= Solution
{c}
If two solutions exist, their difference $w$ satisfies $\nabla^2w-w=0$ inside $B$ and $w=0$ on the boundary. Suppose $w$ has a positive value. <Continuity> on the closed ball gives a positive <maximum> at an interior point, where the <Hessian matrix> is negative semidefinite and hence $\nabla^2w\leq0$. But the equation gives $\nabla^2w=w>0$, a contradiction. A negative <minimum> similarly contradicts $\nabla^2w=w<0$.
Thus $\boxed{w=0}$, proving uniqueness by the <maximum principle for the modified Helmholtz equation>. Here a classical solution is understood to be $C^2$ inside and continuous up to the boundary; mere first differentiability would not define the displayed <Laplacian> classically. With enough boundary regularity, the alternative <Green's first identity> proof is $0=\int_Bw(\nabla^2w-w)=-\int_B(|\nabla w|^2+w^2)$.
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