The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation

Abstract

We consider the existence and nonexistence of positive solution for the following Brézis-Nirenberg problem with logarithmic perturbation: \begin{equation*} \begin{cases} -Δu={\left|u\right|}^{{2}^{\ast }-2}u+λu+μu\log {u}^{2} &x\in Ω, \quad \;\:\, u=0& x\in \partial Ω, \end{cases} \end{equation*} where $\Omega$ $\subset$ ${\mathbb{R}}^N$ is a bounded smooth domain, $\lambda ,\mu \in \mathbb{R}$, $N\ge 3$ and $2^{\ast }:=\frac{2N}{N-2}$ is the critical Sobolev exponent for the embedding $H_0^1(\Omega )\hookrightarrow L^{2^{\ast }}(\Omega )$. The uncertainty of the sign of $s\log s^2$ in $(0,+\infty )$ has some interest in itself. We will show the existence of positive ground state solution which is of mountain pass type provided $\lambda \in \mathbb{R},\mu >0$ and $N\ge 4$. While the case of $\mu <0$ is thornier. However, for $N=3,4$ $\lambda \in (-\infty ,{\lambda }_1(\Omega ))$, we can also establish the existence of positive solution under some further suitable assumptions. And a nonexistence result is also obtained for $\mu <0$ and $-\frac{(N-2)\mu }{2}+\frac{(N-2)\mu }{2}\log (-\frac{(N-2)\mu }{2})+\lambda -{\lambda }_1(\Omega )\ge 0$ if $N\ge 3$. Comparing with the results in Brézis, H. and Nirenberg, L. (Comm. Pure Appl. Math. 1983), some new interesting phenomenon occurs when the parameter $\mu$ on logarithmic perturbation is not zero.