# Exponential decay for the gradient of a solution

Dear all, I would like to prove the exponential decay of the derivatives of a solution to the following equation in $\mathbb{R}^N$: $$\sqrt{-\Delta+m^2} u +u= f(u),$$ where I can assume that $m \neq 0$, that $f$ is smooth and that the solution $u \in H^{1/2}(\mathbb{R}^N)$ is at least Hölder-continuous. I also know that $u(x) = O(e^{-C|x|})$ at infinity. If the equation were local (like $-\Delta u + u = f(u)$), the usual approach would consist in using interior Schauder estimates for $\nabla u$ (or some Harnack-like estimate); I did not find any precise reference for Schauder theory of non-local equations, except in the case $m=0$ (the fractional laplace equation).

Any help is welcome.

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I think it would be helpful to define what you mean by $\sqrt{−\Del+m^2}$ (via some integral, Fourier transform, etc.) for preciseness and to best facilitate a response. The theory of regularity for operators like the fractional Laplacian have been treated relatively recently (last 5-10 years, from what I know) by Caffarelli, Silvestre, and perhaps their techniques apply? –  Daniel Spector Dec 8 '12 at 16:07
The usual way to do something like this is to try to adapt the proof that $u$ is exponentially decaying to $\nabla u$. What's convenient here is that $\nabla$ commutes with your operator, so the vector-valued function $V = (u, \nabla u)$ satisfies roughly the same equation as $u$ itself, except that it is a system instead of a single equation. You would also use elliptic regularity to establish a priori regularity of $\nabla u$. –  Deane Yang Dec 8 '12 at 16:14
@DanielSpector It seems to me that the pure fractional laplacian is rather different, for example solutions decay like powers. Moreover, many results by Silvestre are based on the representation via a convolution-like integral, which is not the case for "my" operator, which is often defined by functional calculus. –  Siminore Dec 8 '12 at 17:02
Hmm. If you are in $\mathbb{R}^N$, this should be equivalent (functional calculus or integral), since the fractional Laplacian's symbol is multiplication by the Fourier variable? I guess my question is then this: What do you mean when you write the above equation? –  Daniel Spector Dec 8 '12 at 17:52
I mean: Silvestre defines $$(-\Delta)^u(x) = \operatorname{PV}\int \frac{u(y)}{|x-y|^{n+2s}}dy$$ but I do not think such a definition can be extended to $(-\Delta +1)^{1/2}$. And I am unable to use Fourier analysis to produce Silvestre's regularity results. –  Siminore Dec 8 '12 at 18:20

An exemplary result reads: Suppose that the nonlinearity $f$ is a polynomial in $u$ and that $u\in \langle x \rangle^{-\delta} H^s({\mathbb R}^N)$, where $s>N/2$, $\delta>0$, is a solution of the semilinear elliptic equation shown above. Then $u$ belongs to the Gelfand-Shilov space $S_1^1({\mathbb R}^N)$. Amongst others, this implies that there is an $\varepsilon>0$ so that, for all $\alpha\in{\mathbb N}_0^N$, $$(\partial^\alpha u)(x)= O(e^{-\varepsilon\,|x|}) \enspace \text{as |x|\to\infty.}$$