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# Nonexistence of nontrivial solutions for the p(x)- Laplacian equations and systems in unbounded domains of ℝ n

Boundary Value Problems20112011:50

https://doi.org/10.1186/1687-2770-2011-50

• Accepted: 30 November 2011
• Published:

## Abstract

In this paper, we are interested on the study of the nonexistence of nontrivial solutions for the p(x)-Laplacian equations, in unbounded domains of n . This leads us to extend these results to m-equations systems. The method used is based on pohozaev type identities.

## Keywords

• Bounded Domain
• Elliptic Equation
• Dirichlet Problem
• Nontrivial Solution
• Laplacian Equation

## 1 Introduction

Several works have been reported by many authors, comprise results of nonexistence of nontrivial solutions of the semilinear elliptic equations and systems, under various situations, see [18]. The Pohozaĕv identity [1] published in 1965 for solutions of the Dirichlet problem proved absence of nontrivial solutions for some elliptic equations of the form
when Ω is a star shaped bounded open domain in n and f is a continuous function on satisfying
$\begin{array}{c}\hfill \left(n-2\right)F\left(u\right)-2nuf\left(u\right)>0,\hfill \end{array}$
1. A.
Hareux and B. Khodja [2] established under the assumption
$\begin{array}{c}\hfill f\left(0\right)=0,\hfill \\ \hfill 2F\left(u\right)-uf\left(u\right)\le 0.\hfill \end{array}$

that the problems
admit only the null solution in H2(J × ω) ∩ L(J × ω). where J is an interval of and ω is a connected unbounded domain of N such as

(n(x) is the outward normal to ∂ω at the point x)

In this work we are interested in the study of the nonexistence of nontrivial solutions for the p(x)-laplacian problem
(1.1)
with
where
$\begin{array}{c}\hfill {\Delta }_{p\left(x\right)}u=div\left(\mid \nabla u{\mid }^{p\left(x\right)-2}\nabla u\right)\hfill \end{array}$
Ω is bounded or unbounded domains of n , f is a locally lipshitzian function, H and p are given continuous real functions of $C\left(\stackrel{̄}{\mathrm{\Omega }}\right)$ verifying
(1.4)

(., .) is the inner product in n .

We extend this technique to the system of m-equations
(1.6)
with
Where {f k } are locally lipshitzian functions verify
$\begin{array}{c}{f}_{k}\left({s}_{1},...,{s}_{k-1},0,{u}_{k+1},...,{s}_{m}\right)=0,\left(0\le k\le m\right),\\ \exists {F}_{m}:{ℝ}^{m}\to ℝ:\frac{\partial {F}_{m}}{\partial {s}_{k}}\left({s}_{1},...,{s}_{m}\right)={f}_{k}\left({s}_{1},...,{s}_{m}\right).\end{array}$
H is previously defined and p k functions of ${C}^{1}\left(\overline{\mathrm{\Omega }}\right)$ class, verify
$\begin{array}{c}{p}_{k}\left(x\right)>1,\left(x,\nabla {p}_{k}\left(x\right)\right)\ge 0,\forall x\in \stackrel{̄}{\mathrm{\Omega }}.\\ {a}_{k}=\underset{x\in \stackrel{̄}{\mathrm{\Omega }}}{sup}\left(1-\frac{n}{{p}_{k}\left(x\right)}+\frac{\left(x,\nabla {p}_{k}\left(x\right)\right)}{{p}_{k}^{2}\left(x\right)}\right)\end{array}$
(1.9)

## 2 Integral identities

Let
with the norm
$\begin{array}{c}\hfill \mid u{\mid }_{{L}^{p\left(x\right)}\left(\mathrm{\Omega }\right)}=\mid u{\mid }_{p\left(x\right)}=inf\left\{\lambda >0:\underset{\mathrm{\Omega }}{\int }\mid \frac{u\left(x\right)}{\lambda }{\mid }^{p\left(x\right)}dx\le 1\right\},\hfill \end{array}$
and
$\begin{array}{c}\hfill {W}^{1,p\left(x\right)}\left(\mathrm{\Omega }\right)=\left\{u\phantom{\rule{0.5em}{0ex}}\in {L}^{p\left(x\right)}\left(\mathrm{\Omega }\right):\mid \nabla u\mid \in {L}^{p\left(x\right)}\left(\mathrm{\Omega }\right)\right\},\hfill \end{array}$
with the norm
$\parallel u{\parallel }_{{W}^{1,p\left(x\right)}\left(\mathrm{\Omega }\right)}=\mid u{\mid }_{{L}^{p\left(x\right)}\left(\mathrm{\Omega }\right)}+\mid \nabla u{\mid }_{{L}^{p\left(x\right)}\left(\mathrm{\Omega }\right)}.$

