[Paper Review] Multiplicity of solutions for fractional Hamiltonian systems with Liouville-Weyl fractional derivative

Amado Méndez, César E. Torres Ledesma|arXiv (Cornell University)|Sep 2, 2014

Nonlinear Partial Differential Equations18 references3 citations

TL;DR

This paper establishes the existence of infinitely many solutions for a class of fractional Hamiltonian systems involving Liouville-Weyl fractional derivatives using critical point theory and genus properties. Under subquadratic growth conditions on the potential and a non-coercive but strictly positive lower bound on the matrix $ L(t) $, it proves the existence of infinitely many nontrivial solutions, improving upon prior results by removing coercivity assumptions on $ L(t) $. The key contribution is a new existence criterion for infinitely many solutions in the subquadratic case without requiring $ L(t) \to \infty $.

ABSTRACT

In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: \begin{eqnarray}\label{eq00} _{t}D_{\infty}^α(_{-\infty}D_{t}^αu(t)) + L(t)u(t) = & abla W(t,u(t))\\ u\in H^α(\mathbb{R}, \mathbb{R}^{N}). onumber \end{eqnarray} where $α\in (1/2, 1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^{n}$, $L\in C(\mathbb{R}, \mathbb{R}^{n^2})$ is a symmetric and positive definite matrix for all $t\in \mathbb{R}$, $W\in C^{1}(\mathbb{R} imes \mathbb{R}^{n}, \mathbb{R})$, and $ abla W$ is the gradient of $W$ at $u$. The novelty of this paper is that, assuming there exists $l\in C(\mathbb{R}, \mathbb{R})$ such that $(L(t)u,u)\geq l(t)|u|^{2}$ for all $t\in \mathbb{R}$, $u\in \mathbb{R}^{n}$ and the following conditions on $l$: $\inf_{t\in \mathbb{R}}l(t) >0$ and there exists $r_{0}>0$ such that, for any $M>0$ $$ m(\{t\in (y-r_{0}, y+r_{0})/\;\;l(t)\leq M\}) o 0\;\;\mbox{as}\;\;|y| o \infty. $$ are satisfied and $W$ is of subquadratic growth as $|u| o +\infty$, we show that ( ef{eq00}) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in [Z. Zhang and R. Yuan, Solutions for subquadratic fractional Hamiltonian systems without coercive conditions, Math. Methods Appl. Sci., DOI: 10.1002/mma.3031] are significantly improved.

Motivation & Objective

• To investigate the existence of infinitely many solutions for fractional Hamiltonian systems with Liouville-Weyl fractional derivatives.
• To extend recent results on subquadratic Hamiltonian systems by removing the coercivity assumption on $ L(t) $, which previously required $ L(t) \to \infty $ as $ |t| \to \infty $.
• To establish a new existence criterion for infinitely many nontrivial solutions using genus properties in critical point theory.
• To analyze the behavior of the energy functional and the convergence of critical values $ c_j \to 0^- $ as $ j \to \infty $.

Proposed method

• Formulates the fractional Hamiltonian system using the composition of left and right Liouville-Weyl fractional derivatives: $ {}_{t}D_{\infty}^{\alpha}({}_{-\infty}D_{t}^{\alpha}u(t)) + L(t)u(t) = \nabla W(t,u(t)) $.
• Works in the fractional Sobolev space $ H^{\alpha}(\mathbb{R}, \mathbb{R}^N) $, equipped with a norm involving the fractional derivative and $ L^2 $-norm.
• Applies genus theory from critical point theory to prove the existence of infinitely many critical points of the energy functional $ I(u) $.
• Imposes conditions on $ L(t) $: $ (L(t)u,u) \geq l(t)|u|^2 $, with $ \inf l(t) > 0 $ and $ l(t) $ satisfying a vanishing measure condition at infinity.
• Assumes $ W $ is subquadratic: $ |\nabla W(t,u)| = o(|u|) $ as $ |u| \to 0 $, and $ W(t,u) \geq a(t)|u|^\theta $ with $ \theta < 2 $, $ b \in L^{2/(2-\theta)}(\mathbb{R}) $.
• Uses compact embedding and weak convergence arguments to show $ \beta_j = \sup_{\|u\|_{X^\alpha}=1, u \in Z_j} \|u\|_{L^2} \to 0 $ as $ j \to \infty $, crucial for proving $ c_j \to 0^- $.

Experimental results

Research questions

• RQ1 Can the existence of infinitely many solutions be established for fractional Hamiltonian systems with Liouville-Weyl derivatives under subquadratic growth of the potential?
• RQ2 Can the coercivity assumption on $ L(t) $, previously required in earlier works, be removed or significantly weakened?
• RQ3 What conditions on $ L(t) $ and $ W(t,u) $ ensure that the critical values $ c_j $ of the energy functional satisfy $ c_j \to 0^- $ as $ j \to \infty $?
• RQ4 How can genus theory be applied effectively to fractional Hamiltonian systems with non-periodic, non-autonomous coefficients?

Key findings

• The system admits infinitely many nontrivial solutions under the subquadratic growth condition on $ W $ and the relaxed condition on $ L(t) $, where $ \inf l(t) > 0 $ and $ l(t) $ satisfies a vanishing measure condition at infinity.
• The critical values $ c_j $ of the energy functional satisfy $ c_j \to 0^- $ as $ j \to \infty $, indicating that the solutions accumulate near zero in energy.
• The coercivity of the energy functional $ I(u) $ is established via Hölder’s inequality and the boundedness of $ b(t) $ in $ L^{2/(2-\theta)} $, ensuring $ I(u) \to \infty $ as $ \|u\|_{X^\alpha} \to \infty $.
• The proof relies on constructing a sequence of symmetric sets $ S_j^\delta $ with genus $ j $, and using odd homeomorphisms to show $ \gamma(I^{-\epsilon}) \geq j $, which implies infinitely many critical points.
• The result improves upon earlier work by Zhang and Yuan by removing the requirement that $ l(t) \to \infty $, thus allowing non-coercive but strictly positive $ L(t) $.
• The vanishing measure condition on $ l(t) $, i.e., $ m(\{t \in (y-r_0,y+r_0) : l(t) \leq M\}) \to 0 $ as $ |y| \to \infty $, ensures the necessary decay for compactness in the variational setting.