While providing examples of p-admissible weights in the book "Nonlinear Elliptic Theory of degenerate elliptic equation" the author used the weight to be $w(x)=1$ and the measure to be lebesgue. I am struck proving the second property which is if $D$ is an open set, $\phi_i\in C^{\infty}(D)$ then, $\int|\phi_i|^p dx\to 0$ and $\int|\nabla \phi_i-v|^p dx\to 0$ as $i\to\infty$ implies $v=0$.
Any help is appreciated.