一类Schr（?）dinger-Maxwell方程和序列分数阶微分方程的解
|
摘要
非线性问题来源于几何学,天文学,流体力学,弹性力学,物理学,化学,生物,控制论,图像处理和经济学等许多学科.非线性泛函分析起源于一百多年前,它的目的就是建立一些抽象的拓扑方法和变分方法来研究这些应用学科中的非线性问题.虽然它是一个比较新的领域,但是已经取得了显著的进展,尤其是近三十年取得了快速发展,它已经成为现代数学分析的主流研究领域.非线性泛函分析的理论和方法源于数学的许多领域,比如常微分方程,偏微分方程,变分学,动力系统,微分几何,Lie群,代数拓扑,线性泛函分析,测度论,调和分析,凸分析,博弈论,优化理论等等.目前非线性泛函分析的主要内容包括拓扑度理论,临界点理论,半序方法,局部和全局分歧理论等.许多数学家为非线性泛函分析学科的发展做出了重要贡献,例如L. Lusternick, L. Schnirelman, M. Morse, R. S. Palais, S. Smale, E. Rothe, M. A. Krasonsel'skii, H. Amann, A. Ambrosetti, P. Rabinowitz, I. Ekeland, H. Brezis, L. Nirenberg.国户的张恭庆教授,郭大钧教授,陈文源教授,孙经先教授等在非线性泛函分析的许多领域都取得了非常出色的成就(参见文献[2,3,7,15,16,32,35,41,45,47,48,58,64,73,78,91,94]).
在本论文中,我们首先用变分法得到了Schrodinger-Maxwell方程解的存在性结果,然后分别用上下解方法和不动点定理讨论了一类序列分数阶微分方程解的存在性.Schrodinger-Maxwell方程是在寻找非线性Schrodinger方程与一个未知静电场相互作用的驻波解时得到的.更多关于它的物理背景可以参考[29,39]及其中的参考文献.近年来,Schrodinger-Maxwell驻波解的存在性被许多学者广为研究,可以参见文献[17,19,3639,60,82,93,99,100].分数阶微分方程出现在许多研究领域,例如物理,化学,空气动力学,复杂介质电动力学,聚合物流变学,控制动力系统等等.近来,许多学者关注分数阶微分方程的初值和边值问题解的存在性,参见[13,42,43,57,61,63,75,80,96,98].分数阶微积分理论的进展可以参见[10,44,62,70,72,74,76,85].
在第一章中,我们首先介绍了变分法的发展历程、基本思想以及一些最新进展,然后介绍了Sobolev空间的定义和相应的嵌入定理.第三节中引入了一个椭圆方程边值问题弱解的定义, (P.S.)条件的定义以及几个极小极大定理.第四节中介绍了几个常用不等式和Lebesgue积分理论中几个基本定理.最后一节介绍了Riemann-Liouville分数阶积分和导数的定义和几个基本性质.
在第二章中,我们考虑如下的非线性静态的Schrodinger-Maxwell方程其中对于位势y,我们假设y∈C(R3,R),infx∈R3V(x)≥a1>0,a1>0而且对任意M>0,meas{x∈R3|V(x)≤M}<∞.对于非线性项f,我们没有假设所谓的"Ambrosetti-Rabinowitz"型条件,所以(P.S.)条件的验证变得非常复杂.为了克服这一困难,我们利用邹文明在文献[W. Zou Variant fountain theorems and their applications, Manu. Math.104 (2001) 343-358]中得到的变化的喷泉定理来证明问题(0.1)无穷多个高能量解的存在性.
在第三章中,我们研究了如下带有Riemann-Liouville序列分数阶导数的微分方程的初值问题其中0 在第四章中,我们研究如下非线性序列分数阶微分方程的初值问题其中0<α≤1,f：[0,1]×R×R→R是连续的.我们利用Leray-Schauder型不动点定理和Banach压缩映像原理分别得到了问题(0.3)解的存在和唯一性.不同于前一章的单调迭代方法,我们不需要假设问题(0.3)存在一对上下解.
Many nonlinear problems have their roots in geometry, astronomy, fluid and elastic mechanics, physics, chemistry, biology, control theory, image processing and economics. The main purpose of nonlinear functional analysis is to develop abstract topological and variational methods to study nonlinear problems arising in these applied subjects. Although nonlinear functional analysis is a rather recent field, initiated about one hun-dred years ago, remarkable advances have been made. Especially, in the past thirty years it has undergo rapid growth. And it has become part of the mainstream research fields in contemporary mathematical analysis. The theories and methods in nonlin-ear functional analysis stem from many areas of mathematics:ordinary differential equations, partial differential equations, the calculus of variations, dynamical systems, differential geometry, Lie groups, algebraic topology, linear functional analysis, mea-sure theory, harmonic analysis, convex analysis, game theory, optimization theory, etc. Now the main ingredients of nonlinear analysis consist of topological degree theory, critical point theory, partial order theory, local and global bifurcation theory, etc.
Many mathematicians have made significant contributions to nonlinear functional analysis, e.g. L. Lusternick, L. Schnirelman, M. Morse, R. S. Palais, S. Smale, E. Rothe, M. A. Krasonsel'skii, H. Amann, A. Ambrosetti, P. Rabinowitz, I. Ekeland, H. Brezis, L. Nirenberg. Many well known Chinese mathematicians, e.g. Kung-Ching Chang, Dajun Guo, Wenyuan Chen, Jingxian Sun, have also done good works in various fields in nonlinear functional analysis (see [2,3,7,15,16,32,35,41,45,47,48,58,64,73, 78,91,94]).
