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On the interior spike solutions for some singular perturbation problems. (English) Zbl 0944.35021

The paper deals with positive solutions to singularly perturbed semilinear elliptic problems of the following form:

$-{\epsilon }^{2}{\Delta }u=f\left(u\right)\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega }u=0\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{or}\phantom{\rule{1.em}{0ex}}\frac{\partial u}{\partial \nu }=0\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega }·$

Here ${\Omega }\subseteq {ℝ}^{n}$ is a bounded domain with smooth boundary $\partial {\Omega }$ and $f:ℝ\to ℝ$ is smooth; a typical choice is $f\left(u\right)=-u+{u}^{p}\left(p>1\right)$. The purpose of the paper is to investigate the existence of single interior spike solutions. A family of solutions $\left\{{u}_{\epsilon }\right\}$ is called single peaked if: $\left(i\right)$ the energy is bounded for any $\epsilon >0$; $\left(ii\right)$ ${u}_{\epsilon }$ has only one local maximum point ${P}_{\epsilon }$, ${P}_{\epsilon }\to {P}_{0}$, ${u}_{\epsilon }\to 0$ in ${C}_{\text{loc}}^{1}\left(\overline{{\Omega }}\setminus {P}_{0}\right)$ and ${u}_{\epsilon }\left({P}_{\epsilon }\right)\to \alpha >0$ as $\epsilon \to 0$. The limiting point ${P}_{0}$ is called a boundary spike if ${P}_{0}\in \partial {\Omega }$, respectively an interior spike if ${P}_{0}\in {\Omega }$. In both cases the problem of existence and location of spikes has stimulated quite a few papers in recent years. The paper provides a unified approach to this problem under rather general assumptions on the function $f$ (both for Dirichlet and for Neumann homogeneous boundary conditions); in particular, necessary and sufficient conditions for the existence of interior spike solutions are given. A variety of methods is used to this purpose, including in particular weak convergence of measures and Lyapunov-Schmidt reduction.

MSC:
 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35B25 Singular perturbations (PDE) 35B65 Smoothness and regularity of solutions of PDE