Semi-classical states for nonlinear Schrodinger equations

We consider existence and asymptotic behavior of solutions for an equation of the form epsilon(2) Delta u - V(x) u + f(u) = 0, u>0, u is an element of H-0(1)(Omega), (*) where Omega is a smooth domain in R-N, not necessarily bounded. We assume that the potential V is positive and that it possesses a topologically nontrivial critical value c, characterized through a min-max scheme. The function f is assumed to be locally Holder continuous having a subcritical, superlinear growth. Further we assume that f is such that the corresponding limiting equation in R-N has a unique solution, up to translations. We prove that there exists epsilon(0) so that for all 0<epsilon<epsilon(0), Eq. (*) possesses a solution having exactly one maximum point x(epsilon) is an element of Omega, such that V(x(epsilon)) --> c and del V(x(epsilon)) --> 0 as epsilon --> 0. (C) 1997 Academic Press.