On isolated singularities for fractional Lane-Emden equation in the Serrin's critical case

TL;DR

This paper classifies isolated singularities of positive solutions to the fractional Lane-Emden equation at the critical exponent, introducing a novel approach that transforms the problem into a subcritical case involving the fractional Hardy operator.

Contribution

It develops an innovative, self-contained method to analyze singularities in fractional elliptic equations at the Serrin's critical exponent, applicable to both fractional and classical Laplacian cases.

Findings

Classified non-removable singularities of solutions.

Constructed singular solutions using special subsolutions.

Established a sequence of singular solutions parameterized by blow-up coefficients.

Abstract

In this paper, we solve the fractional Lane-Emden equation in the Serrin's critical case for the fractional Laplacian by developing an innovative and self-contained approach that also applies to the classical setting ( Laplacian). We give a classification of the isolated singularities of positive solutions to the semilinear fractional elliptic equations $$(E) \qquad\qquad (-\Delta)^s u = u^{\frac{N}{N-2s}}\quad {\rm in}\ \ \Omega\setminus\{0\},\qquad u\geq 0\quad{\rm in}\ \ \mathbb{R}^N\setminus\Omega,\qquad\qquad\quad $$ where $s\in(0,1)$, $\Omega$ is a bounded domain containing the origin in $\mathbb{R}^N$ with $N>2s$ and $\frac{N}{N-2s}$ is the Serrin's critical exponent. We use an initial asymptotic at infinity to transform the critical case into a subcritical case where the underlying equation involves the fractional Hardy operator. The construction of singular solutions is based on the fact that some special functions are subsolutions of the original problem near the origin. We also classify the non-removable singularities of the solutions of $(E)$ and show the existence of a sequence of isolated singular solutions parameterizing the coefficients of the second order blow up term. To the best of our knowledge, this idea has also been first used in this work.