Positive solutions of critical Hardy-H\'{e}non equations with logarithmic term

Abstract

We consider the existence, non-existence and multiplicity of positive solutions to the following critical Hardy-H\'{e}non equation with logarithmic term \begin{equation*}\label{eq11}\left\{ \begin{array}{ll} -\Delta u =|x|^{\alpha}|u|^{2^*_{\alpha}-2}\cdot u+\mu u\log u^2+\lambda u, &x\in \Omega,\\ u=0, &x\in \partial \Omega,\\ \end{array} \right.\end{equation*} where $ \Omega=B$ for $\alpha\geq 0$, $ \Omega=B\setminus\{0\}$ for $\alpha\in(-2,0)$, $B\subset\mathbb{R}^N$ is an unit ball, $\lambda, \mu \in \mathbb{R}$, $N\geq 3, \alpha>-2$, $2^*_{\alpha}:=\frac{2(N+\alpha)}{N-2}$ is the critical exponent for the embedding $H_{0,r}^{1}( \Omega)\hookrightarrow L^p( \Omega;|x|^\alpha)$, and which can be seen as a Br\'{e}zis-Nirenberg problem. When $N \geq 4$ and $\mu>0$, we will show that the above problem has a positive Mountain pass solution, which is also a ground state solution. At the same time, when $\mu<0$, under some assumptions on the $N$, $\mu$, $\lambda$ and $\alpha$, we will show that the above problem has at least a positive least energy solution and at least a positive Mountain pass solution, respectively. What's more, when certain inequality related to $N \geq 3$, $\mu<0 $ and $\alpha\in(-2,0]$ holds, we will demonstrate the non-existence of positive solutions to the above-mentioned problem. The presence of logarithmic term brings some new and interesting phenomena to this problem.