Generalized linking theorem with an application to a semilinear schrödinger equation

Abstract

Consider the semilinear Schrödinger equation (*) -δu + V (x)u = f(x, u), u ∈ H1(RN). It is shown that if f, V are periodic in the x-variables, f is superlinear at u = 0 and ±∞ and 0 lies in a spectral gap of -δ + V , then (*) has at least one nontrivial solution. If in addition f is odd in u, then (*) has infinitely many (geometrically distinct) solutions. The proofs rely on a degree-theory and a linking-type argument developed in this paper.