Boundedness of the extremal solution of some p-Laplacian problems

Abstract

In this article we consider the p-Laplace equation on a smooth bounded domain of with zero Dirichlet boundary conditions. Under adequate assumptions on f we prove that the extremal solution of this problem is in the energy class independently of the domain. We also obtain Lq and W1,q estimates for such a solution. Moreover, we prove its boundedness for some range of dimensions depending on the nonlinearity f.