= Solution
The required functional, with the sign appropriate to a positive divergence operator, is
$$
\boxed{\mathcal F(u)=\frac12Q(u,u)+\int_\Omega fu\,dx.}
$$
It is differentiable on $W_0^{1,2}$, with $D\mathcal F(u)[v]=Q(u,v)+\int fv$. A critical point consequently satisfies the weak equation. We now prove existence by the <direct method in the calculus of variations>, rather than invoke the representation theorem again.
The ellipticity and Poincare estimates imply
$$
\mathcal F(u)\geq\frac\lambda2\|Du\|_2^2-C_P\|f\|_2\|Du\|_2
\geq\frac\lambda4\|Du\|_2^2-\frac{C_P^2}\lambda\|f\|_2^2.
$$
Thus the infimum is finite and any <minimizing sequence> is bounded in the <Sobolev norm>. A bounded sequence in a <Hilbert space> has a <weakly convergent> subsequence; write $u_j\rightharpoonup u$ in $W_0^{1,2}$. The equivalent $Q$ norm has the same continuous linear functionals and hence the same <weak convergence>. A Hilbert norm is <Weak lower semicontinuity>, giving $Q(u,u)\leq\liminf_jQ(u_j,u_j)$. The term $\int fu_j$ converges to $\int fu$ because it is a <bounded linear functional>. Therefore
$$
\mathcal F(u)\leq\liminf_j\mathcal F(u_j)=\inf\mathcal F.
$$
The limit attains the infimum. Differentiating $t\mapsto\mathcal F(u+tv)$ at its minimum, for every $v$, gives $D\mathcal F(u)[v]=0$. This produces the desired <weak solution>. Strict positivity of $Q$ also makes the functional <strictly convex>, consistent with uniqueness.
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