### Multiple positive solutions for a singularly perturbed Dirichlet problem in "geometrically trivial" domains

#### Abstract

In this paper we consider the singularly perturbed Dirichlet problem

(P$_{\varepsilon}$), when the potential $a_{\varepsilon}(x)$, as $\varepsilon$ goes to

$0$, is concentrating round a point $x_0\in\Omega$.

Under suitable growth assumptions on $f$, we prove that (P$_{\varepsilon}$)

has at least three distinct solutions whatever $\Omega$ is and that at least

one solution is not a one-peak solution.

(P$_{\varepsilon}$), when the potential $a_{\varepsilon}(x)$, as $\varepsilon$ goes to

$0$, is concentrating round a point $x_0\in\Omega$.

Under suitable growth assumptions on $f$, we prove that (P$_{\varepsilon}$)

has at least three distinct solutions whatever $\Omega$ is and that at least

one solution is not a one-peak solution.

#### Keywords

Singularity perturbet elliptic problem; variational methods; critical point; Lusternik-Schnirelman theory; (LS)-category

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