In this thesis, we mainly explore the existence results for a class of Schrodinger-Maxwell equations using variational methods, and the existence results for a class of fractional differential equations by upper and lower solutions method and fixed point theorem respectively. Schrodinger-Maxwell equations is obtained while looking for existence of standing waves for the nonlinear Schrodinger equations interacting with an unknown electrostatic field. For more of its physical background we refer to [29,39] and the references therein. Recently, the existence results of the standing wave solutions for the Schrodinger-Maxwell equations have been widely studied by many researchers, see e.g. [17,19,36-39,60,82,93,99,100]. Differential equations of fractional order occur more frequently in different research areas and engineering, such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, control of dynamical systems, etc. Recently, many researchers paid attention to existence result of solution of the initial value problem and boundary value problem for fractional differential equations, such as [13,42,43,57,61,63,75,80,96,98]. Some recent contributions to the theory of fractional differential equations can be seen in [10,44,62,70,72,74,76,85].
In Chapter 1, we first introduce the history, basic idea and recent progress of variational methods. Then in section 2, we introduce Sobolev space and correspond-ing embedding theorems. In section 3, we present the definition of weak solution of a boundary value problem for elliptic PDE, (P.S.) condition and several minimax theo-rems. In section 4, we recall some important inequalities and several basic theorems of Lebesgue integral theory. In the last section, we introduction the definitions of Riemann-Liouville fractional integral and derivative and their several properties.
In Chapter 2, we study the nonlinear stationary Schrodinger-Maxwell equations where for the potential V, we assume V∈C(R3,R), infx∈R3 V(x)≥a1>0, a1>0 and for any M>0, meas{x∈R3|V(x)≤M}<∞. For the nonlinear term f, we don't assume the so called "Ambrosetti-Rabinowitz" type condition, so the verification of (P.S.) condition becomes complicated. In order to overcome this difficulty, we use the variant fountain theorem introduced by Zou [W. Zou Variant fountain theorems and their applications, Manu. Math.104 (2001) 343-358] to get infinitely many large solutions for system (0.1).
In Chapter 3, we consider the following initial value problem for fractional differ-ential equation involving Riemann-Liouville sequential fractional derivative where 0< T<+∞and f∈C([0, T]×R×R). By discuss the properties of the well known Mittag-Leffler function we get a comparison result which is important to obtain the main result. Under the assumptions of the existence of upper and lower solutions of problem (0.2) and suitable assumptions on f, we get the existence and uniqueness result of problem (0.2) by using monotone iterative method.
In Chapter 4, we consider initial value problem of the following nonlinear sequential fractional differential equation where 0<α≤1,f:[0, 1]×R×R→R is continuous. We make use of the Leray-Schauder type fixed point theorem and Banach contraction principle to get the exis-tence and uniqueness of solutions for problem (0.3). Unlike the monotone iterative method in the previous chapter, we don't need to assume the existence of a pair of upper and lower solutions for problem (0.3).
在本论文中,我们首先用变分法得到了Schrodinger-Maxwell方程解的存在性结果,然后分别用上下解方法和不动点定理讨论了一类序列分数阶微分方程解的存在性.Schrodinger-Maxwell方程是在寻找非线性Schrodinger方程与一个未知静电场相互作用的驻波解时得到的.更多关于它的物理背景可以参考[29,39]及其中的参考文献.近年来,Schrodinger-Maxwell驻波解的存在性被许多学者广为研究,可以参见文献[17,19,3639,60,82,93,99,100].分数阶微分方程出现在许多研究领域,例如物理,化学,空气动力学,复杂介质电动力学,聚合物流变学,控制动力系统等等.近来,许多学者关注分数阶微分方程的初值和边值问题解的存在性,参见[13,42,43,57,61,63,75,80,96,98].分数阶微积分理论的进展可以参见[10,44,62,70,72,74,76,85].
在第一章中,我们首先介绍了变分法的发展历程、基本思想以及一些最新进展,然后介绍了Sobolev空间的定义和相应的嵌入定理.第三节中引入了一个椭圆方程边值问题弱解的定义, (P.S.)条件的定义以及几个极小极大定理.第四节中介绍了几个常用不等式和Lebesgue积分理论中几个基本定理.最后一节介绍了Riemann-Liouville分数阶积分和导数的定义和几个基本性质.
在第二章中,我们考虑如下的非线性静态的Schrodinger-Maxwell方程其中对于位势y,我们假设y∈C(R3,R),infx∈R3V(x)≥a1>0,a1>0而且对任意M>0,meas{x∈R3|V(x)≤M}<∞.对于非线性项f,我们没有假设所谓的"Ambrosetti-Rabinowitz"型条件,所以(P.S.)条件的验证变得非常复杂.为了克服这一困难,我们利用邹文明在文献[W. Zou Variant fountain theorems and their applications, Manu. Math.104 (2001) 343-358]中得到的变化的喷泉定理来证明问题(0.1)无穷多个高能量解的存在性.
