非线性椭圆方程的非平凡解
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摘要
本文应用临界点理论和变分方法研究了几类非线性微分方程非平凡解的存在性和多重性.全文主要由四部分组成.
在第二章中,我们应用Morse理论和极小极大方法研究了半线性椭圆方程的多解性.这里,Ω是R~N中边界光滑的有界区域,f∈C~1((?)×R,R)满足一定的增长性条件.
在第三章中,我们研究了MCS方程(Modified Capillary Surface Equation)的一个正解和一个负解的存在性,其中p>1,Ω是R~N中边界光滑的有界区域.
在第四章中,我们应用变分研究了GKP方程(generalized Kadomtsev-Petviashviliequation)的非平凡孤立波解的存在性.
在最后一章,我们在Orlicz-Sobolev函数空间框架下研究了非齐次拟线性椭圆方程的无穷多解的存在性,其中Ω是R~N中边界光滑的有界区域,μ,λ∈R是两个参数.
This dissertation is concerned with the existence and multiplicity of nontrivial solutions for a few kinds of nonlinear differential equations by appying critical point theory and variational methods. It mainly consists of four parts.
In Chapter 2, we study by applying Morse theory and minimax methods the multiplicity of nontrivial solutions of semilinear elliptic boundary value problemwhereΩ(?) R~N is a bounded domain with smooth boundary (?)Ωand f∈C~1((?)×R,R) possesses certain increase conditions.
In Chapter 3, a negative and a positive solution are obtained for the following modified capillary surface equationwhere p > 1,Ωis a bounded domain in R~N with smooth boundary.
In Chapter 4, we study by variational methods the existence of nontrivial solitary waves of the generalized Kadomtsev-Petviashvili equation
In the final chapter, we study a non-homogeneous quasilinear elliptic equation on the Orlicz-Sobolev space setting and obtain the existence of infinitely many solutions.
在第二章中,我们应用Morse理论和极小极大方法研究了半线性椭圆方程的多解性.这里,Ω是R~N中边界光滑的有界区域,f∈C~1((?)×R,R)满足一定的增长性条件.
在第三章中,我们研究了MCS方程(Modified Capillary Surface Equation)的一个正解和一个负解的存在性,其中p>1,Ω是R~N中边界光滑的有界区域.
在第四章中,我们应用变分研究了GKP方程(generalized Kadomtsev-Petviashviliequation)的非平凡孤立波解的存在性.
在最后一章,我们在Orlicz-Sobolev函数空间框架下研究了非齐次拟线性椭圆方程的无穷多解的存在性,其中Ω是R~N中边界光滑的有界区域,μ,λ∈R是两个参数.
This dissertation is concerned with the existence and multiplicity of nontrivial solutions for a few kinds of nonlinear differential equations by appying critical point theory and variational methods. It mainly consists of four parts.
In Chapter 2, we study by applying Morse theory and minimax methods the multiplicity of nontrivial solutions of semilinear elliptic boundary value problemwhereΩ(?) R~N is a bounded domain with smooth boundary (?)Ωand f∈C~1((?)×R,R) possesses certain increase conditions.
In Chapter 3, a negative and a positive solution are obtained for the following modified capillary surface equationwhere p > 1,Ωis a bounded domain in R~N with smooth boundary.
In Chapter 4, we study by variational methods the existence of nontrivial solitary waves of the generalized Kadomtsev-Petviashvili equation
In the final chapter, we study a non-homogeneous quasilinear elliptic equation on the Orlicz-Sobolev space setting and obtain the existence of infinitely many solutions.
引文
[1]R.Adams,Sobolev spaces,Academic Press,New York,1975.
[2]A.Ambrosetti and P.H.Rabinowitz,Dual variational methods in critical point theory and applications,J.Funct.Anal.,14(1973)349-381.
[3]A.Ambrosetti,H.Brezis and G.Cerami,Combined effects of concave and convex nonlin-earities in some elliptic problems,J.Funct.Anal.,122(1994)519-543.
