Existence of standing waves of nonlinear Schrodinger equations with potentials vanishing at infinity

For a singularly perturbed nonlinear elliptic equation epsilon(2) Delta u - V(x)u + u(p) = 0, x is an element of R(N), we prove the existence of bump solutions concentrating around positive critical points of V when nonnegative V is not identically zero for p is an element of (N/N-2, N+2/N-2) or nonnegative V satisfies lim inf(vertical bar x vertical bar ->infinity) V(x)vertical bar x vertical bar(2) log vertical bar x vertical bar > 0 p = N/N-2. (C) 2011 Elsevier Inc. All rights reserved.