}$. In addition, $\{u_{n}\}_{n\geq 1}$ is bounded in $W_{0}^{1,\overrightarrow{p}}(\Omega )$.

Proof

We define $A_{n}:X_{n}\to X_{n}^{\star }$ by

$$ \begin{aligned} &\bigl\langle A_{n}(u),v\bigr\rangle _{X_{n}} \\ &\quad =\bigl\langle (-\Delta _{\overrightarrow{p}}+\mu \Delta _{ \overrightarrow{q}} ) (u),v\bigr\rangle _{W_{0}^{1,\overrightarrow{p}}( \Omega )}- \int _{\Omega }f\bigl(x,\phi \star u(x),\nabla \bigl(\phi \star u(x) \bigr)\bigr)v(x)\,dx \end{aligned} $$

for all $u,v\in X_{n}$ and all $n\in \mathbb{N}$. The operator $A_{n}$ is continuous (by Lemma 3.1) and

$$ \begin{aligned} &\bigl\langle A_{n}(v),v\bigr\rangle _{X_{n}} \\ &\quad =\overset{N}{\underset{i=1}{\sum}} \int _{\Omega } \biggl( \biggl\vert \frac{\partial v}{\partial x_{i}} \biggr\vert ^{p_{i}}-\mu \biggl\vert \frac{\partial v}{\partial x_{i}} \biggr\vert ^{q_{i}} \biggr)\,dx - \int _{ \Omega }f\bigl(x,\phi \star v(x),\nabla \bigl(\phi \star v(x) \bigr)\bigr)v(x)\,dx \\ &\quad \geq \overset{N}{\underset{i=1}{\sum}} \Vert v \Vert _{W_{0}^{1,{p_{i}}}( \Omega )}^{p_{i}} -\mu \overset{N}{\underset{i=1}{\sum}} \vert \Omega \vert ^{ \frac{p_{i}-q_{i}}{p_{i}}} \Vert v \Vert _{W_{0}^{1,{p_{i}}}(\Omega )}^{q_{i}}- \Vert \sigma \Vert _{L^{\gamma ^{\prime}}(\Omega )} \Vert v \Vert _{L^{\gamma}(\Omega )} \\ & \qquad {}-c_{1} \Vert \phi \star v \Vert ^{p^{+}-1}_{L^{p^{+}}(\Omega )} \Vert v \Vert _{L^{p^{+}}( \Omega )}-c_{2}\overset{N}{\underset{i=1}{ \sum}} \Vert \phi \star v \Vert _{W_{0}^{1,{p_{i}}}( \Omega )}^{p_{i}-1} \Vert v \Vert _{L^{p_{i}}(\Omega )} \end{aligned} $$

for all $v\in X_{n}$, by ($H_{1}$) and the Hölder inequality. Now (2.1), (2.3), and Sobolev embedding show that

