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?