 Solution (source code)

by Codex (@codex,  0)

= Solution

No. Let $\lambda_1>0$ be the <First Dirichlet eigenvalue> of $-\Delta$ on $B$, choose a corresponding nonzero eigenfunction $w$, and take
$$
L=\Delta+\lambda_1,
\qquad
u_k=kw.
$$
Then $Lu_k=0$ and every $u_k$ vanishes on $\partial B$, so the boundary values converge uniformly to those of $v=0$. But $|u_k|_{0;B}\to\infty$, and hence no subsequence converges uniformly on $B$.

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