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Another argument perhaps worth mentioning uses the maximum principle. If $u_0$ is a (positive) principal eigenfunction, then we may assume after multiplying by a positive constant that $u_0$ either touches $u$ from below in $\Omega$, or lies below $u$ and is tangent to $u$ at a boundary point. Applying the strong maximum principle in the first case or the Hopf lemma in the second one to the nonnegative supersolution $u - u_0$, we conclude that $u = u_0$. (A drawback here is that we need sufficient regularity of $\partial \Omega$, e.g. $C^2$, to apply the Hopf lemma).