1. I was going through the paper of Brezis and Merle [1]. In theorem 3 step 2 it's written that the boundedness of $f_n=V_ne^{u_n}$ in $L_{loc}^p(\Omega)$ 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 conclusion follows i.e. why this implies the uniform convergence of $v_n\to v$ on compact subsets of $\Omega $.

2. At the very beginning they prove a basic inequality which they use later. In the last line of the proof they mention that for $y\in B_R$ we have $$\DeclareMathOperator{\dmu}{d\!} \int\limits_{B_R}\left(\frac{2R}{|x-y|}\right)^{2-\frac{\delta}{2\pi}} \dmu x\leq\int_{B_R}\left(\frac{2R}{|x|}\right)^{2-\frac{\delta}{2\pi}}\dmu x $$ Does this result follows from translation? Otherwise $|x-y|\ngeq |x|$ in general. And also why $$\DeclareMathOperator{\diam}{diam} \int\limits_{B_R}\left(\frac{2R}{|x|}\right)^{2-\frac{\delta}{2\pi}}\dmu x=\frac{4\pi^2}{\delta}\big(\diam(\Omega)\big)^2 $$ Also this is not coming from radial integration as one term $2^{2-\frac{\delta}{2\pi}}$ survives.

Any idea would be very much helpful.

Reference

[1] Haïm Brézis, Frank Merle, "Uniform estimates and blow-up behavior for solutions of $−Δu=V(x)e^u$ in two dimensions" Communications in Partial Differential Equations 16, No. 8-9, 1223-1253 (1991), MR1132783, Zbl 0746.35006.