### Multiple positive symmetric solutions of a singularly perturbed elliptic equation

DOI: http://dx.doi.org/10.12775/TMNA.2001.022

#### 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).

$$

-\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).

#### Keywords

Singularly perturbed elliptic problems; symmetric solutions; nonlinear PDE's

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