The Pohozaev identity for mixed local-nonlocal operator

Abstract

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

15 pages