$$\begin{aligned}& \bigl\langle A_{n}(v) ,v\bigr\rangle _{X_{n}} \\& \quad = \overset{N}{\underset{i=1}{\sum}} \int _{\Omega } \biggl( \biggl\vert \frac{\partial v}{\partial x_{i}} \biggr\vert ^{p_{i}}-\mu \biggl\vert \frac{\partial v}{\partial x_{i}} \biggr\vert ^{q_{i}} \biggr)\,dx \\& \qquad {} - \int _{\Omega }f\bigl(x,\phi \star v(x),\nabla \bigl(\phi \star v(x) \bigr)\bigr)v(x)\,dx \\& \quad \geq \overset{N}{\underset{i=1}{\sum}} \Vert v \Vert _{W_{0}^{1,{p_{i}}}( \Omega )}^{p_{i}} -\mu \overset{N}{\underset{i=1}{\sum}} \vert \Omega \vert ^{ \frac{p_{i}-q_{i}}{p_{i}}} \Vert v \Vert _{W_{0}^{1,{p_{i}}}(\Omega )}^{q_{i}}- \Vert \sigma \Vert _{L^{\gamma ^{\prime}}(\Omega )} \Vert v \Vert _{L^{\gamma}(\Omega )} \\& \qquad {} -c_{1} \Vert \phi \Vert ^{p^{+}-1}_{L^{1}(\mathbb{R}^{N})} \Vert v \Vert ^{p^{+}}_{L^{p^{+}}( \Omega )}-c_{2}\overset{N}{ \underset{i=1}{\sum}} \Vert \phi \Vert ^{p_{i}-1}_{L^{1}( \mathbb{R}^{N})} \Vert v \Vert _{W_{0}^{1,{p_{i}}}(\Omega )}^{p_{i}-1} \Vert v \Vert _{L^{p_{i}}( \Omega )} \\& \quad \geq \overset{N}{\underset{i=1}{\sum}} \Vert v \Vert _{W_{0}^{1,{p_{i}}}( \Omega )}^{p_{i}} -\mu \overset{N}{\underset{i=1}{\sum}} \vert \Omega \vert ^{ \frac{p_{i}-q_{i}}{p_{i}}} \Vert v \Vert _{W_{0}^{1,{p_{i}}}(\Omega )}^{q_{i}}-S_{ \gamma} \Vert \sigma \Vert _{L^{\gamma ^{\prime}}(\Omega )} \Vert v \Vert _{W_{0}^{1, \overrightarrow{p}}(\Omega )} \\& \qquad {} -c_{1}S_{p^{+}} \Vert \phi \Vert ^{p^{+}-1}_{L^{1}(\mathbb{R}^{N})} \sum_{i=1}^{N} \Vert v \Vert ^{p_{i}}_{W_{0}^{1,p_{i}}(\Omega )}-c_{2} \overset{N}{\underset{i=1}{\sum }} S'_{p_{i}} \Vert \phi \Vert ^{p_{i}-1}_{L^{1}( \mathbb{R}^{N})} \Vert v \Vert _{W_{0}^{1,{p_{i}}}(\Omega )}^{p_{i}} \\& \quad \geq \overset{N}{\underset{i=1}{\sum}} \Vert v \Vert _{W_{0}^{1,{p_{i}}}( \Omega )}^{p_{i}} -\mu \overset{N}{\underset{i=1}{\sum}} \vert \Omega \vert ^{ \frac{p_{i}-q_{i}}{p_{i}}} \Vert v \Vert _{W_{0}^{1,{p_{i}}}(\Omega )}^{q_{i}}-S_{ \gamma} \Vert \sigma \Vert _{L^{\gamma ^{\prime}}(\Omega )} \Vert v \Vert _{W_{0}^{1, \overrightarrow{p}}(\Omega )} \\& \qquad {} - \bigl( \Vert \phi \Vert _{L^{1}(\mathbb{R}^{N})}^{p^{+}-1} c_{1}S_{p^{+}}+c_{2} \Pi \bigr)\overset{N}{ \underset{i=1}{\sum}} \Vert v \Vert _{W_{0}^{1,{p_{i}}}( \Omega )}^{p_{i}}, \end{aligned}$$

(3.5)

for all $x\in X_{n}$, where $|\Omega |$ is the Lebesgue measure of Ω.

Assume $\lambda _{1,\overrightarrow{p}}>0$ denotes the first eigenvalue of the negative anisotropic p-Laplacian on $W_{0}^{1,\overrightarrow{p}}(\Omega )$ that is given by

$$ \lambda _{1,\overrightarrow{p}}=\min \biggl\{ \frac{\overset{N}{\underset{i=1}{\sum}}\int _{\Omega} \vert \frac{\partial u}{\partial x_{i}} \vert ^{p_{i}}\,dx}{ \Vert u \Vert _{L^{p^{+}}(\Omega )}^{p^{+}}}: u\in W_{0}^{1,\overrightarrow{p}}(\Omega )\backslash \{0\} \biggr\} . $$

(3.6)

See [15, Theorem 3] or [17, Theorem 2] for more details. By (1.4) (recall that $S_{p^{+}}=\lambda _{1,\overrightarrow{p}}^{-\frac{1}{p^{+}}}$) and $p_{i}>q_{i}>1$ and $p^{+}>q^{+}>1$, for $i=1,\ldots ,N$, for $R = R(n)>0$ sufficiently large we obtain

$$ \bigl\langle A_{n}(v),v\bigr\rangle _{X_{n}}\geq 0\quad \text{whenever } v\in X_{n} \text{ with } \Vert v \Vert _{W_{0}^{1,\overrightarrow{p}}(\Omega )}=R. $$

As a consequence of Brouwer’s fixed-point theorem (see, e.g., [29, p. 37]) (since $X_{n}$ is a finite-dimensional space) there exists $u_{n} \in X_{n}$ solving the equation $A_{n}(u_{n})=0$ and this shows that $u_{n}\in X_{n}$ is a solution for problem (3.4).

