The fundamental solution of the fractional p-laplacian

TL;DR

This work derives explicit fundamental solutions for the fractional $p$-Laplacian, providing the power-type form $v_\beta(x)=|x|^\beta$ for $ps\neq N$ and the logarithmic form for $ps=N$, with precise conditions and signs for the associated constant $\mathcal{C}(\beta)$. Using these fundamental solutions, the authors prove two Liouville-type theorems: (i) in the subcritical regime $N\le ps$, nonnegative supersolutions are constant, and (ii) in the nonlocal nonlinear setting with a zero-order nonlinearity $u^q$, the subcritical case forces $u\equiv0$ while the supercritical case admits a positive solution. The development of Hadamard-type properties via comparison with modified fundamental solutions yields sharp asymptotic controls for nonnegative supersolutions in the regime $N>ps$, and together with weak/viscosity solution frameworks, broadens Liouville theory to nonlocal, nonlinear operators with potential implications for regularity and blow-up analysis in fractional PDEs.

Abstract

In this article, we find the fundamental solution of the fractional p-laplacian and use them to prove two different Liouville-type theorems. A non-existence classical Liouville-type theorem for p-superharmonic and a Louville type results for an Emden-Folder type equation with the fractional p-laplacian.

Theorems & Definitions (32)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Theorem 1.2: First Liouville-type theorem
• Remark 1.3
• Theorem 1.3: Second Liouville-type theorem
• Remark 2.1
• Definition 2.1: Weak solution
• Definition 2.2: Viscosity Solution
• Theorem 3.1