Multiplicity of Positive Solutions for a Semilinear Elliptic System with Strongly Coupled Critical Terms and Concave Nonlinearities

Abstract

In this paper, the following semilinear elliptic system involving strongly coupled critical terms and concave nonlinearities is considered

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\displaystyle \frac{\eta _1\alpha _1}{2^*}|u|^{\alpha _1-2}|v|^{\beta _1}u+\frac{\eta _2\alpha _2}{2^*}|u|^{\alpha _2-2}|v|^{\beta _2}u+a_1\frac{|u|^{q-2}u}{|x|^{\gamma }},&{}x\in \Omega ,\\ -\Delta v=\displaystyle \frac{\eta _1\beta _1}{2^*}|u|^{\alpha _1}|v|^{\beta _1-2}v+\frac{\eta _2\beta _2}{2^*}|u|^{\alpha _2}|v|^{\beta _2-2}v+a_2\frac{|v|^{q-2}v}{|x|^{\gamma }},&{}x\in \Omega ,\\ u,v>0,&{}x\in \Omega ,\\ u=v=0,&{} x\in \partial \Omega , \end{array}\right. } \end{aligned}$$

where $\Omega \subset \mathbb {R}^N(N\ge 3)$ is a bounded domain with smooth boundary and $0\in \Omega , 1<q<2$, $\eta _1+\eta _2>0, 0\le \eta _i<+\infty ,a_i>0, 0\le \gamma <N+q-\frac{qN}{2}, \alpha _i,\beta _i>1, \alpha _i+\beta _i=2^*=\frac{2N}{N-2}(i=1,2)$ is the critical Sobolev exponent. By the Nehari method and variational method, two positive solutions are obtained which generalizes and improves some corresponding results in the literature.