# Nontrivial u such that $\Delta u(x) = u(x)$ on a compact domain with zero Dirichlet condition?

Let $u(x)$ be a solution to the problem $\Delta u(x) = u(x)$ on a compact domain with smooth boundary. Furthermore demand that $u(x)=0$ on the boundary. Is there an easy argument why $u(x)$ has to be zero everywhere? I can prove it using the spectral theorem but it seems to be overkill and not so enlightening to me.

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Multiply by (the complex conjugate) $\bar{u}$ (if $u$ is real valued, just take $\bar{u} = u$) and integrate (by parts)
$$0 \geq -\int_\Omega |\nabla u|^2 dx = \int_\Omega \bar{u} \triangle u dx = \int u^2 \geq 0$$