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Multi-peak solutions for some singular perturbation problems. (English) Zbl 0974.35041
The paper presents an analysis of multi-peak solutions of the following singularly perturbed problem $$\cases \varepsilon^2\Delta u- u+ f(u)=0\quad &\text{in }\Omega,\\ u> 0\text{ in }\Omega,\ u=0\quad &\text{on }\partial\Omega,\endcases$$ where $\Omega$ is a smooth domain in $\bbfR^N$ ($\Omega$ does not have to be bounded) and $\varepsilon$ is small parameter; the term $f(u)$ is a superlinear, subcritical nonlinearity. The analysis is based on a variational method. By modifying the nonlinearity and adding a penalization term the authors introduce a new penalized energy functional and analyze its critical points. Section 1 of the paper includes the analysis of a single peak case and Section 2 treats the general multi-peak case.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B25 Singular perturbations (PDE) 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE) 35A15 Variational methods (PDE)
##### Keywords:
singular perturbation; multi-peak solutions
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