Liouville theorems for mixed local and nonlocal indefinite equations

TL;DR

This paper studies positive solutions to mixed local and nonlocal equations, establishing maximum principles, monotonicity, and nonexistence results, and extends these findings to parabolic equations with dual nonlocal features.

Contribution

It introduces a novel adaptation of the method of moving planes for mixed local-nonlocal problems, deriving Liouville-type and qualitative results without standard decay assumptions.

Findings

Proved maximum principles and strict monotonicity for mixed elliptic operators.

Derived nonexistence results for the operator (-Δ)^s - Δ.

Extended qualitative results to parabolic equations with fractional time derivatives.

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 $ (-\Delta)^s - \Delta$ 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.