Existence of infinitely many solutions for a class of fractional Hamiltonian systems

摘要

In this paper we study solutions of the following nonperiodic fractional Hamiltonian systems: -tD∞α(-∞Dtαx(t))-L(t).x(t)+?W(t,x(t))=0,x∈Hα(R,RN),documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$begin{aligned} left{ begin{array}{lllll} -_{t}D^{alpha }_{infty }(_{-infty }D^{alpha }_{t}x(t))- L(t).x(t)+nabla W(t,x(t))=0, xin H^{alpha }(mathbb {R}, mathbb {R}^{N}), end{array} right. end{aligned}$$end{document}where α∈12,1,t∈R,x∈RN,-∞Dtαdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$alpha in left( {1over {2}}, 1right] , tin mathbb {R}, xin mathbb {R}^N, _{-infty }D^{alpha }_{t}$$end{document} and tD∞αdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$_{t}D^{alpha }_{infty }$$end{document} are the left and right Liouville-Weyl fractional derivatives of order αdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$alpha $$end{document} on the whole axis Rdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathbb {R}$$end{document} respectively, L:R?R2Ndocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L:mathbb {R}longrightarrow mathbb {R}^{2N}$$end{document} and W:R×RN?Rdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$W: mathbb {R}times mathbb {R}^{N}longrightarrow mathbb {R}$$end{document} are suitable functions. Applying a new Fountain Theorem established by W. Zou, we prove the existence of infinitely many solutions for the above system in the case where the matrix L(t) is not required to be either uniformly positive definite or coercive, and W∈C1(R×RN,R)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$W in C^{1}(mathbb {R}times mathbb {R}^{N},mathbb {R})$$end{document} is superquadratic at infinity in x but does not needed to satisfy the Ambrosetti-Rabinowitz condition.