# $W_{loc}^{1,2}$ regularity of nonnegative subharmonic functions

I'm trying to solve the following excercise:

Let $$v \in C(D)$$ be a nonnegative subharmonic function in an open set $$D$$. Prove that $$v \in W_{loc}^{1,2}(D)$$

The problem has the following hint:

Mollifications $$v_\epsilon$$ of $$v$$ satisfy the inequality $$\int_D \nabla v_\epsilon \cdot \nabla \phi \;dx \leq 0$$ for any nonnegative $$\phi \in C_0^\infty(D)$$. Take $$\phi = v_\epsilon \zeta^2$$ with $$\zeta \in C_0^\infty(D)$$ and let $$\epsilon \to 0+$$.

I got confused trying to integrate by parts and use the Poincaré inequality to prove a uniform bound, but I guess and don't really understand how to use the hint. I know that under the hypotheses the convergence $$v_\epsilon \to v$$ is in $$L_{loc}^2(D)$$ and uniform on compact subsets of $$D$$, but not more than that.

I will really appreciate any comment.