Abstract: Using the domain variation estimate, we give a new proof of the local Pohozaev identities for weak solutions of elliptic equations involving $p$-Laplacian operators under only $C^1$-regularity of the solutions. As an application, we obtain the Pohozaev identities for $C^1$ solution of $p$-Laplacian equation in $\mathbb{R}^N$ with $1<p<N$.

Key words: $p$-Laplacian equation, domain variation, Pohozaev identity