在第三章中,我们研究了如下带有Riemann-Liouville序列分数阶导数的微分方程的初值问题其中0
Many nonlinear problems have their roots in geometry, astronomy, fluid and elastic mechanics, physics, chemistry, biology, control theory, image processing and economics. The main purpose of nonlinear functional analysis is to develop abstract topological and variational methods to study nonlinear problems arising in these applied subjects. Although nonlinear functional analysis is a rather recent field, initiated about one hun-dred years ago, remarkable advances have been made. Especially, in the past thirty years it has undergo rapid growth. And it has become part of the mainstream research fields in contemporary mathematical analysis. The theories and methods in nonlin-ear functional analysis stem from many areas of mathematics:ordinary differential equations, partial differential equations, the calculus of variations, dynamical systems, differential geometry, Lie groups, algebraic topology, linear functional analysis, mea-sure theory, harmonic analysis, convex analysis, game theory, optimization theory, etc. Now the main ingredients of nonlinear analysis consist of topological degree theory, critical point theory, partial order theory, local and global bifurcation theory, etc.
Many mathematicians have made significant contributions to nonlinear functional analysis, e.g. L. Lusternick, L. Schnirelman, M. Morse, R. S. Palais, S. Smale, E. Rothe, M. A. Krasonsel'skii, H. Amann, A. Ambrosetti, P. Rabinowitz, I. Ekeland, H. Brezis, L. Nirenberg. Many well known Chinese mathematicians, e.g. Kung-Ching Chang, Dajun Guo, Wenyuan Chen, Jingxian Sun, have also done good works in various fields in nonlinear functional analysis (see [2,3,7,15,16,32,35,41,45,47,48,58,64,73, 78,91,94]).
In this thesis, we mainly explore the existence results for a class of Schrodinger-Maxwell equations using variational methods, and the existence results for a class of fractional differential equations by upper and lower solutions method and fixed point theorem respectively. Schrodinger-Maxwell equations is obtained while looking for existence of standing waves for the nonlinear Schrodinger equations interacting with an unknown electrostatic field. For more of its physical background we refer to [29,39] and the references therein. Recently, the existence results of the standing wave solutions for the Schrodinger-Maxwell equations have been widely studied by many researchers, see e.g. [17,19,36-39,60,82,93,99,100]. Differential equations of fractional order occur more frequently in different research areas and engineering, such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, control of dynamical systems, etc. Recently, many researchers paid attention to existence result of solution of the initial value problem and boundary value problem for fractional differential equations, such as [13,42,43,57,61,63,75,80,96,98]. Some recent contributions to the theory of fractional differential equations can be seen in [10,44,62,70,72,74,76,85].
In Chapter 1, we first introduce the history, basic idea and recent progress of variational methods. Then in section 2, we introduce Sobolev space and correspond-ing embedding theorems. In section 3, we present the definition of weak solution of a boundary value problem for elliptic PDE, (P.S.) condition and several minimax theo-rems. In section 4, we recall some important inequalities and several basic theorems of Lebesgue integral theory. In the last section, we introduction the definitions of Riemann-Liouville fractional integral and derivative and their several properties.
In Chapter 2, we study the nonlinear stationary Schrodinger-Maxwell equations where for the potential V, we assume V∈C(R3,R), infx∈R3 V(x)≥a1>0, a1>0 and for any M>0, meas{x∈R3|V(x)≤M}<∞. For the nonlinear term f, we don't assume the so called "Ambrosetti-Rabinowitz" type condition, so the verification of (P.S.) condition becomes complicated. In order to overcome this difficulty, we use the variant fountain theorem introduced by Zou [W. Zou Variant fountain theorems and their applications, Manu. Math.104 (2001) 343-358] to get infinitely many large solutions for system (0.1).
In Chapter 3, we consider the following initial value problem for fractional differ-ential equation involving Riemann-Liouville sequential fractional derivative where 0< T<+∞and f∈C([0, T]×R×R). By discuss the properties of the well known Mittag-Leffler function we get a comparison result which is important to obtain the main result. Under the assumptions of the existence of upper and lower solutions of problem (0.2) and suitable assumptions on f, we get the existence and uniqueness result of problem (0.2) by using monotone iterative method.
In Chapter 4, we consider initial value problem of the following nonlinear sequential fractional differential equation where 0<α≤1,f:[0, 1]×R×R→R is continuous. We make use of the Leray-Schauder type fixed point theorem and Banach contraction principle to get the exis-tence and uniqueness of solutions for problem (0.3). Unlike the monotone iterative method in the previous chapter, we don't need to assume the existence of a pair of upper and lower solutions for problem (0.3).
引文
[1]陈文源,非线性泛函分析,兰州：甘肃人民出版社,1982.
[2]郭大钧,非线性泛函分析中的半序方法,济南：山东科学技术出版社,2000.
[3]郭大钧,非线性泛函分析,济南：山东科学技术出版社,2003.
[4]郭大钧,黄春朝,梁方豪,韦忠礼,实变函数与泛函分析(上、下),济南：山东大学出版社,2005.
[5]陆文端,微分方程中的变分方法,北京：科学出版社,2003.
[6]孙经先,非线性泛函分析及其应用,北京：科学出版社,2008.
[7]张恭庆,临界点理论及其应用,上海：上海科学技术出版社,1986.
[8]张恭庆,林源渠,泛函分析讲义(上册),北京：北京大学出版社,1987.
[9]R. A. Adams, Sobolev space, Acad. Press,1975.
