(a) Consider variable-coefficient operators of the form

$$P u=-\sum_{j, k=1}^{n} a_{j k} \partial_{j} \partial_{k} u+\sum_{j=1}^{n} b_{j} \partial_{j} u+c u$$

whose coefficients are defined on a bounded open set $\Omega \subset \mathbb{R}^{n}$ with smooth boundary $\partial \Omega$. Let $a_{j k}$ satisfy the condition of uniform ellipticity, namely

$$m\|\xi\|^{2} \leqslant \sum_{j, k=1}^{n} a_{j k}(x) \xi_{j} \xi_{k} \leqslant M\|\xi\|^{2} \quad \text { for all } x \in \Omega \text { and } \xi \in \mathbb{R}^{n}$$

for suitably chosen positive numbers $m, M$.

State and prove the weak maximum principle for solutions of $P u=0$. [Any results from linear algebra and calculus needed in your proof should be stated clearly, but need not be proved.]

(b) Consider the nonlinear elliptic equation

$$-\Delta u+e^{u}=f$$

for $u: \mathbb{R}^{n} \rightarrow \mathbb{R}$ satisfying the additional condition

$$\lim _{|x| \rightarrow \infty} u(x)=0 .$$

Assume that $f \in \mathcal{S}\left(\mathbb{R}^{n}\right)$. Prove that any two $C^{2}$ solutions of (1) which also satisfy (2) are equal.

Now let $u \in C^{2}\left(\mathbb{R}^{n}\right)$ be a solution of $(1)$ and (2). Prove that if $f(x)<1$ for all $x$ then $u(x)<0$ for all $x$. Prove that if $\max _{x} f(x)=L \geqslant 1$ then $u(x) \leqslant \ln L$ for all $x$.