带权的非线性椭圆方程的多解性问题
|
摘要
本文应用临界点理论(光滑或非光滑)、不动点定理等变分与非变分的方法讨论了几类椭圆边值问题在不同条件下解的存在性和多解性问题.全文共分三个部分,主要内容如下：
在第一部分中,我们讨论了一类含散度型算子的椭圆边值问题多解的存在性问题,这里p>N≥2,a与g是正的径向对称函数.我们首先利用极大极小原理得到该问题在一定条件下的非径向对称最小能量解的存在性,再利用Ljusternik-Schnirelmann畴数定理得到了该问题的一个非径向对称的多解的结果.
在第二部分中,我们研究了两类含奇异性算子的非光滑边值问题和其中是一个可测函数,并且关于u是局部Lipschitz的,8F(x,u)是函数u→F(x, u)广义梯度(因此8F(x,u)不一定连续),f是一可测函数(不一定连续).我们利用非光滑临界点理论证明了这两类边值问题在一定条件下存在两个解.
前两部分,无论是光滑还是非光滑的模型,我们的处理方法都是变分的,在第三部分中,我们利用一个不动点定理证明了下列无界区域上含奇异性项的非光滑椭圆方程的解的存在性,这里N≥3,H是Heaviside函数,即
In this thesis, by applying variational or non-variational methods, such as critical point theory (smooth or nonsmooth), fixed point theorem, we discuss the existence and multiplicity of solutions for several kinds of elliptic boundary value problems. This thesis is composed by three parts, the main contents are as follows:
In Part 1, we discuss a class of elliptic boundary value problems involving diver-gence type operator where p> N≥2,αand g are positive radial continuous functions. Using the minimax principle, we first obtain the existence of non-radial ground state solutions for the above problem, and then by applying the Ljusternik-Schnirelmann category theorems, we get the existence of multiple solutions for the problems.
In Parts 2, we study two kinds of boundary value problems involving singular and nonsmooth nonlinearities and where is measurable and locally Lipschitz in the u-variable,(?) F(x, u) denotes the generalized gradient of the function u→F(x, u) (hence (?)F(x, u) may not be continuous)and f is a measurable function (probably not continuous). By applying nonsmooth critical point theory, under some conditions, we obtain the existence of at least two solutions for both above boundary value problems.
We point out that our methods dealing with boundary value problems (smooth or nonsmooth) in above parts are variational, in Part 3, by apply a fixed point theorem, we show that the existence of solutions for the following elliptic equation on unbounded domain RN involving singular and nonsmooth nonlinearities: Multiple solutions for elliptic equations with weights Abstract where N≥3, H is the Heaviside function, namely
在第一部分中,我们讨论了一类含散度型算子的椭圆边值问题多解的存在性问题,这里p>N≥2,a与g是正的径向对称函数.我们首先利用极大极小原理得到该问题在一定条件下的非径向对称最小能量解的存在性,再利用Ljusternik-Schnirelmann畴数定理得到了该问题的一个非径向对称的多解的结果.
在第二部分中,我们研究了两类含奇异性算子的非光滑边值问题和其中是一个可测函数,并且关于u是局部Lipschitz的,8F(x,u)是函数u→F(x, u)广义梯度(因此8F(x,u)不一定连续),f是一可测函数(不一定连续).我们利用非光滑临界点理论证明了这两类边值问题在一定条件下存在两个解.
前两部分,无论是光滑还是非光滑的模型,我们的处理方法都是变分的,在第三部分中,我们利用一个不动点定理证明了下列无界区域上含奇异性项的非光滑椭圆方程的解的存在性,这里N≥3,H是Heaviside函数,即
In this thesis, by applying variational or non-variational methods, such as critical point theory (smooth or nonsmooth), fixed point theorem, we discuss the existence and multiplicity of solutions for several kinds of elliptic boundary value problems. This thesis is composed by three parts, the main contents are as follows:
In Part 1, we discuss a class of elliptic boundary value problems involving diver-gence type operator where p> N≥2,αand g are positive radial continuous functions. Using the minimax principle, we first obtain the existence of non-radial ground state solutions for the above problem, and then by applying the Ljusternik-Schnirelmann category theorems, we get the existence of multiple solutions for the problems.
In Parts 2, we study two kinds of boundary value problems involving singular and nonsmooth nonlinearities and where is measurable and locally Lipschitz in the u-variable,(?) F(x, u) denotes the generalized gradient of the function u→F(x, u) (hence (?)F(x, u) may not be continuous)and f is a measurable function (probably not continuous). By applying nonsmooth critical point theory, under some conditions, we obtain the existence of at least two solutions for both above boundary value problems.
