Multiple positive symmetric solutions of a singularly perturbed elliptic equation

Mónica Clapp, Gustavo Izquierdo


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).


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

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