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

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.

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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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