We point out that our methods dealing with boundary value problems (smooth or nonsmooth) in above parts are variational, in Part 3, by apply a fixed point theorem, we show that the existence of solutions for the following elliptic equation on unbounded domain RN involving singular and nonsmooth nonlinearities: Multiple solutions for elliptic equations with weights Abstract where N≥3, H is the Heaviside function, namely
引文
[1]Kristaly A, Varga C, On a class of quasilinear eigenvalue problems in RN. Math. Nachr.,278(2005),1756-1765.
[2]Kristaly A, Infinitely many solutions for a differential inclusion problem in RN. J. Diff. Equat.,220(2006),511-530.
[3]Figueiredo G M, Furtado M F, Multiplicity of positive solutions for a class of elliptic equations in divergence form. J. Math. Anal. Appl.,321(2006),705-721.
[4]Figueiredo G M, Furtado M F, Positive solutions for some quasilinear equations with critical and supercritical growth of the equation. Nonlin. Anal.,66(2007), 1600-1616.
[5]Lions P L, Symeetrie et compacite dans les espaces Sobolev. J. Funct. Anal. 49(1982),315-334.
[6]Poblo De Napoli, M.C.Mariani, Mountain pass solutions to equations of p-Laplacian type. Nonlin. Anal.54(2003),1205-1219.
[7]Palis R, The principle of symmetric criticality. Comm. Math. Phys.69(1979), 19-30.
[8]Willem M, Minimax theorems, Journal of mathematical analysis and applica-tions. Birkhauser,1996.
[9]Adams R A, Sobolev Spaces, Academic Press, New York,1975.
[10]Li S J, Wang Z Q, Ljusternik-Schnirelmann theory in partially ordered Hilbert spaces. Transactions of the American mathematical society,354(2002),3207-3227.
[11]张恭庆,临界点理论及其应用,上海科技出版社,1986.
[12]Diaz J I, Nonlinear partial differential equations and free boundaries Elliptic equations. Pitman. Adv. Publ., Boston etc.,1986.
[13]Allegretto W, Principal eigenvalues for indefinite weighted elliptic problems in RN. Proc. Amer. Math. Soc.,116(1992),701-706.
[14]Brown K J, Cosner C, Fleckinger J, Principal eigenvalues for problems with indefinite weight function on RN. Proc. Amer. Math. Soc.,109(1990),147-155.
[15]Brown K J, Tertikas A, The existence of principal eigenvalues for problems with indefinite weight function on RN. Proc. Royal Soc. Edinb.,123A(1993),561-569.
[16]Szulkin A, Willem M, Eigenvalue problems with indefinite weight. Studin Mat.,
2(135) (1999),191-201. [17] Costea N, Mihailescu M, Nonlinear, degenerate and singular eigenvalue problems on RN. Nonlin. Anal.71(2009),1153-1159.
[18]Carl S, Heikkila S, Elliptic problems with lack of compactness via a new fixed point theorem. J. Diff. Equat.,186(2002),122-140.
[19]Clarke F H, A new approach to Lagrange multipliers. Math. Oper. Res.,1(1976) 165-174.
[20]Gasinski L, Papageorgiou N S, Nonsmooth critical point theory and nonlinear boundary value problems. Chapman Hall and CRC Press, Boca Raton,2005.
[21]Rudin W, Functional Analysis. McGraw-Hill Com. Inc.,1991.
[22]Chang K C, Variational methods for non-defferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl.,80(1981) 102-129.
[23]Xuan B J, The eigenvalue problem of a singular quasilinear elliptic equation, Electron. J. Diff. Equat.,16(2004) 1-11.
[24]Xuan B. J, Multiple solutions to a Caffarelli-Kohn-Nirenberg type equation with asymptiontically linear term. Rev. Colombiana Mat.,37(2003) 65-79.
[25]Iturriaga L, Existence and multiplicity results for some quasilinear elliptic equa-tion with weights. J. Math. Anal. Appl.,339(2008) 1084-1102.
[26]Chen C H, Wang H, Ground state solutions for singular p-Laplacian equation on RN. J. Math. Anal. Appl.,351(2009) 773-780.
[27]Ghergu M, Radulescu V, Singular elliptic problems with lack of compactness. Annali di Matematica,185(2006) 63-79.
[28]Toan H Q, Ngo Q A, Multiplcity of weak solutions for a class of nonuniformly elliptic equations of p-Laplacian type. Nonlin. Anal.70(2009) 1536-1546.
[29]Costa D G, Goncalves J V, Critical point theory for nondifferentiable functionals and applications. J. Math. Anal. Appl.,153(1990) 470-485.
