= Solution
For $s\in\mathbb R$, the <Sobolev space> $H^s(\mathbb R^n)$ consists of tempered distributions $u$ such that
$$
\|u\|_{H^s}^2
=\int_{\mathbb R^n}
(1+|\xi|^2)^s|\widehat u(\xi)|^2\,d\xi
<\infty.
$$
For a multi-index $\alpha$,
$$
\widehat{D^\alpha u}(\xi)=(i\xi)^\alpha\widehat u(\xi).
$$
Since
$$
(1+|\xi|^2)^{s-|\alpha|}|\xi|^{2|\alpha|}
\leq(1+|\xi|^2)^s,
$$
we obtain the <Sobolev derivative estimate>
$$
\boxed{
\|D^\alpha u\|_{H^{s-|\alpha|}}
\leq\|u\|_{H^s}
}.
$$
Thus differentiation is bounded and linear between the stated spaces.
Taking the <Fourier transform> of
$$
-\Delta u+u=f
$$
gives
$$
(1+|\xi|^2)\widehat u(\xi)=\widehat f(\xi).
$$
The only possible solution is therefore
$$
\widehat u(\xi)=\frac{\widehat f(\xi)}{1+|\xi|^2}.
$$
Moreover,
$$
\|u\|_{H^{s+2}}^2
=\int(1+|\xi|^2)^{s+2}
\frac{|\widehat f(\xi)|^2}{(1+|\xi|^2)^2}\,d\xi
=\|f\|_{H^s}^2.
$$
This proves existence, uniqueness, and bounded dependence. Conversely, $u\mapsto(-\Delta+1)u$ maps $H^{s+2}$ boundedly to $H^s$, so
$$
\boxed{
-\Delta+1:H^{s+2}(\mathbb R^n)\longrightarrow H^s(\mathbb R^n)
}
$$
is a linear isomorphism with inverse multiplier $(1+|\xi|^2)^{-1}$. This is the <massive Laplacian isomorphism on Sobolev spaces>.
Finally, the assumed estimate and <Plancherel theorem> imply
$$
\int_{\mathbb R^n}
(1+|\xi|^2)^2|\widehat u_j(\xi)|^2\,d\xi
\leq
C\bigl(\|u_j\|_2^2+\|\Delta u_j\|_2^2\bigr)
\leq C K^2.
$$
Thus $(u_j)$ is bounded in $H^2(\mathbb R^n)$. The functions and all their first derivatives are supported in the fixed bounded set $\overline\Omega$. Their $H^1$ bounds give uniform translation estimates
$$
\|v(\,\cdot+h)-v\|_2
\leq|h|\,\|\nabla v\|_2
$$
for $v=u_j$ and $v=D_\ell u_j$. Tight support and the <Rellich-Kondrashov compactness theorem for H01>, equivalently the Fourier compactness criterion, therefore provide a common subsequence for which $u_j$ and every first derivative converge strongly in $L^2$. Hence
$$
\boxed{u_{j_k}\longrightarrow u\quad\text{strongly in }H^1(\mathbb R^n)}.
$$
This is <compactness from bounded support and an H2 bound>.
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