[4]A.Anane,Simplicity et isolation de la premere valeur propre du p-Laplacian avec poids,C.R.Acad.Sci.Paris,305(1987)725-728.
[5]T.Bartsch and M.Willem,On an elliptic equation with concave and convex nonlinearities,Proc.Amer.Math.Soc,123(1995)3555-3561.
[6]T.Bartsch and S.J.Li,Critical point theory for asympotically quadratic functional and application to problems with resonance,Nonlinear Anal.,28 (1997)419-441.
[7]T.Bartsch,K.C.Chang and Z.-Q.Wang,On the Morse indices of sign changing solution of nonlinear elliptic problem,Math.Z.,233 (2000) 655-677.
[8]T.Bartsch and Y.Ding,Deformation theorems on non-metrizable vector spaces and applications to critical point theory,Math.Nachr.,279(2006)1267-1288.
[9]V.Benci and P.H.Rabinowitz,Critical point theorems for indefinite functionals,Invent.Math.,52(1979)241-273.
[10]O.V.Besov,V.P.Il'in and S.M.Nikol'ski(?),Integral representations of functions and imbedding theorems,Vol.Ⅰ,V.H.Winston &;Sons,Washington,D.C,1978.
[11]Jo(?)o Marcos Bezerra do (?),Existence of solutions for Quasilinear elliptic equations,J.Math.Anal.Appl.,207(1997)104-126.
[12]J.Bourgain,On the Cauchy problem for the Kadomtsev-Petviashvili equation,Geom.Funct.Anal.,4(1993) 315-341.
[13] H. Br(?)zis and E. Lieb, A relation between pointwise convergence of functions and convergence of functional,Proc.Amer. Math. Soc, 88(1983)486-490.
[14] H. Br(?)zis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm.Pure Appl.Math., 36(1983)437-477.
[15] H. Br(?)zis and L. Nirenberg, H~1 versus C~1 local minimizers, C. R. Acad. Sci. Paris, 317(1993) 465-475.
[16] H. Br(?)zis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math.,64(1991)939-963.
[17] G. Cerami, Un criterio di esistenza per i punti critici su variet(?) illimitate, Rend. Acad. Sci.Lett.Ist. Lombardo, 112(1978)332-336.
[18] K.C.Chang, Infinite dimensional Morse theory and multiple solution problems, Birkh(a|¨)user,Boston, MA, 1993.
[19] K.C. Chang, S.J. Li and J. Q. Liu, Remarks on multiple solutions for asymptotically linear elliptic boundary value problems, Topo. Meth. Nonl.Anal., 3(1994)179-187.
[20] D. C. Clark, A variant of the Ljusternik-Schnirelmann theory, Indiana Univ. Math. J.,22(1972)65-74.
[21] Ph.Cl(?)ment, M. Garc(?)a-Huidobro, R. Man(?)sevich and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc.Var.,11(2000)33-62.
[22] P. Clement, B. Pagter, G. Sweers and F. Thelin, Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces, Mediterranean Journal of Mathematics, 1(2004)241-267.
[23] P. Concus and R. Finn, The shape of a pendent liquid drop, Philos. Trans. Roy. Soc. London Ser. A, 292(1979)307-340.
[24] A. De Bouard and J. C. Saut, Solitary waves of generalized Kadomtsev-Petviashvili equations, Ann. I. H. Poincare, 14(1997) 211-236.
[25]W.Y.Ding and W.M.Ni,On the existence of positive entire solutions of a semilinear elliptic equation,Arch.Rational Mech.Anal.,91(1986) 283-308.
[26]T.Donaldson,Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces,J.Differential Equations,10(1971)507-528.
[27]I.Ekeland and N.Ghoussoub,Selected new aspects of the the calculus of variations in the large,Bull.Amer.Math.Soc,39(2002)207-265.
[28]R.Finn,Equilibrium capillary surfaces,Springer-Verlag,New York-Berlm-Heiderberg-Tokyo,1986.