[30]Caffarelli L, Kohn R, Nirenberg L, First order interpolation inequalities with weights. Compos. Math.,53(1984) 259-275.
[31]Halidias N, Elliptic problems with discontinuities. J. Math. Anal. Appl., 276(2002) 13-27.
[32]Schechter M, Zou W, Double linking theorem and multiple solutions. J. Func. Anal.,205(2003),37-61.
[33]Palais R S, Critical point theory and the minimax principle. Proc Sympos Pure Math,15,185-202, Amer Math Soc, Providence,1970.
[34]Chang K C, Infinite dimensional Morse theory and multiple solution problems. Boston:Birkhauser,1993..
[35]Palais R S, Lusternik-Schnirelmann theory on Banach manifolds. Topology, 5(1966),115-132.
[36]Rabinowitz P H, Minimax methods in critical point theory with applications to differential equations. CBMS, Regional Conf Ser in Math, Vol.65, Amer Math Soc, Providence, RI,1986.
[37]Jeanean L, Tanak K. A positive solution for an asymptotically linear ellip-tic problem on RN autonomous at infinity. ESAIM Control. Optim. Cal. Var., 7(2002),597-614.
[38]Palais R S, Smale S, A generalized Morse theory. Bull. Amer. Math. Soc., 70(1)(1964),165-172.
[39]Stuart C A. An introduction to elliptic equations on RN. Nonlinear functional analysis and applications to differential equations. ed. Ambrosetti A, Chang K C, Ekeland I. Singapore:World Scientific Publishers,1998.
[40]Costa D G, Miyagaki O H. Nontrivial solutions for perturbations of the p-Laplacian on unbounded domains. J. Math. Anal. Appl.,193(1995),737-755.
[41]Schechter M, A variation of the mountain pass lemma and applications. J. Lon-don Math. Soc.,44(1991),491-502.
[42]William P Z, Weakly differentiable functions. New York:Springer-Verlag,1989, GTM 120.
[43]Reed M, Simon B, Methods of modern mathematical physics. New York:Aca-demic Press,1978, Vol IV.
[44]陆文端,微分方程中的变分方法.成都：四川大学出版社,1996.
[45]孙经先,非线性泛函分析及其应用.科学技术出版社,2008.
[46]沈尧天,严树森,拟线性椭圆型方程的变分方法.华南理工大学出版社,1999.
[47]Sun J X, On some problems about nonlinear operators. Ph. D thesis. Jinan: Shandong University,1984.
[48]Kristaly A, Lisei H, Varga C, Multiple solutions for p-Laplacian type equations. Nonlin. Anal.,68(2008),1375-1381.
[49]Bachar I, Othman S B, Maagli H, Radial solutions for the p-Laplacian equation.
Nonlin. Anal.,70(2009),2198-2205.
[50]GarciaHuidobro M, Yarur C S, Behavior of positive unbounded solutions for a class of nonhomogeneous p-Laplacian like equations. Nonlin. Anal.,69(2008), 349-362.
[51]Papageorgiou N S, Rocha E M, Staicu V, A multiplicity theorem for hemivari-ational inequalities with a p-Laplacian-like differential operator. Nonlin. Anal., 69(2008),1150-1163.
[52]Shen Y T, Chen Z H, Nonlinear degenerate elliptic equation with Hardy potential and critical parameter. Nonlin. Anal.,69(2008),1462-1477.
[53]Hirata J, A positive solution of a nonlinear Schrodinger equations with G-symmetry. Nonlin. Anal.,69(2008),3174-3189.
[54]Ding Y, Non-radial solutions of a semilinear elliptic equation. Acta. Math. Sci., 10(1998),229-239.
[55]Shen Z F, Yang M B, Multiple nodal solutions for quasilinear elliptic problems involving Hardy-Sobolev critical exponents in symmetric domain. J. Math. Anal. Appl.,335(2007),524-542.
[56]Su J B, Wang Z Q, Willem M, Weighted Sobolev embedding with unbounded and decaying radial potentials. J.Diff. Equat.,238(2007),201-219.
[57]Furtado M F, Multiple of nodal solutions for a critical quasilinear equation with symmetry. Nonlin. Anal.,63(2005),1153-1166.
[58]Badiale M, Tarantello G, Existence and moltiplicity results for elliptic problems with critical growth and discontinuous nonlinearities. Nonlin. Anal.,29(1997) 639-677.
[59]Alves C O, A variational approach to discountinuous problems with critical Sobolev exponents. J. Math. Anal. Appl.,265(2002),103-127.
