Nodal solutions for double phase Kirchhoff problems with vanishing potentials

Abstract

We consider the following $\mathrm{(}\mathit{p}\mathrm{,}\mathit{q}\mathrm{)}$ -Laplacian Kirchhoff type problem $$\begin{array}{ll}\hfill & -\mathrm{(}\mathit{a}+\mathit{b}{{\displaystyle \int }}_{{\mathbb{R}}^3}|\nabla \mathit{u}{|}^{\mathit{p}}\mathit{d}\mathit{x}\mathrm{)}{\mathrm{\Delta }}_{\mathit{p}}\mathit{u}-\mathrm{(}\mathit{c}+\mathit{d}{{\displaystyle \int }}_{{\mathbb{R}}^3}|\nabla \mathit{u}{|}^{\mathit{q}}\mathit{d}\mathit{x}\mathrm{)}{\mathrm{\Delta }}_{\mathit{q}}\mathit{u}\\ \hfill & +\mathit{V}\mathrm{(}\mathit{x}\mathrm{)}\mathrm{(}|\mathit{u}{|}^{\mathit{p}-2}\mathit{u}+|\mathit{u}{|}^{\mathit{q}-2}\mathit{u}\mathrm{)}=\mathit{K}\mathrm{(}\mathit{x}\mathrm{)}\mathit{f}\mathrm{(}\mathit{u}\mathrm{)}\text{in }{\mathbb{R}}^3\mathrm{,}\end{array}$$ where $\mathit{a}\mathrm{,}\mathit{b}\mathrm{,}\mathit{c}\mathrm{,}\mathit{d}\mathrm{>}0$ are constants, $\frac{3}{2}\mathrm{<}\mathit{p}\mathrm{<}\mathit{q}\mathrm{<}3$ , $\mathit{V}:{\mathbb{R}}^3\to \mathbb{R}$ and $\mathit{K}:{\mathbb{R}}^3\to \mathbb{R}$ are positive continuous functions allowed for vanishing behavior at infinity, and f is a continuous function with quasicritical growth. Using a minimization argument and a quantitative deformation lemma we establish the existence of nodal solutions.