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Zbl 0947.35075
Dancer, E.N.; Yan, Shusen
A singularly perturbed elliptic problem in bounded domains with nontrivial topology.
(English)
[J] Adv. Differ. Equ. 4, No.3, 347-368 (1999). ISSN 1079-9389

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: $$\cases -\varepsilon^2\Delta u+ u= u^{p- 1},\quad & y\in \Omega,\\ u> 0,\quad & y\in\Omega,\\ u= 0,\quad & y\in\partial\Omega,\endcases$$ where $\Omega$ is a bounded domain in $\bbfR^N$ with smooth boundary, $\varepsilon> 0$ is a small number, $2< p<{2N\over N-2}$ if $N>2$ and $2< p<+\infty$ if $N= 2$.
MSC 2000:
*35J65 (Nonlinear) BVP for (non)linear elliptic equations
35B25 Singular perturbations (PDE)
58E05 Abstract critical point theory
35B30 Dependence of solutions of PDE on initial and boundary data

Keywords: existence; multiplicity

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