The Pohozaev identity for mixed local-nonlocal operator

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