= Solution
In the bounded-domain setting of the <weak maximum principle>, let $u_1,u_2$ be two <classical solutions> with the same forcing and boundary values. Their difference $z=u_1-u_2$ satisfies $Lz=0$ and $z=0$ on the boundary. Apply the <weak maximum principle> first to $z$ and then to $-z$:
$$
\sup_\Omega z\leq0,\qquad\sup_\Omega(-z)\leq0.
$$
Thus \b[$u_1=u_2$], proving uniqueness of the <Dirichlet problem>. The same proof works on an unbounded domain if the difference and its negative satisfy the required condition at infinity. As literally printed without either restriction, uniqueness need not hold: on $\{x_n>0\}$ the two <harmonic functions> $0$ and $x_n$ have the same zero boundary values.
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