Denote ${W}_{0}^{1,p\left(x\right)}\left(\mathrm{\Omega }\right)$ the closure of ${C}_{0}^{\infty }\left(\mathrm{\Omega }\right)$ in W1, p(x)(Ω),

Lemma 1 Let$u\in {W}_{0}^{1,p\left(x\right)}\left(\mathrm{\Omega }\right)\cap {L}^{\infty }\left(\stackrel{̄}{\mathrm{\Omega }}\right)$solution of the equation (1.1) - (1.2), we have
(2.1)
Lemma 2 Let$u\in {W}_{0}^{1,p\left(x\right)}\left(\mathrm{\Omega }\right)\cap {L}^{\infty }\left(\stackrel{̄}{\mathrm{\Omega }}\right)$solution of the equation (1.1) - (1.3), we have
(2.2)
Proof Multiplying the equation (1.1) by $\sum _{j=1}^{n}{x}_{i}\frac{\partial u}{\partial {x}_{i}}$ and integrating the new equation by parts in Ω ∩ B R , B R = B (0, R)
$\begin{array}{c}-\underset{\mathrm{\Omega }\cap {B}_{R}}{\int }div\left(\mid \nabla u{\mid }^{p\left(x\right)-2}\nabla u\right)\left(\sum _{j=1}^{n}{x}_{j}\frac{\partial u}{\partial {x}_{j}}\right)dx\\ =-\sum _{i,j=1}^{n}\underset{\mathrm{\Omega }\cap {B}_{R}}{\int }\frac{\partial }{\partial {x}_{i}}\left(\mid \nabla u{\mid }^{p\left(x\right)-2}\frac{\partial u}{\partial {x}_{i}}\right){x}_{j}\frac{\partial u}{\partial {x}_{j}}dx\\ =\underset{\mathrm{\Omega }\cap {B}_{R}}{\int }\left[\mid \nabla u{\mid }^{p\left(x\right)}+\mid \nabla u{\mid }^{p\left(x\right)-2}\sum _{i,j=1}^{n}{x}_{j}\frac{\partial u}{\partial {x}_{i}}\frac{{\partial }^{2}u}{\partial {x}_{i}\partial {x}_{j}}\right]dx\\ -\sum _{i,j=1}^{n}\underset{\partial \left(\mathrm{\Omega }\cap {B}_{R}\right)}{\int }\mid \nabla u{\mid }^{p\left(x\right)-2}\frac{\partial u}{\partial {x}_{i}}\frac{\partial u}{\partial {x}_{j}}{x}_{j}{\nu }_{i}ds\end{array}$
Introducing the following result
$\begin{array}{c}\hfill \mid \nabla u{\mid }^{p\left(x\right)-2}\sum _{i=1}^{n}\frac{\partial u}{\partial {x}_{i}}\frac{{\partial }^{2}u}{\partial {x}_{i}\partial {x}_{j}}=\frac{1}{p\left(x\right)}\frac{\partial }{\partial {x}_{j}}\left(\mid \nabla u{\mid }^{p\left(x\right)}\right)-\frac{\frac{\partial p}{\partial {x}_{j}}}{{p}^{2}\left(x\right)}\mid \nabla u{\mid }^{p\left(x\right)}ln\left(\mid \nabla u{\mid }^{p\left(x\right)}\right)\hfill \end{array}$
we have
$\begin{array}{c}\underset{\mathrm{\Omega }\cap {B}_{R}}{\int }\left[\mid \nabla u{\mid }^{p\left(x\right)}+\sum _{j=1}^{n}\frac{{x}_{j}}{p\left(x\right)}\frac{\partial }{\partial {x}_{j}}\left(\mid \nabla u{\mid }^{p\left(x\right)}\right)-\sum _{j=1}^{n}\frac{\left(x,\nabla p\left(x\right)\right)}{{p}^{2}\left(x\right)}\mid \nabla u{\mid }^{p\left(x\right)}ln\left(\mid \nabla u{\mid }^{p\left(x\right)}\right)\right]\phantom{\rule{0.3em}{0ex}}dx\\ -\underset{\partial \left(\mathrm{\Omega }\cap {B}_{R}\right)}{\int }\sum _{i,j=1}^{n}\underset{\partial \left(\mathrm{\Omega }\cap {B}_{R}\right)}{\int }{\left|\nabla u\right|}^{p\left(x\right)-2}\frac{\partial u}{\partial {x}_{i}}\frac{\partial u}{\partial {x}_{j}}{x}_{j}{\nu }_{i}ds\\ =\underset{\mathrm{\Omega }\cap {B}_{R}}{\int }\left[1-\frac{n}{p\left(x\right)}+\frac{\left(x,\nabla p\left(x\right)\right)}{{p}^{2}\left(x\right)}\left(1-ln\left(\mid \nabla u{\mid }^{p\left(x\right)}\right)\right)\right]\mid \nabla u{\mid }^{p\left(x\right)}dx\\ -\underset{\partial \left(\mathrm{\Omega }\cap {B}_{R}\right)}{\int }\left(\sum _{i,j=1}^{n}\mid \nabla u{\mid }^{p\left(x\right)-2}\frac{\partial u}{\partial {x}_{i}}\frac{\partial u}{\partial {x}_{j}}{x}_{j}{\nu }_{i}-\sum _{j=1}^{n}\frac{1}{p\left(x\right)}\mid \nabla u{\mid }^{p\left(x\right)}{x}_{j}{\nu }_{j}\right)\phantom{\rule{0.3em}{0ex}}ds\end{array}$
On the other hand
$\begin{array}{c}\underset{\mathrm{\Omega }\cap {B}_{R}}{\int }H\left(x\right)f\left(u\right)\left(\sum _{j=1}^{n}{x}_{j}\frac{\partial u}{\partial {x}_{j}}\right)dx=\sum _{j=1}^{n}\underset{\mathrm{\Omega }\cap {B}_{R}}{\int }{x}_{j}H\left(x\right)\frac{\partial }{\partial {x}_{j}}\left(F\left(u\right)\right)dx\\ =-\underset{\mathrm{\Omega }\cap {B}_{R}}{\int }\left(nH\left(x\right)+\left(x,\nabla H\left(x\right)\right)\right)F\left(u\right)dx+\sum _{j=1}^{n}\underset{\partial \left(\mathrm{\Omega }\cap {B}_{R}\right)}{\int }H\left(x\right)F\left(u\right){x}_{j}{\nu }_{j}ds\end{array}$
these results conduct to the following formula
(2.3)
Multiplying the present equation (1.1) by au and integrating the obtained equation by parts in Ω, we obtain
$\begin{array}{c}\hfill \underset{\left(\mathrm{\Omega }\cap {B}_{R}\right)}{\int }\left[a\mid \nabla u{\mid }^{p\left(x\right)}-auH\left(x\right)f\left(u\right)\right]\phantom{\rule{0.3em}{0ex}}dx=\underset{\partial \left(\mathrm{\Omega }\cap {B}_{R}\right)}{\int }a\mid \nabla u{\mid }^{p\left(x\right)}\frac{\partial u}{\partial \nu }uds=0,\hfill \end{array}$
(2.4)
Combining (2.3) and (2.4) we obtain

On (Ω ∩ ∂B R ) we have ${n}_{i}=\frac{{x}_{i}}{\mid x\mid }$

so the last integral is major by
$\begin{array}{c}\hfill M\left(R\right)=R\underset{\mathrm{\Omega }\cap \partial {B}_{R}}{\int }\left(\left(1+\frac{1}{p\left(x\right)}\right)\mid \nabla u{\mid }^{p\left(x\right)}+\mid H\left(x\right)\mid \phantom{\rule{2.77695pt}{0ex}}\mid F\left(u\right)\mid \right)ds\hfill \end{array}$

We remark that if Ω in bounded, so for R is little greater, we get Ω ∩ ∂B R = ϕ, then M (R) = 0.

