Nodal solutions for double phase Kirchhoff problems with vanishing potentials

TL;DR

This paper proves the existence of nodal solutions for a class of double phase Kirchhoff problems involving $(p,q)$-Laplacian operators with vanishing potentials, using variational methods and deformation techniques.

Contribution

It introduces a novel approach to find nodal solutions for Kirchhoff problems with vanishing potentials and quasicritical growth, expanding the understanding of such nonlinear PDEs.

Findings

Existence of nodal solutions established.

Applicable to problems with vanishing potentials at infinity.

Uses minimization and deformation methods effectively.

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)\Delta_{p}u - \left(c+d\int_{\mathbb{R}^{3}}|\nabla u|^{q}\, dx \right ) \Delta_{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, $\frac{3}{2}< p< q<3$, $V: \mathbb{R}^{3}\rightarrow \mathbb{R}$ and $K: \mathbb{R}^{3}\rightarrow \mathbb{R}$ 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.