Volume 8, Issue 1
Existence and Asymptotic Behavior of Positive Solutions for Variable Exponent Elliptic Systems

Honghui Yin and Zuodong Yang

10.4208/aamm.2013.m322

Adv. Appl. Math. Mech., 8 (2016), pp. 19-36.

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  • Abstract

In this paper, our main purpose is to establish the existence of positive solution of the following system    −∆p(x)u=F(x,u,v), x∈Ω, −∆q(x) v= H(x,u,v), x∈Ω, u=v=0, x∈∂Ω, where Ω = B(0,r) ⊂ R N or Ω = B(0,r2)\B(0,r1) ⊂ R N, 0 < r, 0 < r1 < r2 are constants. F(x,u,v)=λ p(x) [g(x)a(u)+ f(v)], H(x,u,v)=θ q(x) [g1(x)b(v)+h(u)], λ,θ>0 are parameters, p(x), q(x) are radial symmetric functions, −∆p(x)=−div(|∇u| p(x)−2∇u) is called p(x)-Laplacian. We give the existence results and consider the asymptotic behavior of the solutions. In particular, we do not assume any symmetric condition, and we do not assume any sign condition on F(x,0,0) and H(x,0,0) either.

  • History

Published online: 2018-05

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