[60]Carl S, Heikkila S, Fixed point theorems for multifunctions with applications to discontinuous operator abd differential equations. J. Math. Anal. Appl., 297(2004),56-69.
[61]Wu Y X, New fixed point theorems and applications of mixed monotone operator. J. Math. Anal. Appl.,341(2008),883-893.
[62]Marano S A, Bisci G M, Motreanu D, Multiple solutions for a class of elliptic hemivariational inequalities. J. Math. Anal. Appl.,337(2008),85-97.
[63]Papageorgiou N S, Rocha E M, Paris of positive solutions for p-Laplacian equa-
tions with sublinear and superlinear nonlinearities which do not satisfy the AR-condition. Nonlin. Anal.,70(2009),3854-3863.
[64]Motreanu D, Papageorgiou N S, Multiple solutions for nonlinear elliptic equa-tions at resonance with a nonsmooth potential. Nonlin. Anal.,56(2004),.1211-1234.
[65]Wu X, Leng T J, Dirichlet problem with discontinuous nonlinearities super-linear or asymptotically linear at infinity. J. Math. Anal. Appl.,330(2007),245-258.
[66]Jeanjean L, Tanaka K, A remark on least energy solutions in RN. Proc. Amer. Soc.,131(8) (2003),2399-2408.
[67]Miao Q, Yang Z D, Bounded positive entir solutions of singular p-Laplacian equations. Nonlin. Anal.,69(2008),3749-3760.
[68]Sun Y J, Li S J, Some remarks on a superlinear-singular problem:estimates of λ*. Nonlin. Anal.,69(2008),2636-2650.
[69]Denkowski Z, Gasinski L, Papageorgiou N S, Nontrivial solutions for resonant hemivariational inequalities. J. Glob. Opt.,34(2006),317-337.
[70]Kourogenis N, Papageorgiou N S, Three nontrivial solutions for a quasilinear ellptic differential at resonance with discontinuous right hand side. J. Math. Anal. Appl.,238(1999),477-490.
[71]O'Regan D, Papageorgiou N S, The existence of two nontrivial solutions via homological local linking for the non-coercive p-laplacian Neumann problem. Nonlin. Anal.,70(2009),4386-4392.
[72]Marano S A, Motreanu D, Critical points of non-smooth functions with a weak compactness condition. J. Math. Anal. Appl.,358(2009),189-201.
[73]Halidias N, Elliptic problems with discontinuities. J. Math. Anal. Appl., 276(2002),13-27.
[74]Filippakis M E, Papageorgiou N S, Solvability of nonlinear variational-hemivaiational inequalities. J. Math. Anal. Appl.,311(2005),162-181.
[75]Binding P A, Rynne B P, Variational and non-variational eigenvalues of the p-Laplacian. J. Diff. Equat.,244(2008),24-39.
[76]Struwe M, Variational methods:applications to nonlinear partial differential equations and Hamiltonian systems. New York:Springer-Verlag,1991.
[77]Panagiotopoulos P, Hemivariational inequalities:applications to mechanics and engineering. New York:Springer-Verlag.1993.
[78]Bonheure D, Fabry C, Periodic motions in impact oscillators with perfectly elas-tic bounces. Nonlinearity,123(1995),3555-3561.
[79]Budd C J, Dux F, Intermittency in impact oscillators close to resonance. Non-linearity,7(1994),1191-1224.
[80]Qian D B, Torres P, Periodic motions of linear impact oscillatora via the suc-cessor map. SIAM J. Math. Anal.,36(2005).
[81]Jiang M Y, Periodic solutions of second order differential equations with discon-tinuous nonlinearities. J. Dyn. Diff. Equ.,18(2006),1043-1067.
[82]Chang K Q, The spectrum of 1-Laplace operator. Commun. Contemp. Math., 11(2009), 865-894.
[83]Chabrowski J, Degenerate elliptic equation involving a subcritical Sobolev ex-ponent. Port. Math.,53(1996),167-177.
[84]Lazzo M, Existence and multiplicity of solutions for a class of nonlinear elliptic problem in RN. Discrete Contin. Dyn. Syst., (suppl.)(2003),526-535.
[85]Clarke F, Optimization and nonsmooth analysis. New York:John Wiley-Sons, 1983.
[86]Vladimir G. Maz'ja, Sobolev space. New York:Springer-Verlag,1985.
[87]Brezis H, Li S J, Liu J Q, Rabinowitz P H, Morse theory, minimax theory and their applications to nonlinear differential equations. Beijing, Chinese Academy of Sciences,1999.
[88]Gilbarg D, Trudinger N S, Elliptic partial differential equations of second order. New York:Springer-Verlag,1998.