$\{u_{n}\}_{n\geq 1}$ is bounded in $W_{0}^{1,\overrightarrow{p}}(\Omega )$. To show this, let $v=u_{n}\in X_{n}$ in (3.5), then

$$\begin{aligned}& \overset{N}{\underset{i=1}{\sum}} \Vert v \Vert _{W_{0}^{1,{p_{i}}}( \Omega )}^{p_{i}} -c_{1}S_{p^{+}} \Vert \phi \Vert ^{p^{+}-1}_{L^{1}( \mathbb{R}^{N})}\sum_{i=1}^{N} \Vert v \Vert ^{p_{i}}_{W_{0}^{1,p_{i}}(\Omega )} \\& \qquad {} -c_{2}\overset{N}{\underset{i=1}{\sum}} S'_{p_{i}} \Vert \phi \Vert ^{p_{i}-1}_{L^{1}(\mathbb{R}^{N})} \Vert v \Vert _{W_{0}^{1,{p_{i}}}( \Omega )}^{p_{i}} \\& \quad \leq \mu \overset{N}{\underset{i=1}{\sum}} \vert \Omega \vert ^{ \frac{p_{i}-q_{i}}{p_{i}}} \Vert v \Vert _{W_{0}^{1,{p_{i}}}(\Omega )}^{q_{i}}+S_{ \gamma} \Vert \sigma \Vert _{L^{\gamma ^{\prime}}(\Omega )} \Vert v \Vert _{W_{0}^{1, \overrightarrow{p}}(\Omega )}. \end{aligned}$$

Since $p_{i}>q_{i}>1$ and $p^{+}>q^{+}>1$, for $i=1,\ldots ,N$, then (1.4) shows that $\{u_{n}\}_{n\geq 1}$ is bounded in $W_{0}^{1,\overrightarrow{p}}(\Omega )$. □

Now, we can prove the existence of the solution of problem (1.1), i.e., we present the proof of Theorem 1.1.

Proof

Assume $\{u_{n}\}_{n\geq 1}\subset W_{0}^{1,\overrightarrow{p}}(\Omega )$ is given by Proposition 3.2 that is bounded in $W_{0}^{1,\overrightarrow{p}}(\Omega )$ and the reflexively, there exists a subsequence still denoted by $\{u_{n}\}_{n\geq 1}$ that is bounded and

$$ u_{n}\rightharpoonup u \quad \text{in } W_{0}^{1,\overrightarrow{p}}( \Omega ) $$

(3.7)

with some $u\in W_{0}^{1,\overrightarrow{p}}(\Omega )$. The continuity of the operator in (3.3), shows that the sequence $\{f(\cdot ,\phi \star u_{n},\nabla (\phi \star u_{n}))\}_{n\geq 1}$ is bounded in $L^{\overrightarrow{p}^{\prime}}$. Suppose

$$ -\Delta _{\overrightarrow{p}}u_{n}+\mu \Delta _{\overrightarrow{q}}u_{n}-f\bigl( \cdot ,\phi \star u_{n},\nabla (\phi \star u_{n})\bigr)\rightharpoonup \eta\quad \text{in } W^{-1,\overrightarrow{p}^{\prime}}(\Omega ) $$

(3.8)

with some $\eta \in W^{-1,\overrightarrow{p}^{\prime}}(\Omega )$, by the reflexivity of $W^{-1,\overrightarrow{p}^{\prime}}(\Omega )$.

Assume $v\in \bigcup_{n\geq 1}X_{n}$. Fix an integer $m\geq 1$ such that $v\in X_{m}$. Proposition 3.2 provides that (3.4) holds for all $n\geq m$. Letting $n\to \infty$ in (3.4), by means of (3.8) we obtain

$$ \langle \eta ,v\rangle\geq 0 \quad \text{for all} v\in \bigcup _{n\geq 1}X_{n}. $$

By the density of $\bigcup_{n\geq 1}X_{n}$ in $W^{1,\overrightarrow{p}}_{0}(\Omega )$ (see (iii) in the definition of the Galerkin basis), it turns out that $\eta =0$ and so in $W^{-1,\overrightarrow{p}^{\prime}}(\Omega )$ we have

$$ -\Delta _{\overrightarrow{p}}u_{n}+\mu \Delta _{\overrightarrow{q}}u_{n}-f\bigl( \cdot ,\phi \star u_{n},\nabla (\phi \star u_{n})\bigr)\rightharpoonup 0. $$

(3.9)

Letting $v=u_{n}$ in (3.4), we obtain

$$ \langle -\Delta _{\overrightarrow{p}}u_{n}+\mu \Delta _{ \overrightarrow{q}}u_{n},u_{n}\rangle - \int _{\Omega }f\bigl( \cdot ,\phi \star u_{n},\nabla ( \phi \star u_{n})\bigr)\,dx=0 $$

