Multiplicity results for a class of concave-convex elliptic systems involving sign-changing weight functions

Honghui Yin¹ ; Zuodong Yang²

¹Institute 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
²Institute 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 102 (2011) no. 1, pp. 51-71

Résumé

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.

Détail

DOI : 10.4064/ap102-1-5

Keywords: main purpose establish existence weak solutions second order quasilinear elliptic systems cases displaystyle mit delta p lambda q frac alpha alpha beta alpha beta quad mit omega displaystyle mit delta p lambda q frac beta alpha beta alpha beta quad mit omega quad partial mit omega where mit omega subset mathbb bounded smooth domain here lambda lambda geq lambda lambda i sign changing functions where max mit delta hbox div nabla p nabla denotes p laplace operator variational methods