This paper deals with spike-layer solutions of the problem
and in , on . Here, is a suitable function and the assumptions on the smooth domain 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 with smooth boundary and closed subsets , of such that , is a connected and . Let be the distance function to the boundary of . It is assumed that possesses a topologically nontrivial critical point in , characterized through a max-min scheme. Under further assumptions on which are, in particular, satisfied in a local saddle point situation, the authors prove the existence of a family of solutions to (1), with exactly one local maximum point such that , as goes to zero. The similarity between this result and the existence of concentrated bounded states at any topologically nontrivial critical point of the potential 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.