Need source for elliptic regularity on unbounded domains 
I need a source that provides a $W^{2,2}$-regularity result for linear elliptic systems on unbounded domains.



*

*To be more specific, I study the equation $$ -\Delta w - ic \partial_1 w + \left( \frac{c^2}{2}+2 \right)w = h ~~~ \mbox{on}~ \Omega $$ where $\Omega \subset \mathbb{R}^3$ is some unbounded domain, $h$ is $L^2(\Omega, \mathbb{C})$, $w : \Omega \to \mathbb{C}$ and $c \in [0,\sqrt{2}]$. I want to show that if $w$ solves the above equation, then $w \in W^{2,2}(\Omega, \mathbb{C})$.

*Please note again: $\Omega$ is unbounded and by identifying $\mathbb{C} \simeq \mathbb{R}^2$ we can regard $w$ as vector valued (and thus the equation is a  linear elliptic system). I.e. $w : \Omega \to \mathbb{R}^2$ and $h \in L^2(\Omega, \mathbb{R}^2)$.

*This paper suggests that this is an easy step (page 3), but I can't seem to find a source. Also this answer to one of my earlier questions states the same thing, but does not provide a source either (and I did not ask for one back then).

*All sources I can find (e.g. Giaquinta, Taylor, Gilbarg+Trudinger) either don't talk about systems or treat them on bounded domains.
Any hint is much appreaciated! 
 A: I will assume we have good boundary behaviour (at the boundary $\partial \Omega$ and at $\infty$) so I can freely integrate by parts. 
Consider
$$ \begin{align}
|h|^2 &= \left| - \Delta w - i c \partial_1 w + \left( \frac{c^2}{2} + 2\right) w \right| ^2 \\
& = \left| \Delta w \right|^2 + \left| c \partial_1 w \right|^2 + \left| \left( \frac{c^2}{2} + 2\right) w\right|^2 + 2 \Im \left[ c \Delta w \partial_1 \bar{w} - c \left( \frac{c^2}{2} + 2 \right) w \partial_1 \bar{w} \right] - 2 \Re \left[\Delta w \left( \frac{c^2}{2} + 2 \right) \bar{w} \right]
\end{align} $$
Integrating over $\Omega$, and integrating the last term by parts, we have (the norm $\|\cdot\|_k = \|\cdot\|_{W^{k,2}}$)
$$ \|h\|_{0}^2 = \|\Delta w\|^2_0 + c^2 \|\partial_1 w\|^2_0 + \left(\frac{c^2}{2} + 2\right)^2 \|w\|^2_0 + 2 \left( \frac{c^2}{2} + 2\right) \|\nabla w\|^2_0 + 2 c\int \Im \ldots $$
where the 
$$ \ldots = \Delta w \partial_1 \bar{w} - \left( \frac{c^2}{2} + 2\right) w \partial_1 \bar{w} $$
Putting the imaginary parts into absolute values we have
$$ \|h\|_{0}^2 \geq \|\Delta w\|^2_0 + c^2 \|\partial_1 w\|^2_0 + \left(\frac{c^2}{2} + 2\right)^2 \|w\|^2_0 + 2 \left( \frac{c^2}{2} + 2\right) \|\nabla w\|^2_0 - 2 c \left| \int \Im \ldots \right| \tag{*}$$
Now we Cauchy-Schwarz the $\ldots$ with weights. We have that
$$ 2c\left| \int\Im\ldots \right| \leq  A \|\triangle w\|^2_0 + (A^{-1} + B^{-1})c^2 \|\partial_1 w\|^2_0 + B\left(\frac{c^2}{2} + 2\right)^2\|w\|^2_0 $$
for any positive real numbers $A,B$. It remains to choose good $A,B$. 
But now we observe that $\|\nabla w\|^2_0 = \sum \|\partial_i w\|^2_0$ contains a factor of $\|\partial_1w\|^2_0$. We make use of that term. 
Let us set $A = B = (1-\epsilon)$. Plugging in the estimate for the $\Im \ldots$ into (*) we get
$$ \|h\|^2_0 \geq \epsilon \|\Delta w\|^2_0 + \epsilon \left( \frac{c^2}{2} + 2\right) \|w\|^2_0 + \left(2 c^2 + 4 - \frac{2c^2}{1 - \epsilon}\right) \|\partial_1 w\|^2_0 + (c^2 + 4)\left( \|\partial_2 w\|^2_0 + \|\partial_3 w\|^2_0 \right) $$
Now we see that the right hand side $\geq C \|w\|^2_2$ for some constant $C$ which depends on the constant $c$, provided that we can choose $\epsilon$ depending on $c$ such that 
$$ 2 c^2 \left(1 - \frac{1}{1-\epsilon}\right) + 4 > 0 $$
A little algebraic manipulation shows that this only requires
$$ \frac{2}{c^2} > \frac{\epsilon}{1-\epsilon} $$
which can always be satisfied as $\lim_{\epsilon \to 0} \frac{\epsilon}{1-\epsilon} = 0$. In the regime where $c^2 \in [0,2]$ as you supposed, the value $\epsilon = 1/4$ suffices for any such $c$. 
