*(English)*Zbl 0949.35054

This paper deals with spike-layer solutions of the problem

and $u>0$ in ${\Omega}$, $u=0$ on $\partial {\Omega}$. Here, $f$ is a suitable function ${\mathbb{R}}^{+}\to \mathbb{R}$ and the assumptions on the smooth domain ${\Omega}\subset {\mathbb{R}}^{n}$ 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 $d(y,\partial {\Omega})$ 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 $d({x}_{\epsilon},\partial {\Omega})\to c$, 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.