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Liouville type results for the fractional truncated Laplacians in a half-space

Giulio Galise, Hitoshi Ishii

TL;DR

The paper analyzes half-space problems for fully nonlinear integral operators given by the fractional truncated Laplacians $\mathcal I_k^\pm$, focusing on the balance between diffusion along lower-dimensional directions and a power nonlinearity $u^p$. It develops Liouville-type nonexistence results for various regimes of $p$ (including singular $p<0$ and sublinear $0<p\le1$) and derives threshold exponents that separate existence from nonexistence, with refined results for special cases $k=1$ and $s$ in $[\tfrac12,1)$. In parallel, it proves existence of supersolutions in complementary regimes: nonnegative supersolutions for all $p>0$ in the minimal operators, and decaying positive supersolutions for large $p$ in the maximal case, including explicit barrier constructions $u_\gamma$ and $\psi$ that drive the analysis. The results collectively map the feasibility of positive supersolutions across operator families $\mathcal I_k^\pm$ in $\mathbb R^N_+$, contributing to the understanding of nonlocal, degenerate elliptic equations and their connections to stochastic control and Liouville-type phenomena.

Abstract

Existence issues of viscosity supersolutions in the half-space $\mathbb R^N_+$, for a class of fully nonlinear integral equations involving the fractional truncated Laplacians and a power-like nonlinearity in the unknown function, are addressed in this paper, the aim being to obtain estimates on the threshold exponents separating the existence from the nonexistence regimes.

Liouville type results for the fractional truncated Laplacians in a half-space

TL;DR

The paper analyzes half-space problems for fully nonlinear integral operators given by the fractional truncated Laplacians , focusing on the balance between diffusion along lower-dimensional directions and a power nonlinearity . It develops Liouville-type nonexistence results for various regimes of (including singular and sublinear ) and derives threshold exponents that separate existence from nonexistence, with refined results for special cases and in . In parallel, it proves existence of supersolutions in complementary regimes: nonnegative supersolutions for all in the minimal operators, and decaying positive supersolutions for large in the maximal case, including explicit barrier constructions and that drive the analysis. The results collectively map the feasibility of positive supersolutions across operator families in , contributing to the understanding of nonlocal, degenerate elliptic equations and their connections to stochastic control and Liouville-type phenomena.

Abstract

Existence issues of viscosity supersolutions in the half-space , for a class of fully nonlinear integral equations involving the fractional truncated Laplacians and a power-like nonlinearity in the unknown function, are addressed in this paper, the aim being to obtain estimates on the threshold exponents separating the existence from the nonexistence regimes.
Paper Structure (13 sections, 23 theorems, 313 equations)

This paper contains 13 sections, 23 theorems, 313 equations.

Table of Contents

  1. Introduction
  2. Some preliminaries
  3. Liouville-type results
  4. The singular case $p<0$
  5. The sublinear case $0<p\leq1$
  6. Theorems \ref{['th3']}-\ref{['th4']} for $p>1$
  7. The case $\mathcal{I}^+_1$ with $s=\frac{1}{2}$
  8. The case $\mathcal{I}^+_1$ with $\frac{1}{2}<s<1$
  9. Existence of supersolutions
  10. Theorem \ref{['th1']}
  11. Theorem \ref{['thIN']}
  12. Theorem \ref{['th-singular']}
  13. Appendix

Key Result

Theorem 1.1

Let $u\in LSC(\mathbb R^N_+)\cap\mathcal{S}$ be a nonnegative viscosity supersolution of where $s\in\left(0,\frac{1}{2}\right)$ if $k=1$, $s\in(0,1)$ otherwise. If $0<p<1+\frac{2s}{\bar{\gamma}+1}$, then $u(x)\equiv0$ in $\mathbb R^N_+$.

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • ...and 43 more