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

Honghui Yin & Zuodong Yang

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

Published online: 2018-05

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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.

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@Article{AAMM-8-19, author = {Honghui Yin and Zuodong Yang}, title = {Existence and Asymptotic Behavior of Positive Solutions for Variable Exponent Elliptic Systems}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {8}, number = {1}, pages = {19--36}, 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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2013.m322}, url = {http://global-sci.org/intro/article_detail/aamm/12074.html} }
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