Regularity of the extremal solutions associated to elliptic systems (Q743890)

The authors study a semilinear elliptic system with zero Dirichlet boundary depending upon some positive parameters. They prove that for general nonlinearities, the extremal solutions are bounded, provided that the domain is a convex domain in \(\mathbb R^N\) where \(N \leq 3\). In the case of a radial domain, they show the extremal solutions are bounded if \(N < 10\). Certain regularity results are also obtained in higher dimensions in the case of explicit power-like nonlinearities. Such kind of results are widely studied in literature. See, for instance, the contributions by Cabré, Capella, Montenegro, Nedev and Villegas.