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Zbl 0974.35041
del Pino, Manuel; Felmer, Patricio L.; Wei, Juncheng
(Pino, Manuel del)
Multi-peak solutions for some singular perturbation problems.
(English)
[J] Calc. Var. Partial Differ. Equ. 10, No.2, 119-134 (2000). ISSN 0944-2669; ISSN 1432-0835/e

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.
[Alexander Movchan (Liverpool)]
MSC 2000:
*35J65 (Nonlinear) BVP for (non)linear elliptic equations
35B25 Singular perturbations (PDE)
35B05 General behavior of solutions of PDE
35A15 Variational methods (PDE)

Keywords: singular perturbation; multi-peak solutions

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