A singularly perturbed elliptic problem in bounded domains with nontrivial topology. (English) Zbl 0947.35075

From the introduction: The aim of this paper is to study the effect of the domain topology on the existence and multiplicity of multipeak solutions for the following singularly perturbed elliptic problem: \[ \begin{cases} -\varepsilon^2\Delta u+ u= u^{p- 1},\quad & y\in \Omega,\\ u> 0,\quad & y\in\Omega,\\ u= 0,\quad & y\in\partial\Omega,\end{cases} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary, \(\varepsilon> 0\) is a small number, \(2< p<{2N\over N-2}\) if \(N>2\) and \(2< p<+\infty\) if \(N= 2\).


35J65 Nonlinear boundary value problems for linear elliptic equations
35B25 Singular perturbations in context of PDEs
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs