zbMATH — the first resource for mathematics

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 \[ \begin{cases} \varepsilon^2\Delta u- u+ f(u)=0\quad &\text{in }\Omega,\\ u> 0\text{ in }\Omega,\;u=0\quad &\text{on }\partial\Omega,\end{cases} \] where \(\Omega\) is a smooth domain in \(\mathbb{R}^N\) (\(\Omega\) 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.

35J65 Nonlinear boundary value problems for linear elliptic equations
35B25 Singular perturbations in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35A15 Variational methods applied to PDEs
Full Text: DOI