Abstract We consider existence and asymptotic behavior of solutions for an equation of the form ε 2 Δu−V(x) u+f(u)=0, u>0, u∈H 1 0(Δ), ((*)) where Δis a smooth domain in R N , not necessarily bounded. We assume that the potential Vis positive and that it possesses a topologically nontrivialcritical value c, characterized through a min–max scheme. The function fis assumed to be locally Hölder continuous having a subcritical, superlinear growth. Further we assume that fis such that the corresponding limiting equation in R N has a unique solution, up to translations. We prove that there exists ε 0so that for all 0< ε< ε 0, Eq. (*) possesses a solution having exactly one maximum point x ε ∈ Δ, such that V( x ε )→ cand ∇ V( x ε )→0 as ε→0.