Multiplicity results for a class of concave-convex elliptic systems involving sign-changing weight functions
Tom 102 / 2011
Streszczenie
Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems $$ \cases{\displaystyle -{\mit\Delta}_p u+|u|^{p-2}u=f_{1\lambda_1}(x) |u|^{q-2 }u+\frac{2\alpha}{\alpha+\beta}g_\mu|u|^{\alpha-2}u|v|^\beta,\quad x\in {\mit\Omega},\cr \displaystyle -{\mit\Delta}_p v+|v|^{p-2}v=f_{2\lambda_2}(x) |v|^{q-2} v +\frac{2\beta}{\alpha+\beta}g_\mu|u|^\alpha|v|^{\beta-2}v,\quad x\in {\mit\Omega},\cr u=v=0,\quad x\in \partial{\mit\Omega},} $$ where $1< q< p < N$ and ${\mit\Omega}\subset \mathbb{R}^N$ is an open bounded smooth domain. Here $\lambda_1, \lambda_2, \mu\geq0$ and $f_{i\lambda_i}(x)=\lambda_if_{i+}(x)+f_{i-}(x)$ $(i=1,2)$ are sign-changing functions, where $f_{i\pm}(x)=\max\{\pm f_i(x),0\}$, $g_\mu(x)=a(x)+\mu b(x)$, and ${\mit\Delta}_p u=\hbox{div}(|\nabla u|^{p-2}\nabla u)$ denotes the $p$-Laplace operator. We use variational methods.
