Positive solutions for the fractional Laplacian in the almost critical case in a bounded domain

TL;DR

This paper establishes the existence of multiple positive solutions for a fractional scalar field equation in a bounded domain as the exponent approaches the critical Sobolev value, using topological methods.

Contribution

It introduces the use of the photography method to relate domain topology to the number of solutions in the fractional Laplacian context.

Findings

Multiple positive solutions are proven to exist near the critical exponent.

The domain's topology provides a lower bound on the number of solutions.

The photography method effectively links domain topology to solution multiplicity.

Abstract

We prove existence of multiple positive solutions for a {\sl fractional scalar field equation} in a bounded domain, whenever $p$ tends to the critical Sobolev exponent. By means of the "photography method", we prove that the topology of the domain furnishes a lower bound on the number of positive solutions.