I have to prove that the operator $Lu=\Delta u +u$ does not satisfy the weak maximum principle, i.e. if $u$ is such that $Lu=0$, then $u$ can attain its maximum in the interior of the bounded domain $\Omega$ that is considered for $u$.
In the monodimensional case one can simply consider $u(t)=\sin (t)$ in $\Omega=(0,\pi)$ so I tried to adapt it to the general case using for example $u(x_{1}\dots x_{n})=\sum_{i=1}^{n} \sin (x_{i})$ and prove that its maximum is inside the interior of, e.g. $\Omega=(0,\pi)^n$.
Is there a more general way to prove it? Thanks in advance.