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.