[10]Ravi P. Agarwal, M. Benchohra, S. Hamani, A survey on existence result for boundary value of nonlinear fractional differential equations and inclusions, Acta Appl. Math. (2008), doi:10.1007/s10440-008-9356-6.
[11]Ravi P. Agarwal, V. Lakshmikantham, Juan J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal.72 (2010), 2859-2862.
[12]Bashir Ahmad, Juan J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl.58 (2009),1838-1843.
[13]M. A. Al-Bassam, Some existence theorems on differential equations of general-ized order, J. Reine Angew. Math.218 (1965),70-78.
[14]A. Ambrosetti, M. L. Bertotti, Homoclinics for second order convervative sys-tems, Partial differential equations and related subjects (ed M. Miranda), Pitman Reseach Note in Math. Ser.1992.
[15]A. Ambrosetti, A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Prob-lems, Cambridge Studies in Advanced Mathematics 104, Cambridge University Press, Cambridge,2007.
[16]A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point the-ory and applications, J. Funct. Anal.14 (1973),349-381.
[17]A. Ambrosetti, D. Ruiz, Multiple bound states for the Schrodinger-Poisson prob-lem, Commun. Contemp. Math.10 (2008),391-404.
[18]A. Ambrosetti, M. Struwe, Existence of steady vortex rings in an ideal fluid, Arch. Rat. Mech. Anal.108 (1989),97-109.
[19]A. Azzollini, A.Pomponio, Ground state solutions for the nonlinear Schrodinger-Maxwell equations, J. Math. Anal. Appl.345 (2008),90-108.
[20]Z. B. Bai, H. S. Lv, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl.311 (2005),495-505.
[21]P. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and appli-cations to some nonlinear problems with strong resonance at infinity, Nonlinear Anal.7 (1983),981-1012.
[22]T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal.20 (1993),1205-1216.
[23]T. Bartsch, A. Pankov, Z. Q. Wang, Nonlinear Schrodinger equations with steep potential well, Commun. Contemp. Math.3 (2001),549-569.
[24]T. Bartsch, A. Szulkin, M. Willem, Morse theory and nonlinear differential equa-tions, In:Handbook of Global Analysis, D. Krupka and D. Saunders eds., Else-vier, Amsterdam,2008,41-73.
[25]T. Bartsch, Z. Q. Wang, Existence and multiple results for some superlinear elliptic problems on RN, Comm. Partial Diff. Eqns.20 (1995),1725-1741.
[26]T. Bartsch, M. Willem, On an elliptic equation with concave and convex nonlin-earities, Proc. Amer. Math. Soc.123 (1995),3555-3561.
[27]M. Belmekki, Juan J. Nieto, R. Rodriguze-Lopez, Existence of periodic solution for a nonlinear fractional differential equation, Boundary Value Problems (2009), doi:10.1155/2009/324561.
[28]V. Benci, Introduction to Morse theory:a new approach In:Topological nonlinear analysis (Progr. Nonl. Diff. Eq. Appl.15), Birkhauser, Boston,1995,37-177.
[29]V. Benci, D. Fortunato, An eigenvalue problem for the Schrodinger-Maxwell equa-tions, Topol. Methods Nonl. Anal.11 (1998),283-293.
[30]H. Brezis, Analyse Fonctionnelle, Masson, Paris,1983.(中译本：泛函分析—理论及其应用.叶东,周风译.北京：科学出版社,2009.)
[31]H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math.36 (1983),437-477.
[32]H. Brezis, L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math.64 (1991),939-963.
[33]A. M. Candela, A. Salvatore, Multiple solitary waves for non-homogeneous Schrodinger-Maxwell equations, Mediterr. J. Math.3 (2006),483-493.
[34]J. Chabrowski, Weak convergence methods for semilinear elliptic equations, World Scientific Publishing Co., Inc., River Edge, NJ,1999.
[35]K. C. Chang, Infinite dimensional Morse theory and multiple solution problems Prog. Nonl. Diff. Eq. Appl.6 Birkhauser, Boston,1993.
[36]S.-J. Chen, C.-L. Tang, High energy solutions for the Schrodinger-Maxwell equa-tions, Nonlinear Anal. (2009), doi:10.1016/j.na.2009.03.050
[37]G. M. Coclite, A multiplicity result for the nonlinear Schrodinger-Maxwell equa-tions, Commun. Appl. Anal.7 (2003);417-423.
[38]T. D'Aprile, Non-radially symmetric solution of the nonlinear Schrodinger equa-tion coupled with Maxwell equations, Adv. Nonlinear Stud.2 (2002),177-192.
[39]T. D'Aprile, D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrodinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 893-906.
[40]T. D'Aprile, J. Wei, On bound states concentrating on spheres for the Maxwell-Schrodinger equation, SIAM J. Math. Anal.37 (2005),321-342.
[41]K. Deimling, Nonlinear functional analysis, Spring-Verlag, Berlin,1985.
[42]D. Delbosco, L. Rodino, Existence and uniqueness for a nonlinear fractional dif-ferential equation, J. Math. Anal. Appl.204 (1996),609-625.
[43]J. V. Devi, V. Lakshmikantham, Nonsmooth analysis and fractional differential equations, Nonlinear Anal.70 (2009),4151-4157.
[44]K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl.265 (2002),229-248.
[45]P. Drabek and J. Milota, Methods of Nonlinear Analysis. Applications to Differ-ential Equations, Birkhauser, Basel,2007.
