A priori estimates for some elliptic equations involving the $p$-Laplacian
Lucio Damascelli, Rosa Pardo San Gil · arXiv (Cornell University) · 2017
We consider the Dirichlet problem for positive solutions of the equation $-\\Delta_p (u) = f(u)$ in a convex, bounded, smooth domain $\\Omega \\subset\\R^N$, with $f$ locally Lipschitz continuous. \\par We provide sufficient conditions guarantying $L^{\\infty} $ a priori bounds for positive solutions of some elliptic equations involving the $p$-Laplacian and extend the class of known nonlinearities for which the solutions are $L^{\\infty} $ a priori bounded. As a consequence we prove the existence of positive solutions in convex bounded domains.