Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 30, pp. 1-12.

Title: Existence of infinitely many solutions for degenerate and 
       singular elliptic systems with indefinite concave nonlinearities

Author: Nguyen Thanh Chung (Quang Binh Univ., Quang Binh, Vietnam)

Abstract: 
 In this article, we consider degenerate and singular
 elliptic systems of the form
 $$\displaylines{
 - \hbox{div}(h_1(x)\nabla u)
  =  b_1(x)|u|^{r-2}u + F_u(x,u,v) \quad   \hbox{in } \Omega,\cr
 - \hbox{div}(h_2(x)\nabla v)
  =  b_2(x)|v|^{r-2}v + F_v(x,u,v) \quad  \hbox{in } \Omega,
 }$$
 where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $N \geq 2$,
 with smooth boundary $\partial\Omega$;
 $h_i: \Omega \to [0, \infty)$, $h_i \in L^1_{loc}(\Omega)$,
 and are allowed to have ``essential'' zeroes;
 $1 < r < 2$; the weight functions  $b_i: \Omega \to \mathbb{R}$,
 may be sign-changing;  and $(F_u,F_v) = \nabla F$.
 Using variational techniques, a variant of the
 Caffarelli - Kohn - Nirenberg inequality, and a variational
 principle by Clark [9],  we prove the rxistence of
 infinitely many solutions in a weighted  Sobolev space.

Submitted May 14, 2010. Published February 18, 2011.
Math Subject Classifications: 35J65, 35J20.
Key Words: Degenerate and singular Elliptic system; weight function;
           concave nonlinearity; infinitely many solutions.