Existence and multiplicity of positive solutions to a critical elliptic equation with logarithmic perturbation

Abstract: We consider the existence and multiplicity of positive solutions for the following critical problem with logarithmic term: \begin{equation*}\label{eq11}\left{ \begin{array}{ll} -\Delta u={\mu\left|u\right|}^{{2}^{\ast }-2}u+\nu |u|^{q-2}u+\lambda u+\theta u\log {u}^{2}, &x\in \Omega,\ u=0, &x\in \partial \Omega,\ \end{array} \right.\end{equation*} where $\Omega$ $\subset$ $\mathbb{R}^N$ is a bounded smooth domain, $ \nu, \lambda\in \mathbb{R}$, $\mu>0, \theta<0$, $N\ge3$, ${2}^{\ast }=\frac{2N}{N-2}$ is the critical Sobolev exponent for the embedding $H^1_{0}(\Omega)\hookrightarrow L^{2^\ast}(\Omega)$ and $q\in (2, 2^*)$, and which can be seen as a Br$\acute{e}$zis-Nirenberg problem. Under some assumptions on the $\mu, \nu, \lambda, \theta$ and $q$, we will prove that the above problem has at least two positive solutions: One is the least energy solution, and the other one is the Mountain pass solution. As far as we know, the existing results on the existence of positive solutions to a Br$\acute{e}$zis-Nirenberg problem are to find a positive solution, and no one has given the existence of at least two positive solutions on it. So our results is totally new on this aspect.