Consider the functional

$$E(u)=\frac{1}{2} \int_{\Omega}|\nabla u|^{2} d x+\int_{\Omega} F(u, x) d x$$

where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary and $F: \mathbb{R} \times \Omega \rightarrow \mathbb{R}$ is smooth. Assume that $F(u, x)$ is convex in $u$ for all $x \in \Omega$ and that there is a $K>0$ such that

$$-K \leqslant F(v, x) \leqslant K\left(|v|^{2}+1\right) \quad \forall v \in \mathbb{R}, x \in \Omega$$

(i) Prove that $E$ is well-defined on $H_{0}^{1}(\Omega)$, bounded from below and strictly convex. Assume without proof that $E$ is weakly lower-semicontinuous. State this property. Conclude the existence of a unique minimizer of $E$.

(ii) Which elliptic boundary value problem does the minimizer solve?