Sharp a priori estimates for some nonlinear elliptic problems

Abstract

We consider homogeneous Dirichlet problems whose prototype is \[ - \Delta_p u = h | \nabla u |^q + f \quad \mbox{in \(\Omega \),} \quad u = 0 \quad \mbox{in \( \partial \Omega \)}.\] We give existence results focusing our attention on the sharpness of the conditions on the known term \( f \). All the estimates are the result of comparison results obtained via symmetrization methods.