Existence of sign-changing solutions for $p(x)$-Laplacian Kirchhoff type problem in $\mathbb{R}^N$

Abstract

The $p(x)$-Laplacian Kirchhoff type equation involving the nonlocal term $b \int_{\mathbb{R}^N} (1/p(x)) \lvert\nabla u\rvert^{p(x)}dx$ is investigated. Based on the variational methods, deformation lemma and other technique of analysis, it is proved that the problem possesses one least energy sign-changing solution $u_b$ which has precisely two nodal domains. Moreover, the convergence property of $u_b$ as the parameter $b \searrow 0$ is also obtained.