A note on a critical problem with natural growth in the gradient

Abstract

The paper analyzes the influence on the meaning of natural growth in the gradient, of a perturbation by a Hardy potential in some elliptic equations. We obtain a linear differential operator that, in a natural way, is the corresponding gradient for the perturbed elliptic problem.

The main results are: i) Optimal summability of the data to have weak solutions; ii) Optimal linear operator associated, and, iii) Multiplicity and characterization of all solutions in terms of some measures. The results also are new for the Laplace operator perturbed for an inverse-square potential.