weak formulation in a manifold

Question

Let $\mathcal{M}=M\times\mathbb{R}^+$ with the metric $\bar{g}=g+|dx|^2$ where $(M,g)$ is a complete, smooth Riemannian manifold . Consider the problem $$\begin{cases}\nabla_{\bar{g}}.(x^{1-2\gamma}\nabla_{\bar{g}}u)=0 \ \ \text{in} \ \mathcal{M}\\ -x^{1-2\gamma}\partial_xu=f(u) \ \ \ \ \text{on} \ M\times\{0\} \end{cases}$$ for $\gamma\in(0,1)$ . I want to find the weak formulation of the problem . Therefore multiplying by some $\xi\in C_0^\infty(M\times\mathbb{R})$ $$\int_{\mathcal{M}}\xi\nabla_{\bar{g}}.(x^{1-2\gamma}\nabla_{\bar{g}}u)\,dV_{\bar{g}}=0$$ $$\implies\int_\mathcal{M}(x^{1-2\gamma}\Delta_{\bar{g}}u+(1-2\gamma)x^{-2\gamma}\nabla_{\bar{g}}u)\xi\,dV_{\bar{g}}=0$$ Afterwards I'm a bit lost how to apply the by parts to introduce the boundary term . Any help is appreciated .