Ground state solutions of Nehari-Pankov type for Schrodinger equations with local super-quadratic conditions

Consider the semilinear Schrodinger equation {-Delta u + V(x)u = f(x,u) x is an element of R-N, u is an element of H-1 (R-N), where both V(x) and f (x, u) are periodic in x, 0 belongs to a spectral gap of -Delta + V, and f (x, u) is subcritical and allowed to be super-linear at some x is an element of R-N and asymptotically linear at the other x is an element of R-N. In the existing works in the literature, it is commonly assumed that lim(vertical bar u vertical bar ->infinity )integral(u)(0)f(x,s)ds/u(2) = infinity uniformly in x is an element of R-N, to obtain the existence of ground state solutions or infinitely many geometrically distinct solutions. In this paper, for the first time, we prove the existence of ground state solutions and infinitely many geo- metrically distinct solutions under the weaker super-quadratic condition lim(vertical bar u vertical bar ->infinity )integral(u)(0)f(x,s)ds/u(2)= infinity, a.e. x is an element of G just for some domain G subset of R-N. (C) 2019 Elsevier Inc. All rights reserved.