[46]I. Ekeland, On the variational principle, J. Math. Anal. Appl.47 (1974),324-353.
[47]I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, Heidelberg, New York,1990.
[48]I. Ekeland, N. Ghoussoub, Selected new aspects of the calculus of variations in the large, Bull. Amer. Math. Soc. (N.S.) 39 (2002),207-265.
[49]L. C. Evans, Partial Differential Equations, Graduate Studies in Math. Vol.19, Amer. Math. Soc,1998.
[50]N. Ghoussoub, Location, multiplicity and Morse indices of min-max critical points, J. Reine Angew. Math.417 (1991),27-76.
[51]N. Ghoussoub, Duality and perturbation methods in critical point theory, Cam-bridge Tracts in Math.,107 Cambridge University Press, Cambridge,1993.
[52]D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equation of Second Or-der, Springer, New York,2001.
[53]A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York,2003.
[54]N. Heymans, I. Podlubny, Physical interpretation of initial conditions for frac-tional differential equations with Riemann-Liouville fractional derivatives, Rheo-logica Acta,45 (2006),765-771.
[55]R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA,2000.
[56]H. Hofer, E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhauser Advanced Texts, Birkhauser, Basel-Boston-Berlin,1994.
[57]R. W. Ibrahim, M. Darus, Subordination and superordination for univalent so-lutions for fractional differential equations, J. Math. Anal. Appl.345 (2008), 871-879.
[58]Y. Jabri, The Mountain Pass Theorem, Encyclopedia Math. Appl., vol.95, Cam-bridge Univ. Press, Cambridge,2003.
[59]L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on RN, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999),787-809.
[60]H. Kikuchi, On the existence of solution for elliptic system related to the Maxwell-Schrodinger equations, Nonlinear Anal.27 (2007),1445-1456.
[61]A. A. Kilbas, S. A. Marzan, Nonlinear differential equations in weighted spaces of continuous functions, Dokl. Nats. Akad. Nauk Belarusi 47 (2003) 29-35. (in Rusian).
[62]A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Frac-tional Differential Equations, vol.204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands,2006.
[63]N. Kosmatov, Integral equations and initial value problems for nonlinear differ-ential equations of fractional order, Nonlinear Anal.70 (2009),2521-2529.
[64]M. A. Krasonsel'skii, P. P. Zabreiko, Geometrical methods of nonlinear analysis, Springer-Verlag,1984.
[65]V. Lakshmikantham, Theory of fractional functional differential equations, Non-linear Anal.69 (2008),3337-3343.
[66]V. Lakshmikantham, A. S. Vatsala, Basic theory of fractional differential equa-tions, Nonlinear Anal.69 (2008),2677-2682.
[67]P. D. Lax, Functional Analysis, Wiley-Interscience, Wiley & Sons, Hoboken, NJ, 2002.
[68]Junyi Liu, Mingyu Xu, Some exact solutions to Stefan problems with fractional differential equations, J. Math. Anal. Appl.351 (2009),536-542.
[69]Qing-Hua Ma, J. Pecaric, Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations, J. Math. Anal. Appl.341 (2008),894-905.
[70]F. Mainardi, Fractional calculus:some basic problems in continuum and statisti-cal mechanics, in A. Carpinteri and F. Mainardi (Editors), Fractals and Fractional Calculus in Continuum Mechanics. Wien and New York, Springer Verlag (1997), 291-348.
[71]D. McDuff, D. Salamon, Introduction to Symplectic Topology, Oxford University Press, New York,1998.
[72]K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York,1993.
[73]L. Nirenberg, Topics in Nonlinear Functional Analysis, Amer. Math. Soc. and Courant Institute of Math.,2001.
[74]K. B. Oldham, J. Spanier, The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, NY, USA,1974.
[75]E. Pitcher, W. E. Sewell, Existence theorems for solutions of differential equations of non-integral order, Bull. Amer. Math. Soc.44 (1938),100-107.
[76]I. Podlubny, Fractional Differential Equations, vol.198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA,1999.
[77]I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus & Applied Analysis for Theory and Applications,5 (2002),367-386.
[78]P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, A.M.S, Providence, RI,1986.
[79]P. Rabinowitz, Some minimax theorems and applications to nonlinear partial dif-ferential equations, in:L. Cesari, R. Kannan, H.F. Weinberger (Eds.), Nonlinear Analysis (A collection of papers in honor of Erich H. Rothe), Academic Press, New York,1978, pp.161-177.
[80]M. Rivero, L. Rodriguez-Germa, J. J. Trujillo, Linear fractional differential equa-tions with variable coefficients, Appl. Math. Lett.21 (2008),892-897.
[81]D. Ruiz, Semiclassical states for coupled Schrodinger-Maxwell equations concen-tration around a sphere, Math. Models Methods Appl. Sci.15 (2005),141-161.
[82]D. Ruiz, The Schrodinger-Possion equation under the effect of a nonlinear local term, J. Funct. Anal.237 (2006),655-674.
[83]K. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer, Berlin,1987.
[84]D. Salamon, Morse theory, the Conley index and Floer homology, Bull. London Math. Soc.22 (1990),113-140.
[85]S. G. Samko, A. A. Kilbas,O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993.
[86]M. Schechter, Principles of Functional Analysis, Graduate Studies in Math, Vol. 6, AMS,2002.
