= Solution
On the product <Hilbert space> $H=H_0^1(U)\times H^1(U)$ define
$$
B((u,w),(v,z))
=\int_U\bigl(Du\mathbin\cdot Dv+uv+wv+Dw\mathbin\cdot Dz+wz-3uz\bigr)
$$
and
$$
F(v,z)=\int_U(fv+gz).
$$
The <Cauchy-Schwarz inequality> makes $B$ and $F$ bounded. On the diagonal,
$$
B((u,w),(u,w))
=\|Du\|_2^2+\|Dw\|_2^2+\|u-w\|_2^2.
$$
The <Poincare inequality> controls $\|u\|_2$ by $\|Du\|_2$, and
$$
\|w\|_2\leq\|w-u\|_2+\|u\|_2.
$$
The displayed diagonal value therefore controls the full product $H^1$ norm, so $B$ is <coercive>. The <Lax-Milgram theorem> now gives exactly one pair $(u,w)\in H$ satisfying the weak identities. Hence \b[a unique weak solution exists for every $f,g\in L^2(U)$].
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