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On the role of distance function in some singular perturbation problems. (English) Zbl 0949.35054

This paper deals with spike-layer solutions of the problem

${\epsilon }^{2}{\Delta }u-u+f\left(u\right)=0\phantom{\rule{2.em}{0ex}}\left(1\right)$

and $u>0$ in ${\Omega }$, $u=0$ on $\partial {\Omega }$. Here, $f$ is a suitable function ${ℝ}^{+}\to ℝ$ and the assumptions on the smooth domain ${\Omega }\subset {ℝ}^{n}$ are the same as in M. del Pino and P. L. Felmer [J. Funct. Anal. 149, No. 1, 245-265 (1997; Zbl 0887.35058)], namely: there exist an open bounded subset ${\Lambda }$ with smooth boundary and closed subsets $B$, ${B}_{0}$ of ${\Lambda }$ such that $\overline{{\Lambda }}\subset {\Omega }$, $B$ is a connected and ${B}_{0}\subset B$. Let $d\left(y,\partial {\Omega }\right)$ be the distance function to the boundary $\partial {\Omega }$ of ${\Omega }$. It is assumed that $d$ possesses a topologically nontrivial critical point $c$ in ${\Lambda }$, characterized through a max-min scheme. Under further assumptions on $d$ which are, in particular, satisfied in a local saddle point situation, the authors prove the existence of a family ${u}_{\epsilon }$ of solutions to (1), with exactly one local maximum point ${x}_{\epsilon }\in {\Lambda }$ such that $d\left({x}_{\epsilon },\partial {\Omega }\right)\to c$, as $\epsilon$ goes to zero. The similarity between this result and the existence of concentrated bounded states at any topologically nontrivial critical point of the potential $V\left(x\right)$ in loc. cit., is pointed out. The proof is based on the construction of a penalized energy functional and techniques developed by the authors in several recent papers on related topics; one of them was written in collaboration with W.M.NI.

Reviewer: D.Huet (Nancy)
##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B25 Singular perturbations (PDE) 35B40 Asymptotic behavior of solutions of PDE 35J20 Second order elliptic equations, variational methods 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
##### Keywords:
spike-layer; singular perturbations; Dirichlet problem