TITLE: What "standard estimates for the laplacian" do the authors of this paper mean?
QUESTION [6 upvotes]: I am trying to follow the proof of lemma 2.1 in this paper.
The setup. Consider a solution $v$ to the nonlinear equation $$ -\Delta v = ic \partial_1 v + v(1-\vert v\vert^2) ~\mbox{on}~ \mathbb{R}^N$$ with finite energy, i.e., $$\Vert Dv \Vert_{L^2(\mathbb{R}^N)}+ \Vert 1-\vert v \vert^2 \Vert_{L^2(\mathbb{R}^N)} < \infty. $$ Here, $c$ is a real constant and $i$ the imaginary unit.
The question. The authors want to prove the following statement.

There is a constant $K(c,k,N)>0$ such that $\Vert v \Vert_{C^k(\mathbb{R}^N)} \leq K(c,k,N), \forall k \in \mathbb{R}.$

All they say is: "One invokes standard estimates for the laplacian".
What standard estimates do they mean? What, precisely, is the argument here?
More information. I don't know if this helps, but from the first part of the lemma in the paper, we already know the following:

$\Vert 1 - \vert v \vert \Vert_{L^\infty(\mathbb{R}^N)} \leq \max \lbrace 1, \frac{c}{2} \rbrace$
$\Vert \nabla v \Vert_{L^\infty(\mathbb{R}^N)} \leq K(N) \left( 1+ \frac{c^2}{4} \right)^{\frac{3}{2}}$
v is smooth and bounded.

Any help is much appreciated!

REPLY [2 votes]: I suspect they mean the estimates on the Newtonian potential (see Gilbarg-Trudinger, Chapter 4, Thm 4.5, for example). The idea is that if you have some control (in, say, the $L^\infty$ sense) over the right hand side, then you can control the $C^1$ norm of $u$. If you have some control over the right hand side in the $C^\alpha$ sense, then you can control the $C^{2,\alpha}$ norm of $u$. 
Subsequently, to control higher derivatives of $u$, you differentiate the equation to get an elliptic equation involving the Laplacian on appropriate derivatives of $u$, and in this manner iteratively use the estimates on lower order derivatives.