The Pohozaev identity for mixed local-nonlocal operator

TL;DR

This paper establishes a Pohozaev identity for a mixed local-nonlocal operator in bounded domains, leading to results on unique continuation and nonexistence of solutions, advancing understanding of such operators in PDE analysis.

Contribution

It introduces a Pohozaev identity for semilinear problems involving combined local and nonlocal operators, extending previous identities to this mixed setting.

Findings

Proves Pohozaev identity for mixed local-nonlocal operators

Derives unique continuation property for eigenfunctions

Shows nonexistence of solutions in star-shaped domains under certain conditions

Abstract

In this article we prove the Pohozaev identity for the semilinear Dirichlet problem of the form $-\Delta u + a(-\Delta)^s u = f(u)$ in $\Omega$, and $u=0$ in $\Omega^c$, where $a$ is a non-negative constant and $\Omega$ is a bounded $C^2$ domain. We also establish similar identity for systems of equations. As applications of this identity, we deduce a unique continuation property of eigenfunctions and also the nonexistence of nontrivial solutions in star-shaped domains under suitable condition on $f$.