Existence of least energy nodal solution with two nodal domains for a generalized Kirchhoff problem in an Orlicz Sobolev space

TL;DR

This paper proves the existence of a specific type of nodal solution with two nodal domains for a generalized Kirchhoff problem in an Orlicz Sobolev space, using minimization and deformation techniques.

Contribution

It establishes the existence of a least energy nodal solution with two nodal domains for a generalized Kirchhoff equation in an Orlicz Sobolev space, which is a novel result.

Findings

Existence of a nodal solution with two nodal domains.

Solution characterized by minimization and deformation methods.

Applicable to a broad class of Kirchhoff problems in Orlicz spaces.

Abstract

We show the existence of a nodal solution with two nodal domains for a generalized Kirchhoff equation of the type $$ -M\left(\displaystyle\int_\Omega \Phi(|\nabla u|)dx\right)\Delta_\Phi u = f(u) \ \ \mbox{in} \ \ \Omega, \ \ u=0 \ \ \mbox{on} \ \ \partial\Omega, $$ where $\Omega$ is a bounded domain in $\mathbf{R}^N$, $M$ is a general $C^{1}$ class function, $f$ is a superlinear $C^{1}$ class function with subcritical growth, $\Phi$ is defined for $t\in \mathbf{R}$ by setting $ \Phi(t)=\int_0^{|t|}\phi(s)sds$, $\Delta_\Phi$ is the operator $\Delta_\Phi u:=div(\phi(|\nabla u|)\nabla u)$. The proof is based on a minimization argument and a quantitative deformation lemma.