Positive Solution for an Elliptic System with Critical Exponent and Logarithmic Terms

In this paper, we study the existence and nonexistence of positive solutions for the following coupled elliptic system with critical exponent and logarithmic terms: \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\lambda _{1}u+ \mu _1|u|^{2}u+\beta |v|^{2}u+\theta _1 u\log u^2, &{} \quad x\in \Omega ,\\ -\Delta v=\lambda _{2}v+ \mu _2|v|^{2}v+\beta |u|^{2}v+\theta _2 v\log v^2, &{}\quad x\in \Omega ,\\ u=v=0, &{}\quad x \in \partial \Omega , \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u={\lambda }_{1}u+{\mu }_1{|u|}^{2}u+\beta {|v|}^{2}u+{\theta }_{1}ulog{u}^2,\hfill & x\in \mathrm{\Omega },\hfill \\ -\mathrm{\Delta }v={\lambda }_{2}v+{\mu }_2{|v|}^{2}v+\beta {|u|}^{2}v+{\theta }_{2}vlog{v}^2,\hfill & x\in \mathrm{\Omega },\hfill \\ u=v=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \subset {{\mathbb {R}}}^4\) $\mathrm{\Omega }\subset {\mathbb{R}}^4$ is a bounded smooth domain, the parameters \(\lambda _{1},\lambda _{2},\theta _{1},\theta _{2}\in {{\mathbb {R}}}\) ${\lambda }_1,{\lambda }_2,{\theta }_1,{\theta }_2\in \mathbb{R}$ , \(\mu _{1},\mu _2>0\) ${\mu }_1,{\mu }_2>0$ and \(\beta \ne 0\) $\beta \ne 0$ is a coupling constant. Note that the logarithmic term \(s\log s^2\) $slog{s}^2$ has special properties, which makes the problem more complicated. We show that this system has a positive least energy solution for \(|\beta |\) $|\beta |$ small and positive large \(\beta \) $\beta$ if \(\lambda _{1},\lambda _{2}\in {{\mathbb {R}}}\) ${\lambda }_1,{\lambda }_2\in \mathbb{R}$ and \(\theta _{1},\theta _{2}>0\) ${\theta }_1,{\theta }_2>0$ . While the situations for the case \(\theta _{1},\theta _{2}<0\) ${\theta }_1,{\theta }_2<0$ are quite thorny, in this challenging setting we establish the existence result of positive local minimum solutions and nonnegative solutions under various conditions on the parameters. Besides, under some further assumptions, we obtain the nonexistence of positive solutions for both the case where \(\theta _{1}\) ${\theta }_1$ , \(\theta _{2}\) ${\theta }_2$ are negative and the case where they have opposite signs. Comparing our results with those of Chen and Zou (Arch. Ration. Mech. Anal. 205:515–551, 2012), the logarithmic term \(s\log s^2\) $slog{s}^2$ introduces some new interesting phenomenon. Moreover, its presence brings major challenges and make it difficult to use the comparison theorem used in the work of Chen and Zou without new ideas and innovative techniques. To the best of our knowledge, our paper is the first to give a rather complete picture for the existence and nonexistence results to the coupled elliptic system with critical exponent and logarithmic terms. Also, we consider the related single equation \(\begin{aligned} -\Delta u=\lambda u + \mu |u|^{2}u+\theta u\log u^2, ~u\in H_0^1(\Omega ) \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u=\lambda u+{\mu |u|}^{2}u+\theta ulog{u}^2,u\in H_0^1(\mathrm{\Omega })\end{array}$$ with \(\mu >0\) $\mu >0$ , \(\theta <0\) $\theta <0$ , \(\lambda \in {{\mathbb {R}}}\) $\lambda \in \mathbb{R}$ or \(\lambda \in [0,\lambda _{1}(\Omega ))\) $\lambda \in [0,{\lambda }_1(\mathrm{\Omega }))$ and prove the existence of the positive solution under some further suitable assumptions, which is the type of a local minimum or a least energy solution.