The Pohozaev identity for mixed local-nonlocal operator

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 $f$.

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