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.