If Ω is not bounded, such as |u| W1, p(x)(Ω), F(u) L1 (Ω) and $\underset{\mid x\mid \to +\infty }{lim}H\left(x\right)\to 0,$ we should see
$\begin{array}{c}\hfill \underset{0}{\overset{+\infty }{\int }}dr\underset{\mathrm{\Omega }\cap \partial {B}_{R}}{\int }\left(\left(1+\frac{1}{p\left(x\right)}\right)\mid \nabla u{\mid }^{p\left(x\right)}+\mid H\left(x\right)\parallel F\left(u\right)\mid \right)\phantom{\rule{2.77695pt}{0ex}}ds<+\infty \hfill \end{array}$
consequently we can always find a sequence (R n ) n , such as
$\begin{array}{c}\hfill \underset{n\to +\infty }{lim}{R}_{n}\to +\infty \phantom{\rule{1em}{0ex}}\mathsf{\text{and}}\phantom{\rule{1em}{0ex}}\underset{n\to +\infty }{lim}M\left({R}_{n}\right)\to 0.\hfill \end{array}$

In the problem (1.1) - (1.2), u| Ω= 0. Then, $\nabla u=\frac{\partial u}{\partial \nu }n$, we obtain the identity (2.1).

In the problem (1.1) - (1.3), , we obtain the identity (2.2). ■

Lemma 3 Let${u}_{k}\in {W}_{0}^{1,{p}_{k}\left(x\right)}\left(\mathrm{\Omega }\right)\cap {L}^{\infty }\left(\stackrel{̄}{\mathrm{\Omega }}\right)\left(1\le k\le m\right)$, solution of the system (1.6) - (1.7). Then for the constants a k of , we have
(2.5)
Lemma 4 Let${u}_{k}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\cap {L}^{\infty }\left(\stackrel{̄}{\mathrm{\Omega }}\right)\left(1\le k\le m\right)$, solutions of the system (1.6) - (1.8). Then for the constants a k of , we have
$\begin{array}{c}\underset{\mathrm{\Omega }}{\int }\left[\sum _{k=1}^{m}\left(1-\frac{n}{{p}_{k}\left(x\right)}+\frac{\left(x,\nabla {p}_{k}\left(x\right)\right)}{{p}_{k}^{2}\left(x\right)}\left(1-ln\left(\mid \nabla {u}_{k}{\mid }^{{p}_{k}\left(x\right)}\right)\right)-{a}_{k}\right)\mid \nabla {u}_{k}{\mid }^{{p}_{k}\left(x\right)}\right\\ +H\left(x\right)\left(n{F}_{m}\left({u}_{1},...,{u}_{m}\right)-\sum _{k=1}^{m}{a}_{k}{u}_{k}{f}_{k}\left({u}_{1},...,{u}_{m}\right)\right)+\\ +\left(x,\nabla H\left(x\right)\right){F}_{m}\left({u}_{1},...,{u}_{m}\right)\right]\phantom{\rule{2.77695pt}{0ex}}dx\\ =\underset{\partial \mathrm{\Omega }}{\int }\left[\sum _{k=1}^{m}\left(1-\frac{1}{{p}_{k}\left(x\right)}\right)\mid \nabla {u}_{k}{\mid }^{{p}_{k}\left(x\right)}+H\left(x\right){F}_{m}\left({u}_{1},...,{u}_{m}\right)\right]\phantom{\rule{2.77695pt}{0ex}}\left(x,\nu \right)ds\end{array}$
(2.6)
Proof Multiplying the equation (1.6) by $\sum _{j=1}^{n}{x}_{i}\frac{\partial {u}_{k}}{\partial {x}_{i}}$ and integrating the new equation by part in Ω ∩ B R , B R = B (0, R), we get
$\begin{array}{c}\underset{\mathrm{\Omega }\cap {B}_{R}}{\int }\left[1-\frac{n}{{p}_{k}\left(x\right)}+\frac{\left(x,\nabla {p}_{k}\left(x\right)\right)}{{p}_{k}^{2}\left(x\right)}\left(1-ln\left(\mid \nabla {u}_{k}{\mid }^{{p}_{k}\left(x\right)}\right)\right)\right]\mid \nabla {u}_{k}{\mid }^{{p}_{k}\left(x\right)}dx\\ =\underset{\partial \left(\mathrm{\Omega }\cap {B}_{R}\right)}{\int }\left(\sum _{i,j=1}^{n}\mid \nabla {u}_{k}{\mid }^{{p}_{k}\left(x\right)-2}\frac{\partial {u}_{k}}{\partial {x}_{i}}\frac{\partial {u}_{k}}{\partial {x}_{j}}{x}_{j}{\nu }_{i}-\sum _{j=1}^{n}\frac{1}{{p}_{k}\left(x\right)}\mid \nabla {u}_{k}{\mid }^{{p}_{k}\left(x\right)}{x}_{j}{\nu }_{j}\right)\phantom{\rule{0.3em}{0ex}}ds\end{array}$
On the other hand
$\begin{array}{c}\underset{\mathrm{\Omega }\cap {B}_{R}}{\int }H\left(x\right){f}_{k}\left({u}_{1},...,{u}_{m}\right)\left(\sum _{j=1}^{n}{x}_{j}\frac{\partial {u}_{k}}{\partial {x}_{j}}\right)dx\\ =\sum _{j=1}^{n}\underset{\mathrm{\Omega }\cap {B}_{R}}{\int }{x}_{j}H\left(x\right)\frac{\partial {u}_{k}}{\partial {x}_{j}}\frac{\partial }{\partial {u}_{k}}\left({F}_{m}\left({u}_{1},...,{u}_{m}\right)\right)dx\end{array}$
These results conduct to the following formula
Doing the sum on k of 1 to m, we obtain
which leads to the following identity
(2.7)
Now, multiply the equation (1.1) by au and integrating the obtained equation by parts in Ω ∩ B R
$\begin{array}{c}\hfill \underset{\left(\mathrm{\Omega }\cap {B}_{R}\right)}{\int }\left[{a}_{k}\mid \nabla u{\mid }^{{p}_{k}\left(x\right)}-{a}_{k}{u}_{k}H\left(x\right){f}_{k}\left({u}_{1},...,{u}_{m}\right)\right]dx=0\hfill \end{array}$
(2.8)

