= Solution
There is a sign issue in the printed problem. Part (c) constructs the inverse of $\Delta-1$, whereas the second equation printed in part (e) contains $\Delta+1$. The latter operator is invertible with homogeneous Dirichlet data only when $1$ is not a <Dirichlet Laplacian eigenvalue>. Thus the assertion as printed needs this nonresonance hypothesis; with a minus sign it follows directly from parts (c) and (d).
Under either the intended minus sign or the stated nonresonance condition, let $G_0$ and $G_1$ be the bounded Dirichlet solution operators for the two linear equations. Choose $s>n/2$ and work with $u,v\in H^{s+2}(\Omega)\cap H_0^1(\Omega)$. Since $H^s(\Omega)$ is a <Sobolev algebra>,
$$
\|(\partial_{11}v)^2\|_{H^s}\leq C\|v\|_{H^{s+2}}^2,\qquad
\|(\partial_{22}u)^2\|_{H^s}\leq C\|u\|_{H^{s+2}}^2.
$$
Define
$$
\mathcal T(u,v)=
\left(
G_0\big((\partial_{11}v)^2+\varepsilon f\big),
G_1\big((\partial_{22}u)^2\big)
\right).
$$
Elliptic regularity gives, on a ball of radius $R$,
$$
\|\mathcal T(u,v)\|_{H^{s+2}\times H^{s+2}}
\leq C(R^2+\varepsilon\|f\|_{H^s}),
$$
and the difference estimate has Lipschitz constant at most $CR$. Choose $R$ small and then $\varepsilon_0$ so that $C(R^2+\varepsilon_0\|f\|_{H^s})\leq R$. The <contraction mapping theorem> gives a solution for $0\leq\varepsilon<\varepsilon_0$. Repeated elliptic regularity and smoothness of $f$ bootstrap the solution to $C^\infty$.
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