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

Honghui Yin¹ ; Zuodong Yang²

¹ Institute 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
² Institute of Mathematics School of Mathematical Sciences Nanjing Normal University Nanjing, Jiangsu 210046, China and College of Zhongbei Nanjing Normal University Nanjing, Jiangsu 210046, China

Annales Polonici Mathematici, Tome 104 (2012) no. 3, pp. 293-308

Résumé

Our main purpose is to establish the existence of a positive solution of the system $$\begin{cases} -\triangle_{p(x)} u= F(x,u,v),\in \Omega,\\ -\triangle_{q(x)} v= H(x,u,v),\in \Omega,\\ u=v=0,\in\partial\Omega, \end{cases} $$ 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.

Détail

DOI : 10.4064/ap104-3-6

Keywords: main purpose establish existence positive solution system begin cases triangle v omega triangle v omega partial omega end cases where omega subset mathbb bounded domain boundary v lambda u v lambda v lambda parameter functions which satisfy conditions triangle mbox div nabla nabla called laplacian existence results consider asymptotic behavior solutions near boundary assume symmetry conditions system