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Existence of a solution for a nonlocal elliptic system of (p(x),q(x))-Kirchhoff type

  • Ali Taghavi and Horieh Ghorbani EMAIL logo
Published/Copyright: January 10, 2018

Abstract

In this paper, we consider the system

{-M1(Ω|u|p(x)+|u|p(x)p(x)𝑑x)(Δp(x)u-|u|p(x)-2u)=λa(x)|u|r1(x)-2u-μb(x)|u|α(x)-2uin Ω,-M2(Ω|v|q(x)+|v|q(x)q(x)𝑑x)(Δq(x)v-|v|q(x)-2v)=λc(x)|v|r2(x)-2v-μd(x)|v|β(x)-2vin Ω,u=v=0on Ω,

where Ω is a bounded domain in N (N2) with a smooth boundary Ω, M1(t),M2(t) are continuous functions and λ,μ>0. We prove that for any μ>0 there exists λ* sufficiently small such that for any λ(0,λ*) the above system has a nontrivial weak solution. The proof relies on some variational arguments based on Ekeland’s variational principle, and some adequate variational methods.

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Received: 2017-07-20
Revised: 2017-10-01
Accepted: 2017-12-04
Published Online: 2018-01-10
Published in Print: 2018-07-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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