Nodal solutions for double phase Kirchhoff problems with vanishing potentials

Abstract

We consider the following $(p,q)$-Laplacian Kirchhoff type problem \begin{align*} \begin{split} &-\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{p}\, dx \right)Δ_{p}u - \left(c+d\int_{\mathbb{R}^{3}}|\nabla u|^{q}\, dx \right ) Δ_{q}u + V(x) (|u|^{p-2}u + |u|^{q-2}u)= K(x) f(u) \quad \mbox{ in } \mathbb{R}^{3}, \end{split} \end{align*} where $a,b,c,d>0$ are constants, $\frac{3}{2}<p<q<3$, $V:{\mathbb{R}}^3\to \mathbb{R}$ and $K:{\mathbb{R}}^3\to \mathbb{R}$ are positive continuous functions allowed 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.