= Solution
In three dimensions the <Sobolev embedding theorem> gives
$$
H^2(U)\hookrightarrow L^\infty(U),
\qquad
H^2(U)\hookrightarrow W^{1,4}(U).
$$
Consequently, for $w\in H^2(U)\cap H_0^1(U)$,
$$
\||Dw|^2w\|_{L^2}
\leq\|Dw\|_{L^4}^2\|w\|_{L^\infty}
\leq C\|w\|_{H^2}^3.
$$
Thus $f+|Dw|^2w\in L^2(U)$. The <Dirichlet Poisson regularity theorem> on a bounded $C^2$ domain says that
$$
-\Delta v=f+|Dw|^2w,qquad v|_{\partial U}=0,
$$
has a unique $v\in H^2(U)\cap H_0^1(U)$ and
$$
\|v\|_{H^2}\leq C\bigl(\|f\|_2+\||Dw|^2w\|_2\bigr).
$$
Hence $\Phi(w)=v$ is well defined.
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