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.