Singular solutions of fully nonlinear elliptic equations and applications

Details Singular solutions of fully nonlinear elliptic equations and applications Singular solutions of fully nonlinear elliptic equations and applications

2011-04-28

arxiv.org/abs/1104.5338v1

Mathematics

Analysis of PDEs

41 pages, 2 figures

Scientific paper

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.

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