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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

ε 2 Δu-u+f(u)=0inΩ,u>0inΩ,u=0onΩ,

where Ω is a smooth domain in N (Ω does not have to be bounded) and ε 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:
35J65Nonlinear boundary value problems for linear elliptic equations
35B25Singular perturbations (PDE)
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35A15Variational methods (PDE)