Compactness of solutions to nonlocal elliptic equations

TL;DR

This paper proves local boundedness and compactness of solutions to certain nonlocal elliptic equations involving the regional fractional Laplacian, highlighting differences from the standard fractional Laplacian case and analyzing potential behavior at blow-up points.

Contribution

It establishes universal boundedness and solution compactness for critical nonlocal elliptic equations with regional fractional Laplacian and nonnegative potentials, including potential vanishing rate analysis.

Findings

All nonnegative solutions are locally bounded.

Solutions are compact when potentials have non-degenerate zeros.

Potential vanishing rate at blow-up points matches conjectured behavior.

Abstract

We show that all nonnegative solutions of the critical semilinear elliptic equation involving the regional fractional Laplacian are locally universally bounded. This strongly contrasts with the standard fractional Laplacian case. Second, we consider the fractional critical elliptic equations with nonnegative potentials. We prove compactness of solutions provided the potentials only have non-degenerate zeros. Corresponding to Schoen's Weyl tensor vanishing conjecture for the Yamabe equation on manifolds, we establish a Laplacian vanishing rate of the potentials at blow-up points of solutions.