= Maximum principle for the modified Helmholtz equation
{title2=$\Delta u-cu=0,\ c>0\ \Longrightarrow\ \|u\|_{\infty,\overline\Omega}\leq\|u\|_{\infty,\partial\Omega}$}
Let $\Omega$ be a bounded domain, with $u\in C^2(\Omega)\cap C(\overline\Omega)$ real valued and $c>0$. At a positive interior maximum, $\Delta u\leq0$ but $cu>0$, contradicting the equation. A negative interior minimum gives the opposite contradiction. Thus any value with absolute magnitude exceeding the boundary maximum would be impossible, proving the displayed bound. In particular zero boundary data force $u=0$, and subtracting two solutions proves uniqueness for the corresponding inhomogeneous Dirichlet problem.
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