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:$\{\begin{array}{cc}{(-\Delta )}^{s}u=H_v(x,u,v)& \text{in }\Omega ,\\ {(-\Delta )}^{s}v=H_u(x,u,v)& \text{in }\Omega ,\\ u=v=0& \text{in }{R}^n\setminus \Omega .\end{array}$ The method, that can be traced back to the work of De Figueiredo and Felmer [5], 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.