[87]M. Schwarz, Morse Homology, Birkhauser, Basel,1993.
[88]Kehua Shi, Yongjin Wang, On a stochastic fractional partial differential equation driven by a Levy space-time white noise, J. Math. Anal. Appl.364 (2010) 119-129.
[89]M. Struwe, The existence of surfaces of constant mean curvature with free bound-aries, Acta Math.160 (1988),19-64.
[90]M. Struwe, Existence of periodic solutions of Hamiltonian systems on almost every energy surface, Boletim Soc. Bras. Mat.20 (1990),49-58.
[91]M. Struwe, Variational Methods, Springer-Verlag, New York,1996.
[92]G. Talenti, Best constants in Sobolev inequality, Ann. Math. Pura Appl.110 (1976),353-372.
[93]Z. P. Wang, H. S. Zhou, Positive solution for a nonlinear stationary Schrodinger-Poisson system in R3, Discrete. Contin. Dyn. Syst.18 (2007),809-816.
[94]M. Willem, Minimax Theorems, Boston:Birkhauser,1996.
[95]X. J. Xu, D. Q. Jiang, C. J. Yuan, Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Anal.71 (2009),4676-4688.
[96]S. Q. Zhang, The existence of a positive solution for a nonlinear fractional differ-ential equation, J. Math. Anal. Appl.252 (2000),804-812.
[97]S. Q. Zhang, Existence of positive solution for some class of nonlinear fractional differential equations, J. Math. Anal. Appl.278 (2003),136-148.
[98]S. Q. Zhang, Monotone iterative method for intial value problem involving Riemannian-Liouville fractional derivatives, Nonlinear Anal.71 (2009),2087-2093.
[99]L. Zhao, F. Zhao, On the existence of solutions for the Schrodinger-Poisson equa-tions, J. Math. Anal. Appl.346 (2008),155-169.
[100]L. Zhao, F. Zhao, Positive solutions for Schrodinger-Poisson equations with a critical exponent, Nonlinear Anal.70 (2009),2150-2164.
[101]W. Zou, Variant fountain theorems and their applications, Manu. Math.104 (2001),343-358.
[102]W. Zou, Sign-changing Critical Point Theory, Springer, New York,2008.
[103]W. Zou, Schechter M. Critical point theory and its applications, Springer, New York,2006.
[2]郭大钧,非线性泛函分析中的半序方法,济南：山东科学技术出版社,2000.
[3]郭大钧,非线性泛函分析,济南：山东科学技术出版社,2003.
[4]郭大钧,黄春朝,梁方豪,韦忠礼,实变函数与泛函分析(上、下),济南：山东大学出版社,2005.
[5]陆文端,微分方程中的变分方法,北京：科学出版社,2003.
[6]孙经先,非线性泛函分析及其应用,北京：科学出版社,2008.
[7]张恭庆,临界点理论及其应用,上海：上海科学技术出版社,1986.
[8]张恭庆,林源渠,泛函分析讲义(上册),北京：北京大学出版社,1987.
[9]R. A. Adams, Sobolev space, Acad. Press,1975.
[10]Ravi P. Agarwal, M. Benchohra, S. Hamani, A survey on existence result for boundary value of nonlinear fractional differential equations and inclusions, Acta Appl. Math. (2008), doi:10.1007/s10440-008-9356-6.
[11]Ravi P. Agarwal, V. Lakshmikantham, Juan J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal.72 (2010), 2859-2862.
[12]Bashir Ahmad, Juan J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl.58 (2009),1838-1843.
[13]M. A. Al-Bassam, Some existence theorems on differential equations of general-ized order, J. Reine Angew. Math.218 (1965),70-78.
[14]A. Ambrosetti, M. L. Bertotti, Homoclinics for second order convervative sys-tems, Partial differential equations and related subjects (ed M. Miranda), Pitman Reseach Note in Math. Ser.1992.
[15]A. Ambrosetti, A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Prob-lems, Cambridge Studies in Advanced Mathematics 104, Cambridge University Press, Cambridge,2007.
[16]A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point the-ory and applications, J. Funct. Anal.14 (1973),349-381.
[17]A. Ambrosetti, D. Ruiz, Multiple bound states for the Schrodinger-Poisson prob-lem, Commun. Contemp. Math.10 (2008),391-404.
[18]A. Ambrosetti, M. Struwe, Existence of steady vortex rings in an ideal fluid, Arch. Rat. Mech. Anal.108 (1989),97-109.
[19]A. Azzollini, A.Pomponio, Ground state solutions for the nonlinear Schrodinger-Maxwell equations, J. Math. Anal. Appl.345 (2008),90-108.
[20]Z. B. Bai, H. S. Lv, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl.311 (2005),495-505.
[21]P. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and appli-cations to some nonlinear problems with strong resonance at infinity, Nonlinear Anal.7 (1983),981-1012.
[22]T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal.20 (1993),1205-1216.
[23]T. Bartsch, A. Pankov, Z. Q. Wang, Nonlinear Schrodinger equations with steep potential well, Commun. Contemp. Math.3 (2001),549-569.
[24]T. Bartsch, A. Szulkin, M. Willem, Morse theory and nonlinear differential equa-tions, In:Handbook of Global Analysis, D. Krupka and D. Saunders eds., Else-vier, Amsterdam,2008,41-73.
