Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Liouville-type theorems for Lane--Emden inequalities involving nonlocal operators

T. Kim, T. Lee

Abstract

We establish a Liouville-type theorem for nonnegative weak supersolutions to $\mathcal{L}_K u = u^q$ in $\mathbb{R}^n$, where $\mathcal{L}_K$ is a translation-invariant integro-differential operator of order $2s$ with $s \in (0,1)$. The kernel $K$ is assumed to be even and satisfy uniform ellipticity bounds. We prove that the only nonnegative supersolution is the trivial one $u \equiv 0$ in the range $1 < q \le \frac{n}{n-2s}$ for $n > 2s$ (and for all $q > 1$ when $n \le 2s$). Our proof is elementary and relies on a test function method combined with a dyadic decomposition of the nonlocal tail. Notably, our argument does not rely on the maximum principle or the fundamental solution.

Liouville-type theorems for Lane--Emden inequalities involving nonlocal operators

Abstract

We establish a Liouville-type theorem for nonnegative weak supersolutions to in , where is a translation-invariant integro-differential operator of order with . The kernel is assumed to be even and satisfy uniform ellipticity bounds. We prove that the only nonnegative supersolution is the trivial one in the range for (and for all when ). Our proof is elementary and relies on a test function method combined with a dyadic decomposition of the nonlocal tail. Notably, our argument does not rely on the maximum principle or the fundamental solution.
Paper Structure (7 sections, 5 theorems, 61 equations)

This paper contains 7 sections, 5 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $L^q$-growth estimates and a dyadic inequality
  4. Proof of Theorem \ref{['thm:main']}
  5. The subcritical case: $1 < q < \frac{n}{n-2s}$
  6. The critical case: $q = \frac{n}{n-2s}$
  7. The low dimensional case: $n \le 2s$

Key Result

Theorem 1.1

Let $n \ge 1$, $s \in (0,1)$, and let $K$ satisfy eq:K-assump. Assume that $q$ satisfies eq:q-range. Then any nonnegative weak supersolution $u$ of eq:main-ineq is trivial; namely, $u \equiv 0$ in $\mathbb{R}^n$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Proposition 1.2
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: Cutoff estimate
  • proof
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 2 more