An elliptic boundary value problem with fractional nonlinearity

TL;DR

This paper studies the existence and uniqueness of solutions to elliptic boundary value problems involving fractional Laplacians with power nonlinearities, identifying critical powers and employing weighted Sobolev spaces to handle boundary singularities.

Contribution

It introduces a framework for analyzing elliptic problems with fractional nonlinearities, including the critical power threshold and boundary singularity management.

Findings

Identified the critical power separating existence and non-existence regimes.

Established existence and uniqueness results for certain fractional nonlinear elliptic problems.

Developed a weighted Sobolev space approach to handle boundary singularities.

Abstract

We investigate existence and uniqueness of solutions to second-order elliptic boundary value problems containing a power nonlinearity applied to a fractional Laplacian. We detect the critical power separating the existence from the non-existence regimes. For the existence results, we make use of a particular class of weighted Sobolev spaces to compensate boundary singularities which are naturally built in the problem.