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Fractional Lane-Emden Hamiltonian systems

Ignacio Ceresa Dussel, Julián Fernández Bonder, Nicolas Saintier, Ariel Salort

TL;DR

This work develops a variational framework for a fractional Lane-Emden Hamiltonian system with two coupled $s$-order fractional Laplacians on a bounded domain. By constructing a two-parameter energy $J_\theta$ on fractional power spaces $E_s^\theta\times E_s^{2-\theta}$ and seeking $\theta$-weak solutions as critical points, the authors adapt a generalized Mountain Pass approach to overcome strong indefiniteness. Under suitable subcritical growth conditions on the Hamiltonian $H$, they obtain a $\theta$-weak solution, which is shown to be a weak solution, with potential for additional regularity in subcritical regimes. The method is further extended to a wider class of symmetric nonlocal operators and kernels, broadening the applicability of the results to nonlocal Hamiltonian systems.

Abstract

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

Fractional Lane-Emden Hamiltonian systems

TL;DR

This work develops a variational framework for a fractional Lane-Emden Hamiltonian system with two coupled -order fractional Laplacians on a bounded domain. By constructing a two-parameter energy on fractional power spaces and seeking -weak solutions as critical points, the authors adapt a generalized Mountain Pass approach to overcome strong indefiniteness. Under suitable subcritical growth conditions on the Hamiltonian , they obtain a -weak solution, which is shown to be a weak solution, with potential for additional regularity in subcritical regimes. The method is further extended to a wider class of symmetric nonlocal operators and kernels, broadening the applicability of the results to nonlocal Hamiltonian systems.

Abstract

In this work, our interest lies in proving the existence of solutions to the following Fractional Lane-Emden Hamiltonian system: 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.
Paper Structure (8 sections, 11 theorems, 99 equations, 2 figures)

This paper contains 8 sections, 11 theorems, 99 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operator and spaces
  4. Notions of solutions
  5. Interpolation spaces
  6. Variational framework
  7. Proof of Theorem \ref{['teo1']}: Existence of $\theta$-weak solution.
  8. Extensions.

Key Result

Theorem 1.1

Let $s\in (0,1)$ and let $p,q>1$ be such that condpq.0 and condpq.1 hold. Assume that $H$ satisfies H0-H3. Then there exists $\theta=\theta(p,q,n,s)\in (0,2)$ and a weak solution $(u,v)$ to P such that $(u,v)\in H^{s\theta}_0(\Omega)\times H^{s(2-\theta)}_0(\Omega)$ to P.

Figures (2)

  • Figure 1: Range of admissible $p$ and $q$ in terms of $s$ and $n$.
  • Figure 2: Region of admissible $p$ and $q$ (in red) in Corollary \ref{['Corollary_regularity']} for $n=5$ and $s=1/2$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1
  • Proposition 2.2
  • Definition 2.3: Definition 11 in Leonori
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Proposition 2.8
  • ...and 10 more