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.

中文翻译：

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具有临界指数和对数项的椭圆系统的正解：高维情况

摘要

在本文中，我们考虑具有临界指数和对数项的耦合椭圆系统：$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\lambda _{1}u+ \ mu _1|u|^{2p-2}u+\beta |u|^{p-2}|v|^{p}u+\theta _1 u\log u^2, &{} \quad x\in \欧米伽 ,\\ -\Delta v=\lambda _{2}v+ \mu _2|v|^{2p-2}v+\beta |u|^{p}|v|^{p-2}v+\theta _2 v\log v^2, &{}\quad x\in \Omega ,\\ u=v=0, &{}\quad x \in \partial \Omega , \end{array}\right。} \end{aligned}$$其中\(\Omega \subset {\mathbb R}^N\)是有界平滑域，\(2p=2^*=\frac{2N}{N-2}\)是索博列夫临界指数。当\(N \ge 5\)时，对于不同范围的\(\beta ,\lambda _{i},\mu _i,\theta _{i}\) , \(i=1,2\)，我们通过变分方法得到正解存在和不存在的结果。Hajaiej 等人研究了特殊情况\(N=4 \) 。（具有临界指数和对数项的椭圆系统的正解，arXiv：2304.13822，2023）。请注意，对于\(N\ge 5\) ，临界指数由\(2p\in \left( 2,4\right) \)给出；而对于\(N=4\)，它是\(2p=4\)。在高维情况下\(N\ge 5\)带来了新的困难，需要新的思路。此外，我们还研究了对数扰动的 Brézis-Nirenberg 问题$$\begin{aligned} -\Delta u=\lambda u+\mu |u|^{2p-2}u+\theta u \log u^2 \quad \text { in }\Omega , \end{aligned}$$ where \(\mu >0, \theta <0\) , \(\lambda \in {\mathbb R}\)，并获得正数的存在性在某些特定假设下的局部最小和最小能量解。