Multiple solutions for singularly perturbed semilinear elliptic equations in bounded domains (Q2494999)

Summary: We are concerned with the multiplicity of solutions of the following singularly perturbed semilinear elliptic equation in a bounded domain \(\Omega:-\varepsilon^2\Delta u+a(\cdot)u=u|u|^{p-2}\), \(u>0\) 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 the problem has at least three solutions for sufficiently small \(\varepsilon\).