Electron. J. Diff. Equ., Vol. 2012 (2012), No. 122, pp. 1-14.

Multiple symmetric solutions for a singular semilinear elliptic problem with critical exponent

Alfredo Cano, Eric Hernandez-Martinez

Abstract:
Let be $\Gamma$ a closed subgroup of $O(N)$. We consider the semilinear elliptic problem
$$\displaylines{ -\Delta u-\frac{b(x)}{| x|^2}u-a(x)u=f(x)| u|^{2^{\ast }-2}u\quad \hbox{in }\Omega ,\cr u=0 \quad\hbox{on } \partial \Omega , }$$
where $\Omega \subset \mathbb{R}^{N}$ is a smooth bounded domain, $N\geq 4$. We establish the multiplicity of symmetric positive solutions, nodal solutions, and solutions which are $\Gamma$ invariant but are not $\widetilde{\Gamma}$ invariant, where $\Gamma \subset \widetilde{\Gamma}\subset O(N)$.

Submitted March 27, 2012. Published July 20, 2012.
Math Subject Classifications: 35J75, 35J57, 35J60.
Key Words: Critical exponent; singular problem; symmetric solutions.

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Alfredo Cano
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
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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