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Liouville theorems for mixed local and nonlocal indefinite equations

Pengyan Wang, Leyun Wu

Abstract

We investigate the qualitative properties of positive solutions to mixed local-nonlocal equations with indefinite nonlinearities, emphasizing the interaction between classical and fractional Laplacians. We first establish maximum principles and prove strict monotonicity along the $x_1$-direction for mixed elliptic operators. By combining a mollified first eigenfunction with a suitable sub-solution, we derive nonexistence results for the mixed operator $ (-Δ)^s - Δ$ via a contradiction argument. These results are further extended to the parabolic setting, incorporating both the Marchaud-type fractional time derivative and the classical first-order derivative, revealing new qualitative features under dual nonlocality. A key aspect of our approach is a careful adaptation of the method of moving planes to the mixed local-nonlocal context. By addressing the distinct scaling behaviors of local and nonlocal terms, the method yields monotonicity and Liouville-type results without standard decay assumptions, and provides a framework potentially applicable to a broader class of mixed elliptic and parabolic problems.

Liouville theorems for mixed local and nonlocal indefinite equations

Abstract

We investigate the qualitative properties of positive solutions to mixed local-nonlocal equations with indefinite nonlinearities, emphasizing the interaction between classical and fractional Laplacians. We first establish maximum principles and prove strict monotonicity along the -direction for mixed elliptic operators. By combining a mollified first eigenfunction with a suitable sub-solution, we derive nonexistence results for the mixed operator via a contradiction argument. These results are further extended to the parabolic setting, incorporating both the Marchaud-type fractional time derivative and the classical first-order derivative, revealing new qualitative features under dual nonlocality. A key aspect of our approach is a careful adaptation of the method of moving planes to the mixed local-nonlocal context. By addressing the distinct scaling behaviors of local and nonlocal terms, the method yields monotonicity and Liouville-type results without standard decay assumptions, and provides a framework potentially applicable to a broader class of mixed elliptic and parabolic problems.
Paper Structure (4 sections, 9 theorems, 243 equations)

This paper contains 4 sections, 9 theorems, 243 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Liouville theorem for mixed local and nonlocal indefinite elliptic equations
  4. Liouville theorem for mixed local and nonlocal indefinite parabolic equations

Key Result

Theorem 1.1

Let $u(x )\in C^2(\mathbb R^n)\cap {\mathcal{L}}_{2s}$ is a positive bounded classical solution of equation v1. Suppose that $u$ is uniformly continuous and the following conditions hold: (F1):$f$ is strictly increasing in $x_1$ and (F2):$g$ is locally Lipschitz and nondecreasing in $(0,+\infty)$. Moreover, $g'(u)$ is continuous near $u=0$, Then $u(x )$ is strictly monotone increasing in the $

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Lemma 2.1
  • Remark 2.2
  • proof : Proof of Lemma \ref{['lemf4']}
  • ...and 14 more