Multiple solutions for singularly perturbed semilinear elliptic equations in bounded domains

Abstract

We are concerned with the multiplicity of solutions of the following singularly perturbed semilinear elliptic equations in bounded domains $\Omega :-{\varepsilon }^2\Delta u+a(\cdot )u=u{\left|u\right|}^{p-2}$ in $\Omega$, $u>0$ in $\Omega$, $u=0$ on $\partial \Omega$. The main purpose of this paper is to discuss the relationship between the multiplicity of solutions and the profile of $a(\cdot )$ from the variational point of view. It is shown that if $a$ has a “peak” in $\Omega$, then (P) has at least three solutions for sufficiently small $\varepsilon$.