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.
Categories: math.AP