Harnack inequality for singular fully nonlinear operators and some existence results

Abstract

In this article we further advance in the theory of singular fully nonlinear operators modeled on the q-laplacian proving a Harnack inequality. We provide also several applications of this inequality and the ideas used for proving it. In doing so we have left various open questions, all of them related to the fact that the operator is not sub-linear. Mathematics Subject Classification (2000)Primary 35D40 · Secondary 35B45

Eigenfunctions for singular fully nonlinear equations in unbounded domains

“…The proofs that we propose follow the lines in Gilbarg Trudinger [19] and Serrin [25], with some new arguments that make explicite use of the eigenfunction in bounded domains. This extends the result of [14] to the case α > 0, but only in the two dimensional case.…”

Global $$W^{2,\delta }$$ W 2 , δ estimates for a type of singular fully nonlinear elliptic equations

“…I. Birindelli and F. Demengel proved comparison principle [BD04] and C 1,α estimate [BD10]. G. Dávila, P. Felmer and A. Quaas proved Alexandroff-Bakelman-Pucci (ABP for short) estimate [DFQ09] and Harnack inequality [DFQ10]. To the best of our knowledge, W 2,δ estimate for this kind of equation is only known for γ = 0, that is the uniformly fully nonlinear elliptic equation.…”

Estimates on elliptic equations that hold only where the gradient is large

“…Known results. We next explain how results stated in [8,11,3] are related to the ones presented in this paper.…”