[89]Ambrosetti A, Rabinowitz P H, Dual variational methods in critical point theory and applications. J. Func. Anal.,14(1973),349-381.
[90]Ljusternik L, Schnirelman L, Sur le probleme de trois geodesique fermees sur les surfaces de genre 0. C. R. Acad. Sci., Paris,189(1929),269-271.
[91]Ljusternik L, Schnirelman L, Methodes topologiques dans les problemes varia-tionelles. Actualites Sci. Industr.,188, Paris,1934.
[92]Morse M, The calculus of variations in the large. Amer. Math. Soc. Coll. Publ., 18(1934).
[93]Murthy M K V, Stampacchia G, Boundary value problems for some degenerate ellptic operators. Ann. Mat. pura. Appl.,80(1968),1-122.
[94]Clement P, Manasevich R, Mitidieri E, Some existence and non-existence results for a homogeneous quasilinear problem. Asymptot. Anal.,17(2004),119-126.
[95]Ambrosetti A, Malchiodi A, Perturbation methods and semilinear elliptic prob-lem on RN. Berlin:Birkhauser Verlag,2006.
[96]Squassina M, Multiple critical points for perturbed symmetric functionals asso-ciated with quasilinear ellptic problems. Nonlin. Anal.,70(2001),1605-1616.
[97]Zeidler E, Nonlinear Functional Analysis and its Applications. Vol. I, Berlin: Springer-Verlag,1985.
[98]Zeidler E, Nonlinear Functional Analysis and its Applications, Vol. ⅡA/B, New York:Springer-Verlag,1990.
[99]Garca A J P, Peral A I, Hardy inequalities and some critical elliptic and parabolic problems. J. Diff. Equ.,144(2)(1998),441-476.
[100]Badiale M, Semilinear elliptic problems in RN with discontinuous nonlinearities. Rendiconti Sent. Mat. Modena.,123(A)(1993),593-619.
[101]Carl S, Heikkila S, Elliptic equations with discontinuous nonlinearties in RN. Nonlin. Anal.,31(1996),217-227.
[102]Lions P L, The concentration-compactness principle in the calculus of variations, the locally compact case. II Ann Inst H Poincare Anal Non Lineaire,1(1984), 223-283.
[103]Huang Y S, Functional Analysis-An introsuction, Beijing, Science Press,2009.
[104]Troianiello G M, Elliptic Differential Equations and Obstacle Problems. New York:Plenum Press,1987.
[105]Heikkila S, Lakshmikantham V, Monotone Iterative Techniques for Discontinu-ous Nonlin. Diff. Equ., Marcel Dekker, New York, Basel,1994.
[2]Kristaly A, Infinitely many solutions for a differential inclusion problem in RN. J. Diff. Equat.,220(2006),511-530.
[3]Figueiredo G M, Furtado M F, Multiplicity of positive solutions for a class of elliptic equations in divergence form. J. Math. Anal. Appl.,321(2006),705-721.
[4]Figueiredo G M, Furtado M F, Positive solutions for some quasilinear equations with critical and supercritical growth of the equation. Nonlin. Anal.,66(2007), 1600-1616.
[5]Lions P L, Symeetrie et compacite dans les espaces Sobolev. J. Funct. Anal. 49(1982),315-334.
[6]Poblo De Napoli, M.C.Mariani, Mountain pass solutions to equations of p-Laplacian type. Nonlin. Anal.54(2003),1205-1219.
[7]Palis R, The principle of symmetric criticality. Comm. Math. Phys.69(1979), 19-30.
[8]Willem M, Minimax theorems, Journal of mathematical analysis and applica-tions. Birkhauser,1996.
[9]Adams R A, Sobolev Spaces, Academic Press, New York,1975.
[10]Li S J, Wang Z Q, Ljusternik-Schnirelmann theory in partially ordered Hilbert spaces. Transactions of the American mathematical society,354(2002),3207-3227.
[11]张恭庆,临界点理论及其应用,上海科技出版社,1986.
[12]Diaz J I, Nonlinear partial differential equations and free boundaries Elliptic equations. Pitman. Adv. Publ., Boston etc.,1986.
[13]Allegretto W, Principal eigenvalues for indefinite weighted elliptic problems in RN. Proc. Amer. Math. Soc.,116(1992),701-706.
[14]Brown K J, Cosner C, Fleckinger J, Principal eigenvalues for problems with indefinite weight function on RN. Proc. Amer. Math. Soc.,109(1990),147-155.
[15]Brown K J, Tertikas A, The existence of principal eigenvalues for problems with indefinite weight function on RN. Proc. Royal Soc. Edinb.,123A(1993),561-569.
