Existence of solution for Schrodinger equation with discontinuous nonlinearity and asymptotically linear

This paper concerns the existence of a nontrivial solution for the following problem {-Delta u + V(x)u is an element of partial derivative F-u(x,u) a.e in R-N, (P) u is an element of H-1 (R-N), where F(x, t) = integral(t)(0) f(x, s) ds, f is a mensurable function and asymptotically linear at infinity, lambda = 0 is in a spectral gap of -Delta + V, and partial derivative F-t denotes the generalized gradient of F with respect to variable t. Here, by employing Variational Methods for Locally Lipschitz Functionals, we establish the existence of solution when f is periodic and non periodic. (C) 2021 Elsevier Inc. All rights reserved.