I have the following theorem:
Let $u\in C^2(\Omega)\cap W^{2,n}(\Omega)$ satisfy $Lu\geq f$ for $f\in L^n(\Omega)$. Then for any ball $B_{2R} = B_{2R}(y)\subset\Omega$ and $p>0$ where $u$ is positive, we have$$ \sup_{B_R}u\leq C\left\{\left(\frac{1}{|\Omega|}\int_{B_{2R}} (u^+)^p\right)^{1/p}+\frac{R}{\lambda}\|f\|_{n;B_{2R}}\right\} $$where $C = C(n,\gamma,\nu R^2,p)$.
What I want to do is to extend it to the situation $u\in W^{2,n}$.
I know that, since $C^\infty(\Omega)\cap W^{2,n}(\Omega)$ is dense in $W^{2,n}(\Omega)$, there exist some sequence of functions $\{u_j(x)\}_{j\geq1}$ that converges to $u$ locally in the $W_{loc}^{2,n}(\Omega)$-norm.
Still, the details in how to do this seem to allude me.
Edit:
The operator $L$ is a linear 2nd order elliptic operator on the form $$Lu=a^{ij}D_{ij}u+b^iD_iu+cu$$, where the coefficient matrix $[a^{ij}]$ is positive and satisfies $0\leq\lambda<a^{ij}\leq\Lambda,\forall i,j$, and the coefficients $b^i,c$ satisfies $$\left(\frac{|b|}{\lambda}\right)^2,\; \frac{|c|}{\lambda}\leq\nu$$ where $\nu\geq\Lambda/\lambda$