Electron. J. Diff. Equ., Vol. 2010(2010), No. 112, pp. 1-14.

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

Alfredo Cano, Sergio Hernandez-Linares, Eric Hernandez-Martinez

Abstract:
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].

Submitted November 15, 2009. Published August 16, 2010.
Math Subject Classifications: 35J20, 35J25, 49J52, 58E35,74G35.
Key Words: 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.

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Alfredo Cano Rodríguez
Universidad Autónoma del Estado de México
Facultad de Ciencias, Departamento de Matemáticas
Campus El Cerrillo Piedras Blancas
Carretera Toluca-Ixtlahuaca, Km 15.5, Toluca, Estado de México, México
email: calfredo420@gmail.com
Sergio Hernández-Linares
Universidad Autónoma Metropolitana, Cuajimalpa
Departamento de Matemáticas Aplicadas y Sistemas
Artificios No. 40, Col. Hidalgo
Del. Alvaro Obregón, C.P. 01120
México D.F., México} email: slinares@correo.cua.uam.mx
Eric Hernández-Martínez
Universidad Autónoma de la Ciudad de México, Colegio de Ciencia y Tecnología.
Academia de Matemáticas, Calle Prolongación San Isidro No. 151,
Col. San Lorenzo Tezonco, Del. Iztapalapa, C.P. 09790, México D.F., México
email: ebric2001@hotmail.com

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