Solutions in Spectral Gaps for a Nonlinear Equation of Schrödinger Type

AbstractIn this paper we study the existence of a nontrivial H2(RN) solution for an equation of the form −Δu(x) + p(x) u(x) − f(x, u(x)) = λu(x), x ∈ RN, λ ∈ R, where p ∈ L∞(RN) is a periodic function. We assume that the operator −Δ + p − λ : H2(RN) ⊂ L2(RN) → L2(RN) is strongly indefinite and invertible and that ƒ(x, ·):R → R is odd and satisfies some superlinear but subcritical growth conditions. We extend the class of nonlinearities which has been studied up to now. In particular, under standard technical restrictions, the existence of a solution is derived, when lim|x| → ∞, f(x, s) ≡ f̃(s) > 0 exists for all s ∈ R, if we assume that f(x, s) ≥ f(s) for all s ∈ R and a.e. on RN