[16]Szulkin A, Willem M, Eigenvalue problems with indefinite weight. Studin Mat.,
2(135) (1999),191-201. [17] Costea N, Mihailescu M, Nonlinear, degenerate and singular eigenvalue problems on RN. Nonlin. Anal.71(2009),1153-1159.
[18]Carl S, Heikkila S, Elliptic problems with lack of compactness via a new fixed point theorem. J. Diff. Equat.,186(2002),122-140.
[19]Clarke F H, A new approach to Lagrange multipliers. Math. Oper. Res.,1(1976) 165-174.
[20]Gasinski L, Papageorgiou N S, Nonsmooth critical point theory and nonlinear boundary value problems. Chapman Hall and CRC Press, Boca Raton,2005.
[21]Rudin W, Functional Analysis. McGraw-Hill Com. Inc.,1991.
[22]Chang K C, Variational methods for non-defferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl.,80(1981) 102-129.
[23]Xuan B J, The eigenvalue problem of a singular quasilinear elliptic equation, Electron. J. Diff. Equat.,16(2004) 1-11.
[24]Xuan B. J, Multiple solutions to a Caffarelli-Kohn-Nirenberg type equation with asymptiontically linear term. Rev. Colombiana Mat.,37(2003) 65-79.
[25]Iturriaga L, Existence and multiplicity results for some quasilinear elliptic equa-tion with weights. J. Math. Anal. Appl.,339(2008) 1084-1102.
[26]Chen C H, Wang H, Ground state solutions for singular p-Laplacian equation on RN. J. Math. Anal. Appl.,351(2009) 773-780.
[27]Ghergu M, Radulescu V, Singular elliptic problems with lack of compactness. Annali di Matematica,185(2006) 63-79.
[28]Toan H Q, Ngo Q A, Multiplcity of weak solutions for a class of nonuniformly elliptic equations of p-Laplacian type. Nonlin. Anal.70(2009) 1536-1546.
[29]Costa D G, Goncalves J V, Critical point theory for nondifferentiable functionals and applications. J. Math. Anal. Appl.,153(1990) 470-485.
[30]Caffarelli L, Kohn R, Nirenberg L, First order interpolation inequalities with weights. Compos. Math.,53(1984) 259-275.
[31]Halidias N, Elliptic problems with discontinuities. J. Math. Anal. Appl., 276(2002) 13-27.
[32]Schechter M, Zou W, Double linking theorem and multiple solutions. J. Func. Anal.,205(2003),37-61.
[33]Palais R S, Critical point theory and the minimax principle. Proc Sympos Pure Math,15,185-202, Amer Math Soc, Providence,1970.
[34]Chang K C, Infinite dimensional Morse theory and multiple solution problems. Boston:Birkhauser,1993..
[35]Palais R S, Lusternik-Schnirelmann theory on Banach manifolds. Topology, 5(1966),115-132.
[36]Rabinowitz P H, Minimax methods in critical point theory with applications to differential equations. CBMS, Regional Conf Ser in Math, Vol.65, Amer Math Soc, Providence, RI,1986.
[37]Jeanean L, Tanak K. A positive solution for an asymptotically linear ellip-tic problem on RN autonomous at infinity. ESAIM Control. Optim. Cal. Var., 7(2002),597-614.
[38]Palais R S, Smale S, A generalized Morse theory. Bull. Amer. Math. Soc., 70(1)(1964),165-172.
[39]Stuart C A. An introduction to elliptic equations on RN. Nonlinear functional analysis and applications to differential equations. ed. Ambrosetti A, Chang K C, Ekeland I. Singapore:World Scientific Publishers,1998.
[40]Costa D G, Miyagaki O H. Nontrivial solutions for perturbations of the p-Laplacian on unbounded domains. J. Math. Anal. Appl.,193(1995),737-755.
[41]Schechter M, A variation of the mountain pass lemma and applications. J. Lon-don Math. Soc.,44(1991),491-502.
[42]William P Z, Weakly differentiable functions. New York:Springer-Verlag,1989, GTM 120.
[43]Reed M, Simon B, Methods of modern mathematical physics. New York:Aca-demic Press,1978, Vol IV.
[44]陆文端,微分方程中的变分方法.成都：四川大学出版社,1996.
[45]孙经先,非线性泛函分析及其应用.科学技术出版社,2008.
[46]沈尧天,严树森,拟线性椭圆型方程的变分方法.华南理工大学出版社,1999.
[47]Sun J X, On some problems about nonlinear operators. Ph. D thesis. Jinan: Shandong University,1984.
[48]Kristaly A, Lisei H, Varga C, Multiple solutions for p-Laplacian type equations. Nonlin. Anal.,68(2008),1375-1381.
