Electron. J. Diff. Equ., Vol. 2011 (2011), No. 30, pp. 1-12.

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

Nguyen Thanh Chung

In this article, we consider degenerate and singular elliptic systems of the form
 - \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.

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Nguyen Thanh Chung
Department of Mathematics and Informatics
Quang Binh University, 312 Ly Thuong Kiet
Dong Hoi, Quang Binh, Vietnam
email: ntchung82@yahoo.com

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