[29]N.Pukagai,M.Ito and K.Narukawa,Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on R~N,Funkcialaj Ekvacioj,49(2006)235-267.
[30]N.Fukagai and K.Narukawa,On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems,Annali di Matematica,186(2007)539-564.
[31]M.Garcfa-Huidobro,V.K.Le,R.Manasevich and K.Schmitt,On principal eigenvalues for quasilinear elliptic differential operators:an Orlicz-Sobolev space setting,Non.Diff.Equ.Appl.,6(1999)207-225.
[32]J.Gossez,Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients,Trans.Amer.Math.Soc,190(1974)163-205.
[33]J.Gossez,A strongly nonlinear elliptic problem in Orlicz-Sobolev space,Proc.Symp.Pure Math.,45(1986)455-462.
[34]D.Gromoll and W.Meyer,On differential functions with isolated point,Topology,8(1969)361-369.
[35]H.P.Heinz,Free Ljusternik-Schnirelmann theory and the bifurcation diagrams of certain singular nonlinear systems,J.Differential Equations,66(1987)263-300.
[36]N.Hirano,S.J.Li and Z.Q.Wang,Morse theory without (PS) condition at isolated values and strong resonance problems,Calc.Var.,10(2000)223-247.
[37] P. Isaza and J. Mejia, Local and global Cauchy problem for the Kadomtsev-Petviashvili equation in Sobolev spaces of negative indices, Comm. Partial Differntial Equations,26(2001)1027-1054.
[38] M. A. Krasnosel'skii and Ya. B. Rutickii, Convex functions and Orlicz spaces, Noordhoff,Groningen, 1961.
[39] A. Kufner, O.John and S. Fucik,Function spaces,Noordhoff, Leyden, 1997.
[40] E.M. Landesman and A.C.Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19(1970)609-623.
[41] E.M. Landesman, S. Robinson and A. Rumbos, Multiple solutions of semilinear elliptic problems at resonace, Nonlinear Anal., 24(1995)1049-1059.
[42] S.J. Li and M. Willem,Application of local linking to critical point theory, J. Math. Anal.Appl.,189(1995)6-32.
[43] S.J. Li and J.B. Su, Existence of multiple solutions of a two-point boundary value problem at resonance, Topo. Meth. Nonl.Anal., 10(1997)123-135.
[44] S.J. Li and J.Q. Liu, Computations of critical groups at degenerate critical point and application to nonlinear differential equations with resonance, Houston J. Math., 25(1999)563-582.
[45] S. J. Li and Z. Q. Wang, Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, J. Anal. Math., 81(2000)373-396.
[46] S. J. Li and Z. Q. Wang, Dirichlet problem of elliptic equations with strong resonances,Comm. Partial Differntial Equations, 27(2002)2007-2030
[47] S. J. Li, A new Morse theory and strong resonace problems, Topo. Meth. Nonl. Anal.,21(2003)81-100.
[48] P. Lindqvist, On the equation div(|(?)u|~(p-2)(?)u)+λ|u|~(p-2)u=0,Proc.Amer.Math. Soc,109(1990)157-164.
[49] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ann. I. H. Poincare, 1(1984) 109-145 and 223-283.
[50] J. Q. Liu and S. J. Li, Some existence theorems on multiple critical points and their applications, Kexue Tongbao, 17(1984)1025-1027.
[51] J. Q. Liu, A Morse index for a saddle point, Syst. Sci. Math. Sci., 2(1989)32-39.
[52] J. Q. Liu and J. B. Su, Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. Math. Anal. Appl., 258(2001)209-222.
[53] Z. L. Liu and J. X. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172(2001)257-299.
[54] W. Luxemberg,Banach function spaces, PHD thesis, Technische Hogeschool te Delft, The Netherland, 1955.
[55] J. Mawhin and M. Willem,Critical point theory and Hamiltonian systems, Springer, Berlin,1989.
