Multiplicity of Solutions for a Class of Perturbed Fractional Hamiltonian Systems
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Multiplicity of Solutions for a Class of Perturbed Fractional Hamiltonian Systems César Torres1 · Oliverio Pichardo2 Received: 19 February 2019 / Revised: 3 July 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, we consider the following class of fractional Hamiltonian systems α α x D∞ (−∞ D x u(x)) + α N
L(x)u(x) = ∇W (x, u(x)) + f (x), x ∈ R
u ∈ H (R, R ),
2
where α ∈ (1/2, 1), L ∈ C(R, R N ) is a symmetric positive definite matrix, W ∈ C 1 (R × R N , R) is superquadratic and even in u. By using Bolle’s perturbation method in critical point theory, we prove the existence of infinitely many solutions in spite of the lack of the symmetry of this problem. Moreover, we study the spectral properties α ( α of the operator x D∞ −∞ D x ) + L(x). Keywords Perturbed fractional Hamiltonian systems · Fractional Sobolev spaces · Bolle’s perturbation method Mathematics Subject Classification Primary 34C37; Secondary 35A15 · 35B38
Communicated by Shangjiang Guo.
B
César Torres [email protected] Oliverio Pichardo [email protected]
1
Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n., Trujillo, Peru
2
Escuela Académico Profesional de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n., Trujillo, Peru
123
C. Torres, O. Pichardo
1 Introduction Fractional derivatives are nonlocal operators and are historically applied in the study of nonlocal or time-dependent processes. The first and well-established application of fractional calculus in physics was in the framework of anomalous diffusion, which is related to features observed in many physical systems, for example; in dispersive transport in amorphous semiconductor, liquid crystals, polymers, proteins, etc. [5,7– 9]. The fractional calculus of variations is a beautiful and useful field of mathematics that deals with the problems of determining extrema (maxima or minima) of functionals whose Lagrangians contain fractional integrals and/or derivatives. It was born in 1996–1997, when Riewe derived Euler–Lagrange fractional differential equations and showed how nonconservative systems in mechanics can be described using fractional derivatives [17]. More precisely, for y : [a, b] → Rn and α j , β j ∈ [0, 1], i = 1 . . . N , j = 1, . . . , N˜ , he considered the energy functional J (y) =
b
a
β˜
β
F(a Dtα1 y(t), . . . , a Dtα N y(t), t Db 1 y(t), . . . , t Db N y(t), y(t), t)dt,
with n, N , N˜ ∈ N. Using the fractional variational principle he obtained the following Euler–Lagrange equation N
αi t Db [∂i F] +
i=1
N˜
βi a Dt [∂i+N F] + ∂ N˜ +N +1 F
= 0.
(1.1)
i=1
In particular, if 1 1 F = m y˙ 2 − V (y) + γ i 2 2
1 2
a Dt
2 [y]
,
(1.2)
∂ V (y) . ∂y
(1.3)
he obtained the Euler–Lagrange equation m y¨ = −γ i
1
1
2 2 t Db ◦ a Dt
[y] −
Recently, several different approaches have been developed to generalize the least action principle and the Euler–Lagrange equations to include fractional derivatives, for more details
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