Unique continuation property and local asymptotics of solutions to fractional elliptic equations

TL;DR

This paper investigates the local behavior and unique continuation properties of solutions to fractional elliptic equations with Hardy-type potentials, using advanced mathematical tools to describe their asymptotics near singularities.

Contribution

It provides a detailed analysis of solution asymptotics and establishes unique continuation results for fractional elliptic equations with singular potentials.

Findings

Exact asymptotic behavior near singularities

Unique continuation properties proven

Application of Almgren monotonicity formula

Abstract

Asymptotics of solutions to fractional elliptic equations with Hardy type potentials is studied in this paper. By using an Almgren type monotonicity formula, separation of variables, and blow-up arguments, we describe the exact behavior near the singularity of solutions to linear and semilinear fractional elliptic equations with a homogeneous singular potential related to the fractional Hardy inequality. As a consequence we obtain unique continuation properties for fractional elliptic equations.