Fractional Lane-Emden Hamiltonian systems

In this work, our interest lies in proving the existence of solutions to the following Fractional Lane-Emden Hamiltonian system: $$\{\begin{array}{ll}(-\mathrm{\Delta }{)}^{s}u=H_v(x,u,v)& \text{in }\mathrm{\Omega },\\ (-\mathrm{\Delta }{)}^{s}v=H_u(x,u,v)& \text{in }\mathrm{\Omega },\\ u=v=0& \text{in }{\mathbb{R}}^n\setminus \mathrm{\Omega }\mathrm{.}\end{array}$$ The method, that can be traced back to the work of De Figueiredo and Felmer \cite{DF-F}, is flexible enough to deal with more general nonlocal operators and make use of a combination of fractional order Sobolev spaces together with functional calculus for self-adjoint operators.