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

摘要

We show the existence of a nodal solution with two nodal domains for ageneralized 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 theoperator $\Delta_\Phi u:=div(\phi(|\nabla u|)\nabla u)$. The proof is based ona minimization argument and a quantitative deformation lemma.