Concentration of ground state solutions for fractional Hamiltonian systems
Keywords
Fractional Hamiltonian systems, fractional Sobolev space, ground state solution, critical point theory, concentration phenomenaAbstract
We are concerned with the existence of ground states solutions to the following fractional Hamiltonian systems: $$ \begin{cases} - _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla W(t,u(t))=0,\\ u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n), \end{cases} \leqno(\mbox{FHS})_\lambda $$ where $\alpha\in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^n$, $\lambda> 0$ is a parameter, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric matrix for all $t\in \mathbb{R}$, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$ and $\nabla W(t,u)$ is the gradient of $W(t,u)$ at $u$. Assuming 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 $T$ of $\mathbb{R}$, $W(t,u)$ satisfies the Ambrosetti-Rabinowitz condition and some other reasonable hypotheses, we show that (FHS)$_\lambda$ has a ground sate solution which vanishes on $\mathbb{R}\setminus T$ as $\lambda \to \infty$, and converges to $u\in H^{\alpha}(\mathbb{R}, \mathbb{R}^n)$, where $u\in E_{0}^{\alpha}$ is a ground state solution of the Dirichlet BVP for fractional systems on the finite interval $T$. Recent results are generalized and significantly improved.References
O. Agrawal, J. Tenreiro Machado and J. Sabatier, Introduction to Fractional Derivatives and Their Applications, Nonlinear Dynamics 38 (2004), 1–2.
Z.B. Bai and H.S. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005), 495–505.
V. Coti Zelati and P.H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc. 4 (1991), 693–727.
Y.H. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal. 25 (1995), 1095–1113.
M. Du, L. Tian, J. Wang and F. Zhang, Existence of ground state solutions for a superbiquadratic Kirchhoff-type equation with steep potential well, Applicable Analysis (2015), DOI:10.1080/00036811.2015.1022312.
I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer, Berlin, 1990.
V. Ervin and J. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Parial Differential Equations 22 (2006), 58–76.
R. Hilfer, Applications of Fractional Calculus in Physics, World Science, Singapore, 2000.
M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations 219 (2005), 375–389.
M. Izydorek and J. Janczewska, Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential, J. Math. Anal. Appl. 335 (2007), 1119–1127.
W.H. Jiang, The existence of solutions for boundary value problems of fractional differential equatios at resonance, Nonlinear Anal. 74 (2011), 1987–1994.
F. Jiao and Y. Zhou, Existence results for fractional boundary value problem via critical point theory, Internat. J. Bifur. Chaos appl. Sci. Engrg. 22 (2012), 1–17.
A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Vol. 204, North-Holland, Singapore, 2006.
J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989.
A. Mendez and C. Torres, Multiplicity of solutions for fractional Hamiltonian systems with Liouville–Weyl fractional derivatives, Fract. Calc. Appl. Anal. 18 (2015), 875–890.
K. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York, 1993.
W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations 5 (1992), 1115–1120.
H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Gauthier–Villars, Paris, 1897–1899.
I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., Vol. 65, American Mathematical Society, Provodence, 1986.
P. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A. 114 (1990), 33–38.
P. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z. 206 (1991), 473–499.
M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Boston, 1999.
A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications (D. Gao and D. Motreanu, eds.) International Press, Boston, 2010, pp. 597–632.
C. Torres, Existence of solutions for a class of fractional Hamiltonian systems, Electron. J. Differential Equations 2013 (2013), no. 259, 1–12.
C. Torres, Existence of solutions for perturbed fractional Hamiltonian systems, J. Fract. Calc. Appl. 6 (2015), no. 1, 62–70.
C. Torres, Exstence and concentration of solution for a class of fractional Hamiltonian systems with subquadratic potential, Proc. Math. Sci. (in press).
C. Torres, Mountain pass solution for a fractional boundary value problem, J. Fract. Calc. Appl. 1 (2014), no. 1, 1–10.
C. Torres, Ground state solution for differential equations with left and right fractional derivatives, Math. Meth. Appl. Sci. 38 (2015), 5063–5073.
C. Torres, Existence and symmetric result for Liouville–Weyl fractional nonlinear Schrödinger equation, Commun. Nonlinear Sci. Numer Simul. 27 (2015), 314–327.
J.F. Xu, D. O’Regan and K.Y. Zhang, Multiple solutions for a class of fractional Hamiltonian systems, Fract. Calc. Appl. Anal. 18 (2015), no. 1, 48–63.
S.Q. Zhang, Existence of a solution for the fractional differential equation with nonlinear boundary conditions, Comput. Math. Appl. 61 (2011), 1202–1208.
Z.H. Zhang and R. Yuan, Variational approach to solutions for a class of fractional Hamiltonian systems, Meth. Methods Appl. Sci. 37 (2014), no. 13, 1873–1883.
Z.H. Zhang and R. Yuan, Solutions for subquadratic fractional Hamiltonian systems without coercive conditions, Meth. Methods Appl. Sci. 37 (2014), no. 18, 2934–2945.
Z.H. Zhang and C. Torres, Solutions for fractional Hamiltonian systems with a parameter, J. Appl. Math. Comput. 54 (2017), 451–468.
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