For $-\Delta u =0$, the Pohozaev identity on say $B_1$ says $$ \int_{S_1} |u_T|^2 \,d\sigma = \int_{S_1} |u_N|^2 \,d\sigma + (n-2) \int_{B_1} |\nabla u|^2 \ dx$$ Here $u_T$ are the tangential derivative and $u_N$ is the normal derivative.

I am trying to see what is the analogue for solutions of $-\operatorname{div} A(x)\nabla u=0$ where $A(x)$ is symmetric and uniformly elliptic with $\lambda \leq A(x) \leq \Lambda$ (in terms of quadratic forms) and smooth (say $C^{\infty}$) so that weak solutions are also smooth.

How does one go about proving a Pohozaev identity in this setting?

I am more specifically looking for the identity using $\langle x,A(x)\nabla u\rangle$ as a test function. Rather, any other test function is also okay provided all the resulting terms have a specific sign.

Further assumptions: Suppose that all derivatives of $A(x)$ are also uniformly elliptic with some ellipticity constants (in some sense, one can consider the mollification of $A$ as the matrix), then can one get an identity where each of the terms comes with a definite sign?

Clearly, there must be some terms which contain derivatives of $A$ and higher derivatives of $u$, but as long as they are in an expression with a definite sign, then it's okay.

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