Existence and multiplicity of sign-changing solutions for quasilinear Schrödinger equations with sub-cubic nonlinearity

Abstract

In this paper, we consider the quasilinear Schrödinger equation−Δu+V(x)u−uΔ(u2)=g(u),x∈R3, where V and g are continuous functions. Without the coercive condition on V or the monotonicity condition on g, we show that the problem above has a least energy sign-changing solution and infinitely many sign-changing solutions. Our results especially solve the problem above in the case where g(u)=|u|p−2u (2<p<4) and complete some recent related works on sign-changing solutions, in the sense that, in the literature only the case g(u)=|u|p−2u (p≥4) was considered. The main results in the present paper are obtained by a new perturbation approach and the method of invariant sets of descending flow. In addition, in some cases where the functional merely satisfies the Cerami condition, a deformation lemma under the Cerami condition is developed.