Combining (2.7) and (2.8), we get the identities (2.5) and (2.6).

The rest of the proof is similar to the that of lemma 1. ■

## 3 Principal Result

theorem 3.1 If$u\in {W}_{0}^{1,p\left(x\right)}\left(\mathrm{\Omega }\right)\cap {L}^{\infty }\left(\stackrel{̄}{\mathrm{\Omega }}\right)$be a solution of the problem (1.1) - (1.2), Ω is star shaped and that a, H, f and F verify the following assumptions
$nF\left(u\right)-auf\left(u\right)\le 0,\forall x\in \mathrm{\Omega },$
(3.1)
$\left(x,\nabla H\left(x\right)\right)F\left(u\right)\le 0,\forall x\in \mathrm{\Omega }.$
(3.2)

Then, the problem admits only the null solution.

Proof Ω is star shaped, imply that
$\begin{array}{c}\hfill \underset{\partial \mathrm{\Omega }}{\int }\left(1-\frac{1}{p\left(x\right)}\right)\mid \nabla u{\mid }^{p\left(x\right)}\left(x,\nu \right)ds\ge 0.\hfill \end{array}$
(3.3)
On the other hand, the condition (3.1) give
(3.4)
(1.4), (3.3) and (3.4), allow to get
So, the problem (1.1) - (1.2) becomes
(3.5)
Multiplying the equation (3.5) by u and integrating over Ω, we get
$\underset{\mathrm{\Omega }}{\int }\mid \nabla u{\mid }^{p\left(x\right)}dx=0.$
So
$\mid \nabla u\mid =0,$

Hence u = cte = 0, because u| Ω= 0. ■

theorem 3.2 If$u\in {W}_{0}^{1,p\left(x\right)}\left(\mathrm{\Omega }\right)\cap {L}^{\infty }\left(\stackrel{̄}{\mathrm{\Omega }}\right)$solution of the problem (1.1) - (1.3), Ω is a star shaped and that a, H, F and F verify the following conditions
$nF\left(u\right)-auf\left(u\right)\le 0,\forall x\in \mathrm{\Omega },$
(3.6)
$\left(x,\nabla H\left(x\right)\right)F\left(u\right)\le 0,\forall x\in \mathrm{\Omega }.$
(3.7)
$H\left(x\right)F\left(u\right)\ge 0,\forall x\in \partial \mathrm{\Omega }.$
(3.8)

Therefore, the problem admits only the null solution.

