Existence and asymptotic behavior of positive solutions for elliptic systems with nonstandard growth conditions

Volume 104 / 2012

Honghui Yin, Zuodong Yang Annales Polonici Mathematici 104 (2012), 293-308 MSC: Primary 35J25; Secondary 35J60. DOI: 10.4064/ap104-3-6

Abstract

Our main purpose is to establish the existence of a positive solution of the system $$\begin{cases} -\triangle_{p(x)} u= F(x,u,v),&x\in \Omega,\\ -\triangle_{q(x)} v= H(x,u,v),&x\in \Omega,\\ u=v=0,&x\in\partial\Omega, \end{cases} $$ where $\Omega\subset {\mathbb R}^N$ is a bounded domain with $C^2$ boundary, $F(x,u,v)=\lambda^{p(x)}[g(x)a(u)+f(v)]$, $H(x,u,v)=\lambda^{q(x)} [g(x)b(v)+h(u)]$, $\lambda>0$ is a parameter, $p(x), q(x)$ are functions which satisfy some conditions, and $-\triangle_{p(x)}u=-\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)$ is called the $p(x)$-Laplacian. We give existence results and consider the asymptotic behavior of solutions near the boundary. We do not assume any symmetry conditions on the system.

Authors

  • Honghui YinInstitute of Mathematics
    School of Mathematical Sciences
    Nanjing Normal University
    Nanjing, Jiangsu 210046, China
    and
    School of Mathematical Sciences
    Huaiyin Normal University
    Huaian, Jiangsu 223001, China
    e-mail
  • Zuodong YangInstitute of Mathematics
    School of Mathematical Sciences
    Nanjing Normal University
    Nanjing, Jiangsu 210046, China
    and
    College of Zhongbei
    Nanjing Normal University
    Nanjing, Jiangsu 210046, China
    e-mail

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