Multiplicity of Solutions for Fractional Hamiltonian Systems with Liouville-Weyl Fractional Derivatives

Abstract

In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems:tDu ∈ Hwhere α ∈ (1/2, 1), t ∈ ℝ, u ∈ ℝm({t ∈ (y − rare satisfied and W is of subquadratic growth as |u| → +∞, we show that (0.1) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in Z. Zhang and R. Yuan [24] are significantly improved.

Nontrivial Solutions for Time Fractional Nonlinear Schrödinger-Kirchhoff Type Equations

“…Also in view of (V1), (V2), and following the method of proof similar to that of Lemma 2.2 in [15], the embedding…”

“…In recent years, there has been a great interest in studying problems involving fractional Schrödinger equations [1][2][3][4][5], Kirchhoff type equations [6][7][8], fractional Navier-Stokes equations [9,10], and fractional ordinary differential equations and Hamiltonian systems [11][12][13][14][15][16][17], and so forth. For further details and applications, we refer the reader to [18,19] and the references cited therein.…”

Tempered fractional differential equation: variational approach

“…In , Jiao and Zhou were the first who use critical point theory to study the existence of solutions to the fractional boundary value problem: The authors obtained the existence of at least one nontrivial solution by using critical point theory. Motivated by the aforementioned work, more and more authors began considering fractional differential equations with mixed derivatives: see, for example, for boundary value problems and for problems in the real line. For example, in , the author considered the following fractional Hamiltonian systems: where α ∈(1/2,1), is a symmetric and positive definite matrix for all , and ∇ W ( t , u ) is the gradient of W ( t , u ) at u .…”

“…The authors obtained the existence of at least one nontrivial solution by using critical point theory. Motivated by the aforementioned work, more and more authors began considering fractional differential equations with mixed derivatives: see, for example, [22][23][24][25][26] for boundary value problems and [27][28][29][30][31][32][33][34][35][36] for problems in the real line. For example, in [30], the author considered the following fractional Hamiltonian systems:…”

Multiplicity result for a stationary fractional reaction-diffusion equations

“…The idea behind them is trying to find solutions of a given boundary value problem by looking for critical points of a suitable energy functional defined on an appropriate function space. In the last 30 years, the critical point theory has become a wonderful tool in studying the existence of solutions to differential equations with variational structures, we refer the reader to the books due to Mawhin and Willem [15], Rabinowitz [19], Schechter [23] and papers [10,12,13,17,26,27,28,31] for bounded intervals and [16,25,29,30,33,34] in the real line.…”