Let $B_r$ be an open ball in $\mathbb R^3$ with center $0$ and radius $r$. Consider the eigenvalue problem $$-\Delta u=u,~x~\text{in}~B_1,$$ here $\Delta$ is the Laplace operator. Suppose that $u\in C^2(B_1)\cap C(\overline{B_1})$ is a solution of the problem above, and $u$ vanishes at each point $x$ in $B_{1/2}.$ Then, there must hold that $u(x)\equiv0$ for all $x\in B_1.$
I have some stupid trials but failed. We know that if a harmonic function $v$ defined in $B_1$ vanishes in $B_{1/2}$, then it must be zero since it is analytic. However for this problem, it seems that there is no good estimate for $v$'s each $k$-derivative at this stage. Hope someone could give me some hints, and any ideas are welcome, thanks in advance. If I come up with the proof of this problem, I will also post my solution here.