Positive solutions for some Schrِdinger equations having partially periodic potentials

This paper is concerned with the problem of finding positive solutions u ∈ H 0 1 ( Ω ) of the equation − Δ u + ( a ∞ + a ( x ) ) u = | u | q − 2 u , where q is subcritical, Ω is either R N or an unbounded domain which is periodic in the first p coordinates and whose complement is contained in a cylinder { ( x ′ , x ″ ) ∈ R p × R N − p : | x ″ | < R } , a ∞ > 0 , a ∈ C ( R N , R ) is periodic in the first p coordinates, inf x ∈ R N ( 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 .