The effect of the domain topology on the number of positive solutions for an elliptic system

In this paper we prove an existence result of multiple positive solutions for the following system $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u= \frac{2 \alpha _\epsilon }{\alpha _\epsilon +\beta _\epsilon }|u|^{\alpha _\epsilon -2}u |v|^{\beta _\epsilon }&{} \text{ in } \Omega , \\ -\Delta v= \frac{2 \beta _\epsilon }{\alpha _\epsilon +\beta _\epsilon }|u|^{\alpha _\epsilon } |v|^{\beta _\epsilon -2 }v&{} \text{ in } \Omega , \\ u= v =0 &{} \text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a smooth and bounded domain in $${\mathbb {R}}^{N}$$ , $$N\ge 3$$ , $$\alpha _\epsilon = \alpha - \frac{\epsilon }{2}$$ , $$\beta _\epsilon =\beta - \frac{\epsilon }{2}$$ , $$\alpha , \beta >1$$ and $$\alpha +\beta = 2^*$$ , where $$2^{*}=\frac{2N}{N-2}$$ . More specifically, we prove that, for $$\epsilon >0$$ small, the number of positive solutions is estimated below by topological invariants of the domain $$\Omega $$ : the Ljusternick–Schnirelmann category.