On nonlinear fractional field equations

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?

• 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. – CooLee May 17 '15 at 12:00

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.
• @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. – ptf1 Dec 17 '15 at 14:41