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

Nguyen Thanh Chung

Electronic journal of differential equations, Tome 2011 (2011)

Résumé

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.

Détail

Classification : 35J65, 35J20
Keywords: degenerate and singular elliptic system, weight function, concave nonlinearity, infinitely many solutions