Maximum principle for the modified Helmholtz equation ($\Delta u-cu=0,c>0⟹\Vert u{\Vert }_{\infty ,\overline{\Omega }}\le \Vert u{\Vert }_{\infty ,\partial \Omega }$)

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, $\Delta u\le 0$ 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.