Positive solutions for some Schrodinger equations having partially periodic potentials

This paper is concerned with the problem of finding positive solutions u ∈ H01 (Ω) of the equation - Δ u + (a∞ + a (x)) u = | u |q - 2 u, where q is subcritical, Ω is either RN or an unbounded domain which is periodic in the first p coordinates and whose complement is contained in a cylinder {(x′, x″) ∈ Rp × RN - p : | x″ | < R}, a∞ > 0, a ∈ C (RN, R) is periodic in the first p coordinates, infx ∈ RN (a∞ + a (x)) > 0 and a (x′, x″) → 0 as | x″ | → ∞ uniformly in x′. The cases a ≤ 0 and a ≥ 0 are considered and it is shown that, under appropriate assumptions on a, the problem has one solution in the first case and p + 1 solutions in the second case when p ≤ N - 2.