Positive solutions for some Schrödinger equations having partially periodic potentials

This paper is concerned with the problem of finding positive solutions of the equation −Δu+(a∞+a(x))u=|u|q−2u, 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 , 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.