= Solution
After integration by parts, the weak formulation is
$$
\int_\Omega(\nabla u\cdot\nabla\varphi+u\varphi)
=-\int_\Omega f\varphi
\qquad(\varphi\in H_0^1(\Omega)).
$$
The left side is the inner product
$$
(u,\varphi)_*=\int_\Omega(\nabla u\cdot\nabla\varphi+u\varphi),
$$
which is positive definite and induces the usual $H^1$ norm. The right side is bounded in this norm. The Riesz representation theorem, equivalently the <Lax-Milgram theorem>, gives a unique weak solution. This is the <weak Dirichlet problem for the massive Laplacian> with mass one and the signs multiplied by $-1$.
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