Proof Similar to the proof of theorem 1. ■

theorem 3.3 If${u}_{k}\in {W}_{0}^{1,{p}_{k}\left(x\right)}\left(\mathrm{\Omega }\right)\cap {L}^{\infty }\left(\stackrel{̄}{\mathrm{\Omega }}\right)$solution of the system (1.6) - (1.7), Ω is a star shaped and that a k , H, f k and F m verify the following conditions
$n{F}_{m}\left({u}_{1},...,{u}_{m}\right)-\sum _{k=1}^{m}{a}_{k}{u}_{k}{f}_{k}\left({u}_{1},...,{u}_{m}\right)\le 0,\forall x\in \mathrm{\Omega },$
(3.9)
$\left(x,\nabla H\left(x\right)\right){F}_{m}\left({u}_{1},...,{u}_{m}\right)\le 0,\forall x\in \mathrm{\Omega }.$
(3.10)

So, the system admits only the null solutions.

Proof Ω is a star shaped, implies that
$\begin{array}{c}\hfill \underset{\partial \mathrm{\Omega }}{\int }\sum _{k=1}^{m}\left(1-\frac{1}{{p}_{k}\left(x\right)}\right)\mid \nabla {u}_{k}{\mid }^{{p}_{k}\left(x\right)}\left(x,\nu \right)ds\ge 0\phantom{\rule{0.5em}{0ex}}.\hfill \end{array}$
(3.11)
On the other hand, the conditions (3.9) and (3.10), give
(3.12)
(1.4), (3.11) and (3.12), allow to have
So the system (1.6) - (1.7) becomes
(3.13)
Multiplying (3.13) by u k and integrating on Ω, we have
$\underset{\mathrm{\Omega }}{\int }\mid \nabla {u}_{k}{\mid }^{{p}_{k}\left(x\right)}dx=0$
So
$\mid \nabla {u}_{k}\mid =0$

Therefore u k = cte = 0, 1 ≤ km, because u k | Ω= 0. ■

theorem 3.4 If${u}_{k}\in {W}_{0}^{1,{p}_{k}\left(x\right)}\left(\mathrm{\Omega }\right)\cap {L}^{\infty }\left(\stackrel{̄}{\mathrm{\Omega }}\right)$solution of the system (1.6) - (1.8), Ω is a star shaped and that a k , H, f k and F m verify the following conditions
$n{F}_{m}\left({u}_{1},...,{u}_{m}\right)-\sum _{k=1}^{m}{a}_{k}{u}_{k}{f}_{k}\left({u}_{1},...,{u}_{m}\right)\le 0,\forall x\in \mathrm{\Omega },$
(3.14)
$\left(x,\nabla H\left(x\right)\right){F}_{m}\left({u}_{1},...,{u}_{m}\right)\le 0,\forall x\in \mathrm{\Omega },$
(3.15)
$H\left(x\right){F}_{m}\left({u}_{1},...,{u}_{m}\right)\ge 0,\forall x\in \partial \mathrm{\Omega }.$
(3.16)

So, the problem admit only the null solution.

Proof Similar to the that of theorem 3. ■

## 4 Examples

Example 1 Considering in ${W}_{0}^{1,p\left(x\right)}\left(\mathrm{\Omega }\right)\cap {W}_{0}^{1,q}\left(\stackrel{̄}{\mathrm{\Omega }}\right)$ the following problem
$\left\{\begin{array}{c}-div\left(\mid \nabla u{\mid }^{p\left(x\right)-2}\nabla u\right)=\frac{c}{{\left(1+\mid x\mid \right)}^{\mu }}u\mid u{\mid }^{q-1}\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}\mathrm{\Omega },\\ u=0\phantom{\rule{0.5em}{0ex}}on\phantom{\rule{0.5em}{0ex}}\partial \mathrm{\Omega },\end{array}\right\$
(4.1)

where Ω is a bounded domain of n , c, μ > 0, q > 1 and$p\left(x\right)=\sqrt{1+\mid x{\mid }^{2}}>1.$