[25]T. Bartsch, Z. Q. Wang, Existence and multiple results for some superlinear elliptic problems on RN, Comm. Partial Diff. Eqns.20 (1995),1725-1741.
[26]T. Bartsch, M. Willem, On an elliptic equation with concave and convex nonlin-earities, Proc. Amer. Math. Soc.123 (1995),3555-3561.
[27]M. Belmekki, Juan J. Nieto, R. Rodriguze-Lopez, Existence of periodic solution for a nonlinear fractional differential equation, Boundary Value Problems (2009), doi:10.1155/2009/324561.
[28]V. Benci, Introduction to Morse theory:a new approach In:Topological nonlinear analysis (Progr. Nonl. Diff. Eq. Appl.15), Birkhauser, Boston,1995,37-177.
[29]V. Benci, D. Fortunato, An eigenvalue problem for the Schrodinger-Maxwell equa-tions, Topol. Methods Nonl. Anal.11 (1998),283-293.
[30]H. Brezis, Analyse Fonctionnelle, Masson, Paris,1983.(中译本：泛函分析—理论及其应用.叶东,周风译.北京：科学出版社,2009.)
[31]H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math.36 (1983),437-477.
[32]H. Brezis, L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math.64 (1991),939-963.
[33]A. M. Candela, A. Salvatore, Multiple solitary waves for non-homogeneous Schrodinger-Maxwell equations, Mediterr. J. Math.3 (2006),483-493.
[34]J. Chabrowski, Weak convergence methods for semilinear elliptic equations, World Scientific Publishing Co., Inc., River Edge, NJ,1999.
[35]K. C. Chang, Infinite dimensional Morse theory and multiple solution problems Prog. Nonl. Diff. Eq. Appl.6 Birkhauser, Boston,1993.
[36]S.-J. Chen, C.-L. Tang, High energy solutions for the Schrodinger-Maxwell equa-tions, Nonlinear Anal. (2009), doi:10.1016/j.na.2009.03.050
[37]G. M. Coclite, A multiplicity result for the nonlinear Schrodinger-Maxwell equa-tions, Commun. Appl. Anal.7 (2003);417-423.
[38]T. D'Aprile, Non-radially symmetric solution of the nonlinear Schrodinger equa-tion coupled with Maxwell equations, Adv. Nonlinear Stud.2 (2002),177-192.
[39]T. D'Aprile, D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrodinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 893-906.
[40]T. D'Aprile, J. Wei, On bound states concentrating on spheres for the Maxwell-Schrodinger equation, SIAM J. Math. Anal.37 (2005),321-342.
[41]K. Deimling, Nonlinear functional analysis, Spring-Verlag, Berlin,1985.
[42]D. Delbosco, L. Rodino, Existence and uniqueness for a nonlinear fractional dif-ferential equation, J. Math. Anal. Appl.204 (1996),609-625.
[43]J. V. Devi, V. Lakshmikantham, Nonsmooth analysis and fractional differential equations, Nonlinear Anal.70 (2009),4151-4157.
[44]K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl.265 (2002),229-248.
[45]P. Drabek and J. Milota, Methods of Nonlinear Analysis. Applications to Differ-ential Equations, Birkhauser, Basel,2007.
[46]I. Ekeland, On the variational principle, J. Math. Anal. Appl.47 (1974),324-353.
[47]I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag, Berlin, Heidelberg, New York,1990.
[48]I. Ekeland, N. Ghoussoub, Selected new aspects of the calculus of variations in the large, Bull. Amer. Math. Soc. (N.S.) 39 (2002),207-265.
[49]L. C. Evans, Partial Differential Equations, Graduate Studies in Math. Vol.19, Amer. Math. Soc,1998.
[50]N. Ghoussoub, Location, multiplicity and Morse indices of min-max critical points, J. Reine Angew. Math.417 (1991),27-76.
[51]N. Ghoussoub, Duality and perturbation methods in critical point theory, Cam-bridge Tracts in Math.,107 Cambridge University Press, Cambridge,1993.
[52]D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equation of Second Or-der, Springer, New York,2001.
[53]A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York,2003.
[54]N. Heymans, I. Podlubny, Physical interpretation of initial conditions for frac-tional differential equations with Riemann-Liouville fractional derivatives, Rheo-logica Acta,45 (2006),765-771.
[55]R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA,2000.
[56]H. Hofer, E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhauser Advanced Texts, Birkhauser, Basel-Boston-Berlin,1994.
[57]R. W. Ibrahim, M. Darus, Subordination and superordination for univalent so-lutions for fractional differential equations, J. Math. Anal. Appl.345 (2008), 871-879.
[58]Y. Jabri, The Mountain Pass Theorem, Encyclopedia Math. Appl., vol.95, Cam-bridge Univ. Press, Cambridge,2003.
[59]L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on RN, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999),787-809.
[60]H. Kikuchi, On the existence of solution for elliptic system related to the Maxwell-Schrodinger equations, Nonlinear Anal.27 (2007),1445-1456.
[61]A. A. Kilbas, S. A. Marzan, Nonlinear differential equations in weighted spaces of continuous functions, Dokl. Nats. Akad. Nauk Belarusi 47 (2003) 29-35. (in Rusian).
[62]A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Frac-tional Differential Equations, vol.204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands,2006.
[63]N. Kosmatov, Integral equations and initial value problems for nonlinear differ-ential equations of fractional order, Nonlinear Anal.70 (2009),2521-2529.
