Existence and concentration for nonlinear Schrodinger equations with fast decaying potentials

The existence of positive solutions to -epsilon(2) Delta u + Vu = u(p) in R(N), is proved for small epsilon, where N >= 3, V is a nonnegative continuous potential which has a positive local minimum, and N/N-2 < p < N+2/N-2 . No restriction is imposed on the rate of decay of V; this includes in particular the case where V is compactly supported. To cite this article: V. Moroz, J. Van Schaftingen, C. R. Acad. Sci. Paris, Ser. I 347 (2009). (c) 2009 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.