Abstract

We consider existence and asymptotic behavior of solutions for an equation of the form?2?u-V(x)u+f(u)=0,u>0,u?H1 0(?), ((*))where?is a smooth domain in RN, not necessarily bounded. We assume that the potentialVis positive and that it possesses atopologically nontrivialcritical valuec, characterized through a min-max scheme. The functionfis assumed to be locally Hölder continuous having a subcritical, superlinear growth. Further we assume thatfis such that the correspondinglimiting equation inRNhas a unique solution, up to translations. We prove that there exists?0so that for all 0<?<?0, Eq.(*) possesses a solution having exactly one maximum pointx???, such thatV(x?)?cand ?V(x?)?0 as??0. © 1997 Academic Press.