Impact of concave–convex nonlinearities on the multiplicity of solutions for an elliptic critical system

In this paper, we consider the following elliptic systems with critical Sobolev growth \(\begin{aligned} {\left\{ \begin{array}{ll}-\Delta u=\frac{2 \alpha }{\alpha +\beta }\vert u\vert ^{\alpha -2} u\vert v\vert ^{\beta }+\lambda \vert u\vert ^{q-2} u & \text{ in } \Omega , \\ -\Delta v=\frac{2 \beta }{\alpha +\beta }\vert u\vert ^{\alpha }\vert v\vert ^{\beta -2} v+\mu \vert v\vert ^{q-2} v& \text{ in } \Omega , \\ u=v=0 & \text{ on } \partial \Omega ,\end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u=\frac{2\alpha }{\alpha +\beta }{|u|}^{\alpha -2}{u|v|}^{\beta }+\lambda {|u|}^{q-2}u\hfill & \text{in}\mathrm{\Omega },\hfill \\ -\mathrm{\Delta }v=\frac{2\beta }{\alpha +\beta }{|u|}^{\alpha }{|v|}^{\beta -2}v+\mu {|v|}^{q-2}v\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=v=0\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega\) $\mathrm{\Omega }$ is a smooth bounded domain in \({\mathbb {R}}^{N}\) ${\mathbb{R}}^N$ , \(1<q<2\) $1<q<2$ , \(\lambda , \mu \geqslant 0\) $\lambda ,\mu ⩾0$ and \(\lambda +\mu >0\) $\lambda +\mu >0$ , \(\alpha\) $\alpha$ , \(\beta >1\) $\beta >1$ satisfying \(\alpha +\beta =2^{*}\) $\alpha +\beta =2_{}^{\ast }$ , \(2^{*}=\frac{2 N}{N-2}\) $2_{}^{\ast }=\frac{2N}{N-2}$ . We prove that if \(N>2\frac{q+1}{q-1}\) $N>2\frac{q+1}{q-1}$ , then the above problem has two disjoint and infinite sets of solutions.