Introduction and results

Abstract

We study the following singularly perturbed problem

-epsilon(2)Delta u + V(x)u = f(u) in R-N.

Our main result is the existence of a family of solutions with peaks that cluster near a local maximum of V(x). A local variational and deformation argument in an infinite dimensional space is developed to establish the existence of such a family for a general class of nonlinearities f. Earlier works in this direction can be found in [KW, DLY, DY, NY] for f(xi) = xi(p) (1 < p < N+2/N-2 when N >= 3, 1 < p < infinity when N = 1, 2). These papers use the Lyapunov-Schmidt reduction method, where it is essential to have information about the null space of the linearization of a solution of the limit equation -Delta u + u = u(p). Such spectral information is difficult to get and can only be obtained for very special f's. Our new approach in this memoir does not require such a detailed knowledge of the spectrum and works for a much more general class of nonlinearities f.