Existence and nonexistence of solutions for weighted elliptic inequalities involving gradient terms

Abstract

In this paper we prove existence and nonexistence theorems for positive solutions of elliptic inequalities for general quasilinear operators, including $m$-Laplacian, mean curvature and generalized mean curvature operator, in the entire $R^N$ with a reaction involving power type gradient terms and positive weights, possibly singular or degenerate. A complete picture for the exponents involved is given. The proof technique is based on cumbersome integral a priori estimates, in the spirit of the nonlinear capacity method. No maximum principle or growth conditions at infinity for the solutions are required.