On the existence of signed and sign-changing solutions for a class of superlinear Schrödinger equations

This paper deals with a semilinear Schrödinger equation whose nonlinear term involves a positive parameter λ and a real function f (u) which satisfies a superlinear growth condition just in a neighborhood of zero. By proving an a priori estimate (for a suitable class of solutions) we are able to avoid further restrictions on the behavior of f (u) at infinity in order to prove, for λ sufficiently large, the existence of one-sign and sign-changing solutions. Minimax methods are employed to establish this result. © 2007 Elsevier Inc. All rights reserved