(a) Consider the nonlinear elliptic problem

$$\begin{cases}\Delta u=f(u, x), & x \in \Omega \subseteq \mathbb{R}^{d} \\ u=u_{D}, & x \in \partial \Omega\end{cases}$$

Let $\frac{\partial f}{\partial u}(y, x) \geqslant 0$ for all $y \in \mathbb{R}, x \in \Omega$. Prove that there exists at most one classical solution.

[Hint: Use the weak maximum principle.]

(b) Let $\varphi \in \mathcal{C}_{0}^{\infty}\left(\mathbb{R}^{n}\right)$ be a radial function. Prove that the Fourier transform of $\varphi$ is radial too.

(c) Let $\varphi \in \mathcal{C}_{0}^{\infty}\left(\mathbb{R}^{n}\right)$ be a radial function. Solve

$$-\Delta u+u=\varphi(x), \quad x \in \mathbb{R}^{n}$$

by Fourier transformation and prove that $u$ is a radial function.

(d) State the Lax-Milgram lemma and explain its use in proving the existence and uniqueness of a weak solution of

$$\begin{gathered} -\Delta u+a(x) u=f(x), x \in \Omega \\ u=0 \text { on } \partial \Omega \end{gathered}$$

where $\Omega \subseteq \mathbb{R}^{d}$ bounded, $0 \leqslant \underline{a} \leqslant a(x) \leqslant \bar{a}<\infty$ for all $x \in \Omega$ and $f \in L^{2}(\Omega)$.