1)I was going through the paper of Brezis Merle . In theorem 3 step 2 it's written that $f_n=V_ne^{u_n}$ is bounded in $L_{loc}^p(\Omega)$ this implies $\mu\in L^1(\Omega)\cap L_{loc}^p(\Omega)$. Let $v_n$ be a solution of $-\Delta v_n=f_n$ in $\Omega$ and $v_n=0$ in $\partial\Omega$. Then $v_n\to v$ uniformly on compact subsets of $\Omega$ where $v$ is solution of $-\Delta v=\mu$ in $\Omega$ and $v=0$ in $\partial\Omega$. I am not getting how this line follows i.e uniform convergence of $v_n\to v$ on compact subsets of $\Omega $. 

2) In the very beginning he has proved a basic inequality which has been used later. In the last line of the inequality he mentions for $y\in B_R$ we have $\int_{B_R}(\frac{2R}{|x-y|})^(2-\frac{\delta}{2\pi})\leq\int_{B_R}(\frac{2R}{|x|})^(2-\frac{\delta}{2\pi})$ is this follows from translation ? Otherwise $|x-y|\ngeq |x|$ in general. And also $\int_{B_R}(\frac{2R}{|x|})^(2-\frac{\delta}{2\pi})=\frac{4\pi^2}{\delta}(diam(\Omega)^2)$ this is also not coming in radial integration as one term $2^{2-\frac{\delta}{2\pi}}$ survives.

Any idea would be very much helpful.