Show that if $u$ satisfies $(-\Delta + \lambda)u=0$ in a smooth region $\Omega$ where $\lambda$ is a nonnegative real number, then u satisfies the maximum principle.
We can just assume u is smooth, and actually it is true and can be proved. The statement is that any distribution $u$ satisfies the equation must be smooth, which is done in Treves.
But the point here is to prove if $u$ is smooth, then why $u$ satisfies the maximum principle.
Here is my attempt:
The case $\lambda=0$ is trivial, for it is well known that harmonic function satisfies maximum principle. So we assume $\lambda>0 \space.$
If $u$ attains its maximal value in $\Omega$ at $x_0$, then in multivariable calculus we know that $\Delta u(x_0)\le0\space.$ Now we discuss the following two cases:
If $u(x_0)>0$, then $\Delta u=\lambda u>0$ at $x_0$, which contradicts $\Delta u(x_0)\le0\space.$
If $u(x_0)\le0$, then $u\le0$ in $\Omega$, hence $-u\ge0$ in $\Omega$, and $\Delta (-u)=-\lambda u>0$. But it doesn't tell me anything...
Any help would be appreciated. Thanks!