# Boundary singularities of solutions to semilinear fractional equations

## TL;DR

This paper investigates boundary singularities and solutions to semilinear fractional equations, establishing existence results for solutions with prescribed boundary measures and analyzing critical exponents for specific nonlinearities.

## Contribution

It introduces new existence results for solutions with boundary measure data and characterizes critical exponents for power nonlinearities in fractional equations.

## Key findings

- Existence of solutions with prescribed boundary Radon measures.
- Identification of critical exponents for power nonlinearities.
- Analysis of boundary trace for positive moderate solutions.

## Abstract

We prove the existence of a solution of (--$\Delta$) s u + f (u) = 0 in a smooth bounded domain $\Omega$ with a prescribed boundary value $\mu$ in the class of positive Radon measures for a large class of continuous functions f satisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where f (u) = u p and $\mu$ is a Dirac mass, we prove the existence of several critical exponents p.