Multiple positive symmetric solutions of a singularly perturbed elliptic equation

Abstract

This paper is concerned with the multiplicity of positive solutions of the Dirichlet problem $$ -\varepsilon ^{2}\Delta u+u=K( x) \vert u\vert ^{p-2}u \quad\text{in }\Omega, $$ where $\Omega $ is a smooth domain in $\mathbb{R}^{N}$ which is either bounded or has bounded complement (including the case $\Omega =\mathbb{R}^{N}$), $N\geq 3$, $K$ is continuous and $p$ is subcritical. It is known that critical points of $K$ give rise to multibump solutions of this type of problems. It is also known that, in general, the presence of symmetries has the effect of producing many additional solutions. So, we consider domains $\Omega $ which are invariant under the action of a group $G$ of orthogonal transformations of $\mathbb{R}^{N}$, we assume that $K$ is $G$-invariant, and study the combined effect of symmetries and the nonautonomous term $K$ on the number of positive solutions of this problem. We obtain multiplicity results which extend previous results of Benci and Cerami (1994), Cingolani and Lazzo (1997) and Qiao and Wang (1999).