[Paper Review] Existence and concentration of solution for a class of fractional Hamiltonian systems with subquadratic potential

César E. Torres Ledesma|arXiv (Cornell University)|Mar 23, 2015

Nonlinear Partial Differential Equations10 references4 citations

TL;DR

This paper establishes the existence of nontrivial weak solutions for a class of fractional Hamiltonian systems with subquadratic potential using critical point theory, particularly the mountain pass theorem. As the parameter λ → ∞, solutions concentrate in a finite interval I where the matrix L(t) vanishes, converging strongly in H^α(ℝ) to a nontrivial solution of the Dirichlet problem on I.

ABSTRACT

This article study the fractional Hamiltonian systems \begin{eqnarray}\label{00} {_{t}}D_{\infty}^α({_{-\infty}}D_{t}^αu) + λL(t)u = abla W(t, u), \;\;t\in \mathbb{R}, \end{eqnarray} where $α\in (1/2, 1)$, $λ>0$ is a parameter, $L\in C(\mathbb{R}, \mathbb{R}^{n imes n})$ and $W \in C^{1}(\mathbb{R} imes \mathbb{R}^n, \mathbb{R})$. Unlike most other papers on this problem, we require that $L(t)$ is a positive semi-definite symmetric matrix for all $t\in \mathbb{R}$, that is, $L(t) \equiv 0$ is allowed to occur in some finite interval $\mathbb{I}$ of $\mathbb{R}$. Under some mild assumptions on $W$, we establish the existence of nontrivial weak solution, which vanish on $\mathbb{R} \setminus \mathbb{I}$ as $λ o \infty,$ and converge to $ ilde{u}$ in $H^α(\mathbb{R})$; here $ ilde{u} \in E_{0}^α$ is nontrivial weak solution of the Dirichlet BVP for fractional Hamiltonian systems on the finite interval $\mathbb{I}$.

Motivation & Objective

• To establish the existence of nontrivial weak solutions for a fractional Hamiltonian system with a subquadratic potential and a parameter λ.
• To analyze the concentration behavior of solutions as λ → ∞, particularly when L(t) is allowed to vanish on a finite interval.
• To prove that solutions concentrate in a bounded interval I where L(t) ≡ 0, converging to a nontrivial solution of the Dirichlet problem on I.
• To extend the applicability of critical point theory to systems with degenerate or semi-definite potential matrices L(t), including cases where L(t) = 0 on a nontrivial interval.

Proposed method

• Formulate the fractional Hamiltonian system using left and right Liouville-Weyl fractional derivatives of order α ∈ (1/2, 1).
• Define the energy functional I_λ on the fractional Sobolev space X_λ, incorporating the parameter λ and the matrix L(t).
• Apply the mountain pass theorem to prove the existence of a nontrivial critical point u_λ of I_λ for λ > Λ.
• Use the concentration-compactness principle and vanishing lemma to analyze the asymptotic behavior of solutions as λ → ∞.
• Establish convergence of u_λ to a nontrivial weak solution ˜u in E_0^α (the subspace of H^α(ℝ) with compact support in I), under the condition that L(t) ≡ 0 on a closed interval I.
• Prove strong convergence of u_λ to ˜u in X_λ and in L^p(ℝ) for 2 ≤ p < ∞, using uniform boundedness and energy estimates.

Experimental results

Research questions

• RQ1 Under what conditions does the fractional Hamiltonian system with subquadratic potential admit a nontrivial weak solution?
• RQ2 How do solutions behave as the parameter λ → ∞ when L(t) is allowed to vanish on a finite interval?
• RQ3 Can the solution concentrate in a bounded region where L(t) ≡ 0, and if so, to what limit does it converge?
• RQ4 What role does the degeneracy of L(t) (i.e., L(t) = 0 on a set of positive measure) play in the existence and concentration of solutions?
• RQ5 Is the limiting solution ˜u a nontrivial weak solution of the Dirichlet problem on the interval I where L(t) ≡ 0?

Key findings

• For λ > Λ, the energy functional I_λ has a critical point u_λ, which is a nontrivial weak solution of the fractional Hamiltonian system (1.1).
• The solution u_λ satisfies I_λ(u_λ) = c_λ < 0, confirming its nontriviality and existence via the mountain pass geometry.
• As λ → ∞, the sequence of solutions u_λ converges strongly in X_λ to a function ˜u ∈ E_0^α, which is a nontrivial weak solution of the Dirichlet problem on the interval I.
• The limit solution ˜u is supported in I and satisfies the fractional Euler-Lagrange equation on I with zero boundary conditions.
• The convergence u_λ → ˜u holds in H^α(ℝ) and in L^p(ℝ) for all 2 ≤ p < ∞, with strong convergence in the norm of X_λ.
• The solution u_λ vanishes a.e. on ℝ ∖ I as λ → ∞, confirming concentration in the interval I where L(t) ≡ 0.