Singular solutions of fully nonlinear elliptic equations and applications

TL;DR

This paper investigates the behavior of solutions to fully nonlinear elliptic equations near boundary singularities in Lipschitz domains, introducing new methods for classification and establishing sharp boundary principles.

Contribution

It introduces a novel approach for analyzing boundary singularities of nonlinear elliptic equations and classifies isolated boundary singularities with new boundary principles.

Findings

Existence of two positive solutions in each cone of ^n.

Uniqueness of solutions in an appropriate sense.

A sharp Phragme9n-Lindelf6f type result.

Abstract

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of $\mathbb{R}^n$, and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragm\'en-Lindel\"of result as well as a principle of positive singularities in certain Lipschitz domains.