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 $Ω$ 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, $Φ$ is defined for $t\in \mathbf{R}$ by setting $ Φ(t)=\int_0^{|t|}ϕ(s)sds$, $Δ_Φ$ is the operator $Δ_Φu:=div(ϕ(|\nabla u|)\nabla u)$. The proof is based on a minimization argument and a quantitative deformation lemma.