 Solution

Fix $B_r⋐B_R⋐B$. Local uniform convergence bounds $|u_k{|}_{0;B_R}$ uniformly. Interior Schauder estimates therefore bound $u_k$ uniformly in $C^{2,\alpha }(B_r)$. The Arzela-Ascoli theorem gives a subsequence converging in $C^2(B_r)$. Its limit must be the locally uniform limit $v$, and passing to the limit in $L{u}_k=0$ gives $Lv=0$. A diagonal argument over $r\uparrow 1$ proves