Nontrivial Solutions for Schrodinger Equation with Local Super-Quadratic Conditions

This paper is dedicated to studying the semilinear Schrodinger equation - u + V(x) u = f (x, u), x. RN, u. H1(RN), where V. C(RN, R) is sign- changing and either periodic or coercive and f. C(RN xR, R) is subcritical and local super- linear (i. e. allowed to be super- linear at some x. RN and asymptotically linear at other x. RN). Instead of the common condition that lim| t|.8 t 0 f (x, s) ds t2 = 8 uniformly in x. RN, we use a local super- quadratic condition lim| t|.8 t 0 f (x, s) ds t2 =8 a. e. x. G for some domain G. RN to show the existence of nontrivial solutions for the above problem.