Sign-changing solitary waves for a quasilinear Schrödinger equation with general nonlinearity

Abstract

This research is undertaken with the primary objective of exploring a quasilinear Schrödinger equation, a mathematical model of significant importance in describing diverse physical phenomena. Specifically, we direct our focus to the following equation: − Δ u + V ( x ) u − [ Δ ( 1 + u 2 ) 1 / 2 ] u 2 ( 1 + u 2 ) 1 2 = h ( u ) , x ∈ R N , where N ≥ 3 , V is a given positive potential and h represents a general nonlinearity. Employing an innovative perturbation technique and the method of invariant sets in the descending flow, we rigorously establish the existence and multiplicity of sign-changing solutions for the aforementioned problem. In particular, for pure power type nonlinearity h ( u ) = | u | p − 2 u , we are concerned mostly with 2 < p ≤ 12 − 4 6 .