Solvability for a class of Schrödinger 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.