[49]Bachar I, Othman S B, Maagli H, Radial solutions for the p-Laplacian equation.
Nonlin. Anal.,70(2009),2198-2205.
[50]GarciaHuidobro M, Yarur C S, Behavior of positive unbounded solutions for a class of nonhomogeneous p-Laplacian like equations. Nonlin. Anal.,69(2008), 349-362.
[51]Papageorgiou N S, Rocha E M, Staicu V, A multiplicity theorem for hemivari-ational inequalities with a p-Laplacian-like differential operator. Nonlin. Anal., 69(2008),1150-1163.
[52]Shen Y T, Chen Z H, Nonlinear degenerate elliptic equation with Hardy potential and critical parameter. Nonlin. Anal.,69(2008),1462-1477.
[53]Hirata J, A positive solution of a nonlinear Schrodinger equations with G-symmetry. Nonlin. Anal.,69(2008),3174-3189.
[54]Ding Y, Non-radial solutions of a semilinear elliptic equation. Acta. Math. Sci., 10(1998),229-239.
[55]Shen Z F, Yang M B, Multiple nodal solutions for quasilinear elliptic problems involving Hardy-Sobolev critical exponents in symmetric domain. J. Math. Anal. Appl.,335(2007),524-542.
[56]Su J B, Wang Z Q, Willem M, Weighted Sobolev embedding with unbounded and decaying radial potentials. J.Diff. Equat.,238(2007),201-219.
[57]Furtado M F, Multiple of nodal solutions for a critical quasilinear equation with symmetry. Nonlin. Anal.,63(2005),1153-1166.
[58]Badiale M, Tarantello G, Existence and moltiplicity results for elliptic problems with critical growth and discontinuous nonlinearities. Nonlin. Anal.,29(1997) 639-677.
[59]Alves C O, A variational approach to discountinuous problems with critical Sobolev exponents. J. Math. Anal. Appl.,265(2002),103-127.
[60]Carl S, Heikkila S, Fixed point theorems for multifunctions with applications to discontinuous operator abd differential equations. J. Math. Anal. Appl., 297(2004),56-69.
[61]Wu Y X, New fixed point theorems and applications of mixed monotone operator. J. Math. Anal. Appl.,341(2008),883-893.
[62]Marano S A, Bisci G M, Motreanu D, Multiple solutions for a class of elliptic hemivariational inequalities. J. Math. Anal. Appl.,337(2008),85-97.
[63]Papageorgiou N S, Rocha E M, Paris of positive solutions for p-Laplacian equa-
tions with sublinear and superlinear nonlinearities which do not satisfy the AR-condition. Nonlin. Anal.,70(2009),3854-3863.
[64]Motreanu D, Papageorgiou N S, Multiple solutions for nonlinear elliptic equa-tions at resonance with a nonsmooth potential. Nonlin. Anal.,56(2004),.1211-1234.
[65]Wu X, Leng T J, Dirichlet problem with discontinuous nonlinearities super-linear or asymptotically linear at infinity. J. Math. Anal. Appl.,330(2007),245-258.
[66]Jeanjean L, Tanaka K, A remark on least energy solutions in RN. Proc. Amer. Soc.,131(8) (2003),2399-2408.
[67]Miao Q, Yang Z D, Bounded positive entir solutions of singular p-Laplacian equations. Nonlin. Anal.,69(2008),3749-3760.
[68]Sun Y J, Li S J, Some remarks on a superlinear-singular problem:estimates of λ*. Nonlin. Anal.,69(2008),2636-2650.
[69]Denkowski Z, Gasinski L, Papageorgiou N S, Nontrivial solutions for resonant hemivariational inequalities. J. Glob. Opt.,34(2006),317-337.
[70]Kourogenis N, Papageorgiou N S, Three nontrivial solutions for a quasilinear ellptic differential at resonance with discontinuous right hand side. J. Math. Anal. Appl.,238(1999),477-490.
[71]O'Regan D, Papageorgiou N S, The existence of two nontrivial solutions via homological local linking for the non-coercive p-laplacian Neumann problem. Nonlin. Anal.,70(2009),4386-4392.
[72]Marano S A, Motreanu D, Critical points of non-smooth functions with a weak compactness condition. J. Math. Anal. Appl.,358(2009),189-201.
[73]Halidias N, Elliptic problems with discontinuities. J. Math. Anal. Appl., 276(2002),13-27.
[74]Filippakis M E, Papageorgiou N S, Solvability of nonlinear variational-hemivaiational inequalities. J. Math. Anal. Appl.,311(2005),162-181.
