$\; C^3_{loc}$-Convergence Let $\;f:\mathbb R^2 \rightarrow \mathbb R^m\;$ be a bounded function which satisfies the following pde: 
$\;\Delta f - G_f(f)=0\;$ where $\;G\in C^2(\mathbb R^m;\mathbb R_{+})\;$ and $\;G_f=(\frac {\partial G}{\partial f_1}, \dots, \frac {\partial G}{\partial f_m})^{T}\;$
Furthermore for $\;f\;$ holds: 
"$\;f(x_1+y_1,x_2) \to f_1(x_1,x_2)\;$ as $\;y_1 \to +\infty\;$ in $\;C^3_{loc}(\mathbb R^2;\mathbb R^m)\;$ by linear elliptic estimates"
I'm trying to understand what exactly the last sentence means.
I have no clue how this limit follows, so I'm guessing $\; C^3_{loc}$-convergence should be the key here. I haven't seen this kind of convergence before. What is the norm in this case?
In addition, are these "linear elliptic estimates" related somehow to pde? 
I have trouble getting my head around this phrase, so any help would be valuable.
Thanks in advance!
 A: Too long for a comment, but I'm not sure this answers your question.  I can only give some introduction to two of the pieces that might be confusing, but I don't understand your question enough to answer it.
Convergence in $C^3_{loc}$
First, I can explain what $C^3_{loc}$ convergence means.  We say that a sequence of functions $f_n$ converges to $f$ in $C^3_{loc}(\mathbb{R}^2)$ if for every compact set $K\subset \mathbb{R}^2$ and every $\epsilon>0$, there exists $N$ such that for all $n>N$,
$$
\sup_{|\alpha|\leq 3, x\in K} |\partial_\alpha f_n - \partial_\alpha f| < \epsilon.
$$
In other words, if for every compact set $K$, the function and all its derivatives (up to third order) converge uniformly.  
So the statement that $f(x_1+y_1,x_2)\to f_1(x_1,x_2)$ in $C^3_{loc}$ as $y_1\to\infty$ means that, as one slides the function $f$ left, the behavior "near the origin" stabilizes.  As an example of how this could happen, consider the function 
$$
f(x_1,x_2) = \sin(x_2) + (1+|x_1|^2)^{-1}.
$$
The functions $f(x_1+y_1,x_2)$ converge to $f_1(x_1,x_2) = \sin(x_2)$.  Does this make sense?
Elliptic estimates and PDE
In PDE the phrase "elliptic estimate" can mean many things, but let's start simply.  Suppose one has a solution to the Poisson equation
$$
\Delta u = f,
$$
and one knows for example that $f\in L^2$.  One might hope to show that $u$ is in the Sobolev space $H^2$ (that is, that its first and second derivatives are in $L^2$), but a priori the only obvious control is over the specific combination of second derivatives that is the Laplacian.  However it turns out to be true that 
$$
\|\partial_{ij} u\|_{L^2} \leq C \|f\|_{L^2} = C\|\Delta u\|_{L^2}.
$$
In general an elliptic estimate is something that controls individual partial derivatives of a function in terms of an elliptic operator acting on the same function.
