Uniqueness and existence of positive solutions for some semilinear elliptic systems (Q1888598)

The author deals with positive solutions of the following semilinear elliptic system \[ \begin{cases} -\Delta u_i= \prod^N_{j=1} u^{p_{ij}}_j\quad &\text{in }\Omega,\\ u_i= 0\quad &\text{on }\partial\Omega,\quad i= 1,2,\dots, N,\end{cases}\tag{1} \] where \(p_{ij}\) \((1\leq i,\;j\leq N)\) are nonnegative constants and \(\Omega\subset\mathbb{R}^n\) \((n\geq 1)\) denotes a bounded domain of class \(C^{2,\alpha}\), \(\alpha\in (0,1)\). The author shows the uniqueness and existence of positive solutions for (1). To this end, he uses a concept of the M-matrix.