[64]M. A. Krasonsel'skii, P. P. Zabreiko, Geometrical methods of nonlinear analysis, Springer-Verlag,1984.
[65]V. Lakshmikantham, Theory of fractional functional differential equations, Non-linear Anal.69 (2008),3337-3343.
[66]V. Lakshmikantham, A. S. Vatsala, Basic theory of fractional differential equa-tions, Nonlinear Anal.69 (2008),2677-2682.
[67]P. D. Lax, Functional Analysis, Wiley-Interscience, Wiley & Sons, Hoboken, NJ, 2002.
[68]Junyi Liu, Mingyu Xu, Some exact solutions to Stefan problems with fractional differential equations, J. Math. Anal. Appl.351 (2009),536-542.
[69]Qing-Hua Ma, J. Pecaric, Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations, J. Math. Anal. Appl.341 (2008),894-905.
[70]F. Mainardi, Fractional calculus:some basic problems in continuum and statisti-cal mechanics, in A. Carpinteri and F. Mainardi (Editors), Fractals and Fractional Calculus in Continuum Mechanics. Wien and New York, Springer Verlag (1997), 291-348.
[71]D. McDuff, D. Salamon, Introduction to Symplectic Topology, Oxford University Press, New York,1998.
[72]K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York,1993.
[73]L. Nirenberg, Topics in Nonlinear Functional Analysis, Amer. Math. Soc. and Courant Institute of Math.,2001.
[74]K. B. Oldham, J. Spanier, The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, NY, USA,1974.
[75]E. Pitcher, W. E. Sewell, Existence theorems for solutions of differential equations of non-integral order, Bull. Amer. Math. Soc.44 (1938),100-107.
[76]I. Podlubny, Fractional Differential Equations, vol.198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA,1999.
[77]I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus & Applied Analysis for Theory and Applications,5 (2002),367-386.
[78]P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, A.M.S, Providence, RI,1986.
[79]P. Rabinowitz, Some minimax theorems and applications to nonlinear partial dif-ferential equations, in:L. Cesari, R. Kannan, H.F. Weinberger (Eds.), Nonlinear Analysis (A collection of papers in honor of Erich H. Rothe), Academic Press, New York,1978, pp.161-177.
[80]M. Rivero, L. Rodriguez-Germa, J. J. Trujillo, Linear fractional differential equa-tions with variable coefficients, Appl. Math. Lett.21 (2008),892-897.
[81]D. Ruiz, Semiclassical states for coupled Schrodinger-Maxwell equations concen-tration around a sphere, Math. Models Methods Appl. Sci.15 (2005),141-161.
[82]D. Ruiz, The Schrodinger-Possion equation under the effect of a nonlinear local term, J. Funct. Anal.237 (2006),655-674.
[83]K. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer, Berlin,1987.
[84]D. Salamon, Morse theory, the Conley index and Floer homology, Bull. London Math. Soc.22 (1990),113-140.
[85]S. G. Samko, A. A. Kilbas,O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993.
[86]M. Schechter, Principles of Functional Analysis, Graduate Studies in Math, Vol. 6, AMS,2002.
[87]M. Schwarz, Morse Homology, Birkhauser, Basel,1993.
[88]Kehua Shi, Yongjin Wang, On a stochastic fractional partial differential equation driven by a Levy space-time white noise, J. Math. Anal. Appl.364 (2010) 119-129.
[89]M. Struwe, The existence of surfaces of constant mean curvature with free bound-aries, Acta Math.160 (1988),19-64.
[90]M. Struwe, Existence of periodic solutions of Hamiltonian systems on almost every energy surface, Boletim Soc. Bras. Mat.20 (1990),49-58.
[91]M. Struwe, Variational Methods, Springer-Verlag, New York,1996.
[92]G. Talenti, Best constants in Sobolev inequality, Ann. Math. Pura Appl.110 (1976),353-372.
[93]Z. P. Wang, H. S. Zhou, Positive solution for a nonlinear stationary Schrodinger-Poisson system in R3, Discrete. Contin. Dyn. Syst.18 (2007),809-816.
[94]M. Willem, Minimax Theorems, Boston:Birkhauser,1996.
[95]X. J. Xu, D. Q. Jiang, C. J. Yuan, Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Anal.71 (2009),4676-4688.
[96]S. Q. Zhang, The existence of a positive solution for a nonlinear fractional differ-ential equation, J. Math. Anal. Appl.252 (2000),804-812.
[97]S. Q. Zhang, Existence of positive solution for some class of nonlinear fractional differential equations, J. Math. Anal. Appl.278 (2003),136-148.
[98]S. Q. Zhang, Monotone iterative method for intial value problem involving Riemannian-Liouville fractional derivatives, Nonlinear Anal.71 (2009),2087-2093.
[99]L. Zhao, F. Zhao, On the existence of solutions for the Schrodinger-Poisson equa-tions, J. Math. Anal. Appl.346 (2008),155-169.
[100]L. Zhao, F. Zhao, Positive solutions for Schrodinger-Poisson equations with a critical exponent, Nonlinear Anal.70 (2009),2150-2164.
[101]W. Zou, Variant fountain theorems and their applications, Manu. Math.104 (2001),343-358.
[102]W. Zou, Sign-changing Critical Point Theory, Springer, New York,2008.
[103]W. Zou, Schechter M. Critical point theory and its applications, Springer, New York,2006.