Existence and asymptotic behavior of positive solutions for elliptic systems with nonstandard growth conditions
Tom 104 / 2012
Annales Polonici Mathematici 104 (2012), 293-308
MSC: Primary 35J25; Secondary 35J60.
DOI: 10.4064/ap104-3-6
Streszczenie
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.