(3.10)

for all $n\geq 1$, while (3.9) gives

$$ \langle -\Delta _{\overrightarrow{p}}u_{n}+\mu \Delta _{ \overrightarrow{q}}u_{n},u_{n}\rangle - \int _{\Omega }f\bigl( \cdot ,\phi \star u_{n},\nabla ( \phi \star u_{n})\bigr)\,dx\to 0 $$

(3.11)

as $n\to \infty$. Together, (3.10) and (3.11) yield

$$ \langle -\Delta _{\overrightarrow{p}}u_{n}+\mu \Delta _{ \overrightarrow{q}}u_{n},u_{n}-u\rangle - \int _{\Omega }f\bigl( \cdot ,\phi \star u_{n},\nabla ( \phi \star u_{n})\bigr) (u_{n}-u)\,dx\to 0 $$

(3.12)

as $n\to \infty$. Theorem 2.1 and (3.7) imply that $u_{n}\to u$ strongly in $L^{p}(\Omega )$, and since $\{f(\cdot ,\phi \star u_{n},\nabla (\phi \star u_{n}))\}$ is bounded, then

$$ \lim_{n\to \infty} \int _{\Omega }f\bigl(\cdot ,\phi \star u_{n},\nabla ( \phi \star u_{n})\bigr) (u_{n}-u)\,dx= 0. $$

(3.13)

By inserting (3.13) into (3.12) we obtain

$$ \lim_{n\to \infty}\langle -\Delta _{\overrightarrow{p}}u_{n}+ \mu \Delta _{\overrightarrow{q}}u_{n},u_{n}-u\rangle =0. $$

(3.14)

Thus, the conditions of Definition 2.3 are satisfied and this implies that $u\in W_{0}^{1,\overrightarrow{p}}(\Omega )$ is a generalized solution to problem (1.1).

Now, we prove the existence of a weak solution in the case $\mu \leq 0$. Assume u is a generalized solution to problem (1.1) and $\{u_{n}\}_{n\geq 1}$ satisfy the conditions of Definition 2.3 with respect to u. We obtain

$$ \begin{aligned} &\langle -\Delta _{\overrightarrow{p}}u_{n},u_{n}-u \rangle _{W_{0}^{1,\overrightarrow{p}}(\Omega )} \\ &\quad \leq \langle -\Delta _{\overrightarrow{p}}u_{n},u_{n}-u\rangle _{W_{0}^{1, \overrightarrow{p}}(\Omega )} -\mu \langle -\Delta _{ \overrightarrow{q}}u_{n}+\Delta _{\overrightarrow{q}}u,u_{n}-u \rangle _{W_{0}^{1,\overrightarrow{p}}(\Omega )} \\ &\quad = \langle -\Delta _{\overrightarrow{p}}u_{n}+\mu \Delta _{ \overrightarrow{q}}u_{n},u_{n}-u\rangle _{W_{0}^{1,\overrightarrow{p}}( \Omega )} -\mu \langle \Delta _{\overrightarrow{q}}u,u_{n}-u\rangle _{W_{0}^{1, \overrightarrow{p}}(\Omega )} \end{aligned} $$

by the monotonicity of $-\Delta _{\overrightarrow{q}}$ and hence,

$$ \limsup_{n\to \infty}\langle \Delta _{\overrightarrow{p}}u_{n},u_{n}-u \rangle _{W_{0}^{1,\overrightarrow{p}}(\Omega )}\leq 0. $$

Then, $u_{n}\to u$ strongly in $W^{1,\overrightarrow{p}}(\Omega )$ (see, e.g., [21, Proposition 2.72]). The continuity of A (Lemma 3.1), shows $A(u_{n})\to A(u)$ in $W^{-1,\overrightarrow{p}^{\prime}}(\Omega )$ and condition (II) of Definition 2.3, shows $A(u) = 0$. This shows that

$$ \bigl\langle (-\Delta _{\overrightarrow{p}}+\mu \Delta _{ \overrightarrow{q}} ) (u),v\bigr\rangle _{W_{0}^{1,\overrightarrow{p}}( \Omega )}= \int _{\Omega }f\bigl(x,\phi \star u(x),\nabla \bigl(\phi \star u(x) \bigr)\bigr)v(x)\,dx $$

for all $v\in W^{1,\overrightarrow{p}}_{0}(\Omega )$, which means u is a weak solution to problem (1.1). □