Ground state sign-changing solutions for Kirchhoff equations with logarithmic nonlinearity

Abstract

In this paper, we study Kirchhoff equations with logarithmic nonlinearity: \begin{equation*} \begin{cases} -(a+b\int_\Omega|\nabla u|^2)\Delta u+ V(x)u=|u|^{p-2}u\ln u^2, & \mbox{in}\ \Omega,\\ u=0,& \mbox{on}\ \partial\Omega, \end{cases} \end{equation*} where $a,b>0$ are constants, $4<p<2^*$, $\Omega$ is a smooth bounded domain of $\mathbb{R}^3$ and $V:\Omega\to\mathbb{R}$. Using constraint variational method, topological degree theory and some new energy estimate inequalities, we prove the existence of ground state solutions and ground state sign-changing solutions with precisely two nodal domains. In particular, some new tricks are used to overcome the difficulties that $|u|^{p-2}u\ln u^2$ is sign-changing and satisfies neither the monotonicity condition nor the Ambrosetti–Rabinowitz condition.

Keywords

• logarithmic nonlinearity
• ground state solution
• sign-changing solution
• kirchhoff equations