Consider the elliptic Dirichlet problem on $\Omega \subset \mathbb{R}^{n}, \Omega$ bounded with a smooth boundary:

$$\Delta u-e^{u}=f \text { in } \Omega, \quad u=u_{D} \text { on } \partial \Omega .$$

Assume that $u_{D} \in L^{\infty}(\partial \Omega)$ and $f \in L^{\infty}(\Omega)$.

(i) State the strong Minimum-Maximum Principle for uniformly elliptic operators.

(ii) Prove that there exists at most one classical solution of the boundary value problem.

(iii) Assuming further that $f \geqslant 0$ in $\Omega$, use the maximum principle to obtain an upper bound on the solution (assuming that it exists).