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

Summary: 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