Semi-classical States for Nonlinear Schrِdinger Equations

We consider existence and asymptotic behavior of solutions for an equation of the formε2 Δu−V(x) u+f(u)=0, u>0, u∈H10(Δ), ((*))whereΔis 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 existsε0so that for all 0<ε<ε0, Eq. (*) possesses a solution having exactly one maximum pointxε∈Δ, such thatV(xε)→cand ∇V(xε)→0 asε→0.