Fractional Lane-Emden Hamiltonian systems

TL;DR

This paper proves the existence of solutions for a class of fractional Lane-Emden Hamiltonian systems using advanced variational methods and fractional Sobolev spaces, extending previous approaches to more general nonlocal operators.

Contribution

It introduces a flexible method to establish solutions for fractional Hamiltonian systems involving general nonlocal operators, broadening the scope of existing techniques.

Findings

Existence of solutions for fractional Lane-Emden Hamiltonian systems.

Extension of methods to more general nonlocal operators.

Use of fractional Sobolev spaces and functional calculus.

Abstract

In this work, our interest lies in proving the existence of solutions to the following Fractional Lane-Emden Hamiltonian system: $$ \begin{cases} (-\Delta)^s u = H_v(x,u,v) & \text{in }\Omega,\\ (-\Delta)^s v = H_u(x,u,v) & \text{in }\Omega,\\ u=v=0 & \text{in } \R^n\setminus\Omega. \end{cases} $$ 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.