= Solution
Put $H=H_0^1(\Omega)$ with its usual <Hilbert space> inner product, and define
$$
B(u,v)=\int_\Omega\left(\nabla u\mathbin\cdot\nabla v+uv+uA\mathbin\cdot\nabla v\right).
$$
The <Cauchy-Schwarz inequality> shows that this is a <bounded bilinear form>. For $u\in H$, <Young inequality> gives
$$
\left|\int_\Omega uA\mathbin\cdot\nabla u\right|
\leq |A|_2\lVert u\rVert_2\lVert\nabla u\rVert_2
\leq\frac{|A|_2}{2}\left(\lVert u\rVert_2^2+\lVert\nabla u\rVert_2^2\right),
$$
and therefore
$$
B(u,u)\geq\left(1-\frac{|A|_2}{2}\right)\lVert u\rVert_{H^1}^2.
$$
Because $|A|_2<2$, this is a <coercive bilinear form>.
The map $u\mapsto\int_\Omega uf$ is a <bounded linear functional> on $H$. By the <Riesz representation theorem>, it equals $\langle u,F\rangle_H$ for some $F\in H$. Applying part (b) produces one and only one $v\in H$ such that
$$
\boxed{\int_\Omega\left(\nabla u\mathbin\cdot\nabla v+uv+uA\mathbin\cdot\nabla v\right)=\int_\Omega uf
\quad\text{for every }u\in H_0^1(\Omega).}
$$
This is precisely the <weak formulation> of the stated <Dirichlet problem>, obtained by <integration by parts> in the <Laplace operator> term. Thus $v$ is its unique <weak solution>; this is the <constant-drift massive-Laplacian Dirichlet problem>.
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