Another argument perhaps worth mentioning uses the maximum principle. To illustrate the idea we assume that $\Omega$ is a bounded smooth domain, that $L = \Delta$, and that we are dealing with Dirichlet eigenfunctions.
Let $u_0$ be a positive eigenfunction corresponding to the smallest eigenvalue of $L$. 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 $u = u_0$.