Local behavior and global existence of positive solutions of \(au^\lambda\leq -\Delta u\leq u^\lambda\). (Q1861112)

The author deals with the problem \[ au^\lambda\leq -\Delta u\leq u^\lambda \tag{1} \] in various subsets of \(\mathbb{R}^n\), where \(a\in(0,1)\) and \(n>2\). He investigates the behaviour near the origin of \(C^2\) positive solutions \(u(x)\) of (1), in a punctured neighborhood of the origin in \(\mathbb{R}^n\), where \(\frac{n}{n-2} <\lambda< \frac{n+2}{n-2}\). Moreover, he studies the existence of \(C^2\) positive solutions of (1) on \(\mathbb{R}^n\). The author investigates the qualitative behaviour of solutions of (1) depending on the parameter \(a\).