Ground state sign-changing solution for a logarithmic Kirchhoff-type equation in $\mathbb{R}^{3}$

Abstract

We investigate the following logarithmic Kirchhoff-type equation: \begin{equation*} \left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}+V(x)u^{2}dx\right)[-\Delta u+V(x)u]=|u|^{p-2}u\ln |u|,\qquad x\in\mathbb{R}^{3}, \end{equation*} where $a,b>0$ are constants, $4<p<2^{*}=6$. Under some appropriate hypotheses on the potential function $V$, we prove the existence of a positive ground state solution, a ground state sign-changing solution and a sequence of solutions by using the constraint variational methods, topological degree theory, quantitative deformation lemma and symmetric mountain pass theorem. Our results complete those of Gao et al. [Appl. Math. Lett. 139(2023), 108539] with the case of $4<p<6$.

Keywords

• kirchhoff-type equation
• logarithmic nonlinearity
• ground state sign-changing solution
• variational methods
• topological degree theory