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

In this paper, we consider the coupled elliptic system with critical exponent and logarithmic terms: \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\lambda _{1}u+ \mu _1|u|^{2p-2}u+\beta |u|^{p-2}|v|^{p}u+\theta _1 u\log u^2, &{} \quad x\in \Omega ,\\ -\Delta v=\lambda _{2}v+ \mu _2|v|^{2p-2}v+\beta |u|^{p}|v|^{p-2}v+\theta _2 v\log v^2, &{}\quad x\in \Omega ,\\ u=v=0, &{}\quad x \in \partial \Omega , \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u={\lambda }_{1}u+{\mu }_1{|u|}^{2p-2}u+{\beta |u|}^{p-2}{|v|}^{p}u+{\theta }_{1}ulog{u}^2,\hfill & x\in \mathrm{\Omega },\hfill \\ -\mathrm{\Delta }v={\lambda }_{2}v+{\mu }_2{|v|}^{2p-2}v+{\beta |u|}^p{|v|}^{p-2}v+{\theta }_{2}vlog{v}^2,\hfill & x\in \mathrm{\Omega },\hfill \\ u=v=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \subset {\mathbb R}^N\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded smooth domain, \(2p=2^*=\frac{2N}{N-2}\) $2p=2^{\ast }=\frac{2N}{N-2}$ is the Sobolev critical exponent. When \(N \ge 5\) $N\ge 5$ , for different ranges of \(\beta ,\lambda _{i},\mu _i,\theta _{i}\) $\beta ,{\lambda }_i,{\mu }_i,{\theta }_i$ , \(i=1,2\) $i=1,2$ , we obtain existence and nonexistence results of positive solutions via variational methods. The special case \(N=4 \) $N=4$ 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 \(N\ge 5\) $N\ge 5$ , the critical exponent is given by \(2p\in \left( 2,4\right) \) $2p\in \left(2,,,4\right)$ ; whereas for \(N=4\) $N=4$ , it is \(2p=4\) $2p=4$ . In the higher-dimensional cases \(N\ge 5\) $N\ge 5$ brings new difficulties, and requires new ideas. Besides, we also study the Brézis–Nirenberg problem with logarithmic perturbation \(\begin{aligned} -\Delta u=\lambda u+\mu |u|^{2p-2}u+\theta u \log u^2 \quad \text { in }\Omega , \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u=\lambda u+{\mu |u|}^{2p-2}u+\theta ulog{u}^2\text{in}\mathrm{\Omega },\end{array}$$ where \(\mu >0, \theta <0\) $\mu >0,\theta <0$ , \(\lambda \in {\mathbb R}\) $\lambda \in \mathbb{R}$ , and obtain the existence of positive local minimum and least energy solution under some certain assumptions.