Sign-changing solutions for a class of <em>p</em>-Laplacian Kirchhoff-type problem with logarithmic nonlinearity

Abstract

In this paper, we study the existence of ground state sign-changing solutions for following $p$-Laplacian Kirchhoff-type problem with logarithmic nonlinearity $ \begin{equation*} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -(a+ b\int _{\Omega}|\nabla u|^{p}dx)\Delta_p u = |u|^{q-2}u\ln u^2, ~x\in\Omega \\ u = 0, ~ x\in \partial\Omega, \end{array} \right. \end{equation*} $ where $\Omega\subset \mathbb{R}^{N}$ is a smooth bounded domain, $a, b \gt 0$ are constant, $4\leq 2 p \lt q \lt p^*$ and $N \gt p$. By using constraint variational method, topological degree theory and the quantitative deformation lemma, we prove the existence of ground state sign-changing solutions with precisely two nodal domains.