The Pohozaev identity for mixed local-nonlocal operators

Abstract

In this article we prove the Pohozaev identity for the semilinear Dirichlet problem of the form −Δu+a(−Δ)su=f(u) in Ω, and u=0 in Ωc, where a is a non-negative constant and Ω is a bounded C2 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