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

This research is undertaken with the primary objective of exploring a quasilinear Schr & ouml;dinger equation, a mathematical model of significant importance in describing diverse physical phenomena. Specifically, we direct our focus to the following equation: \[ -\Delta u +V(x)u-[\Delta(1+u<^>2)<^>{{1}/{2}}]\frac {u}{2(1+u<^>2)<^>\frac 12}= h(u), \quad x\in \mathbb{R}<^>N, \] -Delta u+V(x)u-[Delta(1+u2)1/2]u2(1+u2)12=h(u),x is an element of RN, where $ N\geq 3 $ 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 $ h(u)=|u|p-2u, we are concerned mostly with $ 2 \lt p\le 12-4\sqrt 6 $ 2<p <= 12-46.