The Pohozaev identity for mixed local-nonlocal operator

TL;DR

The paper develops a sharp Pohozaev-type identity for mixed local–nonlocal Dirichlet problems of the form $- ext{Δ}u + a(- ext{Δ})^s u = f(u)$ in a bounded $C^2$ domain, with $u=0$ outside Ω. It expresses a balance between the energy components $[u]^2_s$ and $[u]^2_1$ and the primitive energy $\,\mathcal{E}(u)$, equated to a boundary integral over $∂Ω$, and provides an equivalent form involving $\int_Ω u f(u)\,dx$ and $\int_Ω F(u)\,dx$. The authors establish the identity first in strictly star-shaped domains using delicate boundary regularity estimates for the local and nonlocal parts and then extend to systems, yielding applications to unique continuation for eigenfunctions and nonexistence results in star-shaped domains for various nonlinearities, including Brezis–Nirenberg-type problems. These results illuminate the interplay between local and nonlocal components in elliptic problems and supply tools for proving nonexistence in supercritical regimes.

Abstract

In this article we prove the Pohozaev identity for the semilinear Dirichlet problem of the form $-Δu + a(-Δ)^s u = f(u)$ in $Ω$, and $u=0$ in $Ω^c$, where $a$ is a non-negative constant and $Ω$ 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 $f$.

Theorems & Definitions (25)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Corollary 1.5
• Theorem 2.1
• Theorem 2.2
• proof