Solvability for a Class of Schrodinger Equations with Periodic Potential

Abstract

In this thesis we study the existence of solutions for a class of semilinear Schr¨odinger equations of the form −∆u + V (x)u = f¯(x, u), x ∈ RN , where N ≥ 2, the potential V is a 1-periodic continuous function. In dimension N ≥ 3, we assume that 0 lies in a spectral gap of the Schr¨odinger operator S = −∆+V and the nonlinearity is from concave and convex type. In dimension N = 2, we assume that 0 lies in a spectral gap or on the boundary of a spectral gap of S and we deal with nonlinearities having exponential growth in the Trudinger-Moser sense. We treat the case where f¯(x, t) is periodic as well as the nonperiodic one. The proofs relies on variational setting, by using linking-type theorems, some Trudinger-Moser inequalities and concentration-compactness principles.