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} -Δu={μ\left|u\right|}^{{2}^{\ast }-2}u+ν|u|^{q-2}u+λu+θu\log {u}^{2}, &x\in Ω,\\ u=0, &x\in \partial Ω,\\ \end{array} \right.\end{equation*} where $\Omega$ $\subset$ ${\mathbb{R}}^N$ is a bounded smooth domain, $\nu ,\lambda \in \mathbb{R}$, $\mu >0,\theta <0$, $N\ge 3$, $2^{\ast }=\frac{2N}{N-2}$ is the critical Sobolev exponent for the embedding $H_0^1(\Omega )\hookrightarrow L^{2^{\ast }}(\Omega )$ and $q\in (2,2^{\ast })$, 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.