[75]Binding P A, Rynne B P, Variational and non-variational eigenvalues of the p-Laplacian. J. Diff. Equat.,244(2008),24-39.
[76]Struwe M, Variational methods:applications to nonlinear partial differential equations and Hamiltonian systems. New York:Springer-Verlag,1991.
[77]Panagiotopoulos P, Hemivariational inequalities:applications to mechanics and engineering. New York:Springer-Verlag.1993.
[78]Bonheure D, Fabry C, Periodic motions in impact oscillators with perfectly elas-tic bounces. Nonlinearity,123(1995),3555-3561.
[79]Budd C J, Dux F, Intermittency in impact oscillators close to resonance. Non-linearity,7(1994),1191-1224.
[80]Qian D B, Torres P, Periodic motions of linear impact oscillatora via the suc-cessor map. SIAM J. Math. Anal.,36(2005).
[81]Jiang M Y, Periodic solutions of second order differential equations with discon-tinuous nonlinearities. J. Dyn. Diff. Equ.,18(2006),1043-1067.
[82]Chang K Q, The spectrum of 1-Laplace operator. Commun. Contemp. Math., 11(2009), 865-894.
[83]Chabrowski J, Degenerate elliptic equation involving a subcritical Sobolev ex-ponent. Port. Math.,53(1996),167-177.
[84]Lazzo M, Existence and multiplicity of solutions for a class of nonlinear elliptic problem in RN. Discrete Contin. Dyn. Syst., (suppl.)(2003),526-535.
[85]Clarke F, Optimization and nonsmooth analysis. New York:John Wiley-Sons, 1983.
[86]Vladimir G. Maz'ja, Sobolev space. New York:Springer-Verlag,1985.
[87]Brezis H, Li S J, Liu J Q, Rabinowitz P H, Morse theory, minimax theory and their applications to nonlinear differential equations. Beijing, Chinese Academy of Sciences,1999.
[88]Gilbarg D, Trudinger N S, Elliptic partial differential equations of second order. New York:Springer-Verlag,1998.
[89]Ambrosetti A, Rabinowitz P H, Dual variational methods in critical point theory and applications. J. Func. Anal.,14(1973),349-381.
[90]Ljusternik L, Schnirelman L, Sur le probleme de trois geodesique fermees sur les surfaces de genre 0. C. R. Acad. Sci., Paris,189(1929),269-271.
[91]Ljusternik L, Schnirelman L, Methodes topologiques dans les problemes varia-tionelles. Actualites Sci. Industr.,188, Paris,1934.
[92]Morse M, The calculus of variations in the large. Amer. Math. Soc. Coll. Publ., 18(1934).
[93]Murthy M K V, Stampacchia G, Boundary value problems for some degenerate ellptic operators. Ann. Mat. pura. Appl.,80(1968),1-122.
[94]Clement P, Manasevich R, Mitidieri E, Some existence and non-existence results for a homogeneous quasilinear problem. Asymptot. Anal.,17(2004),119-126.
[95]Ambrosetti A, Malchiodi A, Perturbation methods and semilinear elliptic prob-lem on RN. Berlin:Birkhauser Verlag,2006.
[96]Squassina M, Multiple critical points for perturbed symmetric functionals asso-ciated with quasilinear ellptic problems. Nonlin. Anal.,70(2001),1605-1616.
[97]Zeidler E, Nonlinear Functional Analysis and its Applications. Vol. I, Berlin: Springer-Verlag,1985.
[98]Zeidler E, Nonlinear Functional Analysis and its Applications, Vol. ⅡA/B, New York:Springer-Verlag,1990.
[99]Garca A J P, Peral A I, Hardy inequalities and some critical elliptic and parabolic problems. J. Diff. Equ.,144(2)(1998),441-476.
[100]Badiale M, Semilinear elliptic problems in RN with discontinuous nonlinearities. Rendiconti Sent. Mat. Modena.,123(A)(1993),593-619.
[101]Carl S, Heikkila S, Elliptic equations with discontinuous nonlinearties in RN. Nonlin. Anal.,31(1996),217-227.
[102]Lions P L, The concentration-compactness principle in the calculus of variations, the locally compact case. II Ann Inst H Poincare Anal Non Lineaire,1(1984), 223-283.
[103]Huang Y S, Functional Analysis-An introsuction, Beijing, Science Press,2009.
[104]Troianiello G M, Elliptic Differential Equations and Obstacle Problems. New York:Plenum Press,1987.
[105]Heikkila S, Lakshmikantham V, Monotone Iterative Techniques for Discontinu-ous Nonlin. Diff. Equ., Marcel Dekker, New York, Basel,1994.