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In a paper by Vazquez I read that an interesting variant of fractional diffusion (where the fractional operator is usually $(-\Delta)^s$) could be $(I-\Delta)^s$. Therefore I am wondering if there are results for the fractional field equation $$(I-\Delta)^s u = g(u) \quad\text{in $\mathbb{R}^N$},$$ where $0<s<1$ and $g \colon \mathbb{R} \to \mathbb{R}$ is some nonlinear function with suitable growth conditions.

I have found only one paper dealing with the square root, namely $s=1/2$, and a few words scattered here and there. Do you have any suggestion for solving such fractional equations?

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  • $\begingroup$ I do not know the method to deal with the special case of $s=1/2$, is there any different between $s=1/2$ and a general $s$? Can we change the differential equation to integral form by using the Fourier transform? Then try to use the fixed point theory. $\endgroup$ – CooLee May 17 '15 at 12:00
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Perhaps this will be useful:

Felmer, Vergara: Scalar field equation with non-local diffusion, NoDEA 22(2015), 1411-1428.

The main result is that, when the nonlinearity $g(u)$ "behaves like $|u|^p$" with $p$ subcritical, then the equation has a solution which is Hölder-continuous, strictly positive and has exponential decay. There is also a nonexistence result for $p$ supercritical.

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  • $\begingroup$ Would you mind describing for us in a few sentences what is in this paper? $\endgroup$ – András Bátkai Dec 17 '15 at 14:32
  • $\begingroup$ @AndrásBátkai The main result is that, when the nonlinearity $g(u)$ "behaves like $|u|^p$" with $p$ subcritical, then the equation has a solution which is Hölder-continuous, strictly positive and has exponential decay. There is also a nonexistence result for $p$ supercritical. $\endgroup$ – ptf1 Dec 17 '15 at 14:41
  • $\begingroup$ I added the comment to your answer. Feel free to edit it if you do not agree. $\endgroup$ – András Bátkai Dec 17 '15 at 15:35

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