By choosing
$\begin{array}{c}\hfill a=\underset{\mathrm{\Omega }}{sup}\left(1-\frac{n+\left(n-1\right)\mid x{\mid }^{2}}{\left(1+\mid x{\mid }^{2}\right)\sqrt{1+\mid x{\mid }^{2}}}\right),\hfill \end{array}$
we obtain
$\begin{array}{c}\left(x,\nabla H\left(x\right)\right)F\left(u\right)=\frac{-c\mu \mid x\mid }{q{\left(1+\mid x\mid \right)}^{\mu +1}}\mid u{\mid }^{q+1}<0,\\ \left(x,\nabla p\left(x\right)\right)=\frac{\mid x{\mid }^{2}}{\sqrt{1+\mid x{\mid }^{2}}}\ge 0,\\ nF\left(u\right)-auf\left(u\right)=\left(\frac{n}{q+1}-a\right)\mid u{\mid }^{q+1}\le 0\phantom{\rule{0.5em}{0ex}}if\phantom{\rule{0.5em}{0ex}}q\ge \frac{n-a}{a}.\end{array}$
So, the problem (4.1) doesn't admit non trivial solutions if
$q\ge \frac{n-a}{a}.$
Example 2 Considering in ${W}_{0}^{1,p\left(x\right)}\left(\mathrm{\Omega }\right)\cap {W}_{0}^{1,\gamma }\left(\stackrel{̄}{\mathrm{\Omega }}\right),$ the following elliptic system
$\left\{\begin{array}{c}-{\Delta }_{p\left(x\right)}u=\frac{c\gamma }{{\left(1+\mid x\mid \right)}^{\mu }}u\mid u{\mid }^{\gamma -1}\mid v{\mid }^{\delta }\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}\mathrm{\Omega },\\ -{\Delta }_{q\left(x\right)}v=\frac{c\delta }{{\left(1+\mid x\mid \right)}^{\mu }}v\mid v{\mid }^{\delta -1}\mid u{\mid }^{\gamma }\phantom{\rule{0.5em}{0ex}}in\phantom{\rule{0.5em}{0ex}}\mathrm{\Omega },\\ u=0\phantom{\rule{0.5em}{0ex}}on\phantom{\rule{0.5em}{0ex}}\partial \mathrm{\Omega }\end{array}\right\$
(4.2)

where Ω is a bounded domain of n , c, μ, γ, δ > 0 and p, q > 1.

By choosing
$\begin{array}{c}{a}_{1}=\underset{x\in \stackrel{̄}{\mathrm{\Omega }}}{sup}\left(1-\frac{n}{p\left(x\right)}+\frac{\left(x,\nabla p\left(x\right)\right)}{{p}^{2}\left(x\right)}\right)\\ and\\ {a}_{2}=\underset{x\in \stackrel{̄}{\mathrm{\Omega }}}{sup}\left(1-\frac{n}{p\left(x\right)}+\frac{\left(x,\nabla q\left(x\right)\right)}{{q}^{2}\left(x\right)}\right)\end{array}$
we obtain
$\begin{array}{c}\left(x,\nabla H\left(x\right)\right)F\left(u,v\right)=\frac{-c\mu }{{\left(1+\mid x\mid \right)}^{\mu +1}}\mid u{\mid }^{\gamma }\mid v{\mid }^{\delta }<0,\\ nF\left(u,v\right)-{a}_{1}u{f}_{1}\left(u,v\right)-{a}_{2}v{f}_{2}\left(u,v\right)=\left(n-\gamma {a}_{1}-\delta {a}_{2}\right)\mid u{\mid }^{\gamma }\mid v{\mid }^{\delta }\end{array}$
So, the system (4.2) doesn't admit non trivial solutions if
$\gamma {a}_{1}+\delta {a}_{2}\ge n$

## Authors’ Affiliations

(1)
Department of mathematics and informatics, Tebessa university, Algeria

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