Positive solutions of critical Hardy-Hénon equations with logarithmic term

Abstract

We consider the existence, non-existence and multiplicity of positive solutions to the following critical Hardy-Hénon equation with logarithmic term \begin{equation*}\label{eq11}\left\{ \begin{array}{ll} -Δu =|x|^α|u|^{2^*_α-2}\cdot u+μu\log u^2+λu, &x\in Ω,\\ u=0, &x\in \partial Ω,\\ \end{array} \right.\end{equation*} where $\Omega =B$ for $\alpha \ge 0$, $\Omega =B\setminus \{0\}$ for $\alpha \in (-2,0)$, $B\subset {\mathbb{R}}^N$ is an unit ball, $\lambda ,\mu \in \mathbb{R}$, $N\ge 3,\alpha >-2$, $2_{\alpha }^{\ast }:=\frac{2(N+\alpha )}{N-2}$ 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 Brézis-Nirenberg problem. When $N\ge 4$ 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 $N\ge 3$, $\mu <0$ 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.