Harnack Inequalities and Bôcher‐Type Theorems for Conformally Invariant, Fully Nonlinear Degenerate Elliptic Equations

Abstract

We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions. © 2014 Wiley Periodicals, Inc.

Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations

“…Although there have been many works on a priori estimates for solutions to (4) and (5) and closely related issues (see e.g. [9,10,14,15,16,17,18,24,25,31,32,33,34,35,38,39,41,43,44]), our theorem above appears to be the first regularity result for viscosity solutions in this context.…”

“…As a consequence of Theorem 1.1, several previously known results for Lipschitz continuous solutions of (5) hold for continuous solutions. This includes the Liouville-type Theorem 1.4, the symmetry results Theorem 1.18 and Theorem 1.23 in [33]; the Bôcher-type Theorems 1.2 and 1.3, the Harnack-type Theorem 1.5, and the asymptotic behavior results Corollary 1.7 and Theorem 1.8 in [34].…”

A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound

“…Nevertheless, Problem 1.1 is largely open for 2 < k < n 2 and Problem 1.2 is largely open for 2 ≤ k < n 2 . In [30], we began our study on Problem 1.2. In that paper, we restricted our attention to a locally conformally flat setting and established various asymptotic behavior near isolated singularities of the degenerate elliptic equation which arises naturally in the study of (5), namely λ(A gu ) ∈ ∂Γ in a punctured ball.…”

“…In this sequel to [30], we study compactness of solutions of (5). We consider the following equation with a more general right hand side:…”

Symmetry, quantitative Liouville theorems and analysis of large solutions of conformally invariant fully nonlinear elliptic equations

“…When = 1 , (5) is a consequence of (3) and (4) (cf. [13,Proposition B.1]). However, this does not have to be the case when = 1 , for example when…”