Positive solution for an elliptic system with critical exponent and logarithmic terms: the higher-dimensional cases

Abstract

In this paper, we consider the coupled elliptic system with critical exponent and logarithmic terms:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\lambda _{1}u+ \mu _1|u|^{2p-2}u+\beta |u|^{p-2}|v|^{p}u+\theta _1 u\log u^2, &{} \quad x\in \Omega ,\\ -\Delta v=\lambda _{2}v+ \mu _2|v|^{2p-2}v+\beta |u|^{p}|v|^{p-2}v+\theta _2 v\log v^2, &{}\quad x\in \Omega ,\\ u=v=0, &{}\quad x \in \partial \Omega , \end{array}\right. } \end{aligned}$$

where $\Omega \subset {\mathbb R}^N$ is a bounded smooth domain, $2p=2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent. When $N \ge 5$, for different ranges of $\beta ,\lambda _{i},\mu _i,\theta _{i}$, $i=1,2$, we obtain existence and nonexistence results of positive solutions via variational methods. The special case $N=4$ was studied by Hajaiej et al. (Positive solution for an elliptic system with critical exponent and logarithmic terms, arXiv:2304.13822, 2023). Note that for $N\ge 5$, the critical exponent is given by $2p\in \left( 2,4\right)$; whereas for $N=4$, it is $2p=4$. In the higher-dimensional cases $N\ge 5$ brings new difficulties, and requires new ideas. Besides, we also study the Brézis–Nirenberg problem with logarithmic perturbation

$$\begin{aligned} -\Delta u=\lambda u+\mu |u|^{2p-2}u+\theta u \log u^2 \quad \text { in }\Omega , \end{aligned}$$

where $\mu >0, \theta <0$, $\lambda \in {\mathbb R}$, and obtain the existence of positive local minimum and least energy solution under some certain assumptions.