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

Abstract

We show the existence of a nodal solution with two nodal domains for a generalized Kirchhoff equation of the type $$ -M\left(\displaystyle\int_ΩΦ(|\nabla u|)dx\right)Δ_Φu = f(u) \ \ \mbox{in} \ \ Ω, \ \ u=0 \ \ \mbox{on} \ \ \partialΩ, $$ 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 $\Phi (t)={\int }_0^{|t|}\varphi (s)sds$, ${\Delta }_{\Phi }$ is the operator ${\Delta }_{\Phi }u:=div(\varphi (|\nabla u|)\nabla u)$. The proof is based on a minimization argument and a quantitative deformation lemma.