Multiple solutions for a singular semilinear elliptic problems with critical exponent and symmetries

Cano, Alfredo ; Hernández-Linares, Sergio ; Hernández-Martínez, Eric

Electronic Journal of Differential Equations, Tome 2010 (2010).

Résumé

Summary: We consider the singular semilinear elliptic equation $$ -\Delta u-\frac{\mu }{| x| ^2}u-\lambda u=f(x)| u| ^{2^{\ast }-1} $$ in $\Omega , u=0$ on $\partial \Omega $, where $\Omega $ is a smooth bounded domain, in $\mathbb{R}^N, N\geq 4, 2^{\ast }:=\frac{2N}{N-2}$ is the critical Sobolev exponent, $f:\mathbb{R} ^N\to \mathbb{R}$ is a continuous function, $0\lambda \lambda _1$, where $\lambda _1$ is the first Dirichlet eigenvalue of $-\Delta -\frac{\mu }{| x| ^2}$ in $\Omega $ and $0\mu \overline{\mu }:=(\frac{N-2}{2})^2$. We show that if $\Omega $ and f are invariant under a subgroup of $O(N)$, the effect of the equivariant topology of $\Omega $ will give many symmetric nodal solutions, which extends previous results of Guo and Niu [8].

Détail

Classification : 35J20, 35J25, 49J52, 58E35, 74G35
Keywords: critical points, critical Sobolev exponent, multiplicity of solutions, invariant under the action of a orthogonal group, palais-Smale condition, singular semilinear elliptic problem, relative category