[56] M. Mih(?)ilescu and V. Radulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting, J. Math. Anal. Appl.,330(2007)416-432.
[57] K. Narukawa and T. Suzuki, Nonlinear eigenvalue problem associated to capillary surfaces,Boll. Un. Mat.Ital.,4-B(1990)223-241.
[58] K. Narukawa and T. Suzuki, Nonlinear eigenvalue problem for a modified capillary surface equation, Funkcialaj Ekvacioj, 37(1994)81-100.
[59] D. Passaseo, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Punct. Anal., 114(1993)97-105.
[60] K. Perera, Critical groups of critical points produced by local linking with applications,Abst. Appl. Anal., 3(1998)437-446.
[61] M. del Pino and J. C. Wei, Supercritcal elliptic problems in domains with small holes, Ann.I. H. Poincar(?),24(2007)507-520.
[62] S. P. Pohozaev, On the eigenfunctions of the equation -Δu+λf(u) =0,Dokl.Akad. Nauk SSSR,165(1965)36-39.
[63] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math. 65, AMS, Providence, R.I., 1986.
[64] P.H. Rabinowitz, J.B. Su and Z.Q. Wang, Multiple solutions of superlinear elliptic equations, Rend. Lincei Mat. Appl,18(2007)97-108.
[65] M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Marcel Dekker, New York, 1991.
[66] S. Robinson, Multiple solutions for semilinear elliptic boundary value problem at resonance,Elec. J. Diff.Equ., 1995(1995)1-14.
[67] J. B. Su, Semilinear elliptic resonant problems at higher eigenvalue with unbounded nonlinear terms, Acta Math. Sinica, New Series, 14(1998)411-419.
[68] J.B. Su and C.L. Tang, Multiplicity results for semilinear elliptic equations with resonance at higher eigenvalues, Nonlinear Anal., 44 (2001)311-321.
[69] J.B. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48(2002)881-895.
[70] J.B. Su, Multiplicity results for asymptotically linear elliptic problems at resonance, J.Math. Anal. Appl., 278(2003)397-408.
[71] J.B. Su and L.G. Zhao, An elliptic resonance problem with multiple solutions, J. Math.Anal. Appl., 319(2006)604-616.
[72] Z. Q. Wang, Multile solutions for indefinite functionals and applications to asymptotically linear problems, Acta Math. Sinica, 5 (1989)101-113.
[73] Z. Q. Wang, On a superlinear elliptic equation, Ann. I. H. Poincare, 8(1991)43-57.
[74] Z. Q. Wang and M. Willem, A multiplicity result for the generalized Kadomtsev-Petviashvili equation, Topo. Meth. Nonl. Anal., 7(1996) 261-270.
[75] H. C. Wente, The symmetry of sessile and pendent drops, Pacific J. Math., 88(1980)387-397.
[76] M. Willem, Minimax theorems, progress in nonlinear differential equations and their applications, Birkhauser,1996.
[77] B. J. Xuan, Nontrivial solitary waves of GKP equation in multi-dimensional spaces, Revista Colombiana de Matematicas, 7(2003) 11-23.
[2]A.Ambrosetti and P.H.Rabinowitz,Dual variational methods in critical point theory and applications,J.Funct.Anal.,14(1973)349-381.
[3]A.Ambrosetti,H.Brezis and G.Cerami,Combined effects of concave and convex nonlin-earities in some elliptic problems,J.Funct.Anal.,122(1994)519-543.
[4]A.Anane,Simplicity et isolation de la premere valeur propre du p-Laplacian avec poids,C.R.Acad.Sci.Paris,305(1987)725-728.
[5]T.Bartsch and M.Willem,On an elliptic equation with concave and convex nonlinearities,Proc.Amer.Math.Soc,123(1995)3555-3561.
[6]T.Bartsch and S.J.Li,Critical point theory for asympotically quadratic functional and application to problems with resonance,Nonlinear Anal.,28 (1997)419-441.
[7]T.Bartsch,K.C.Chang and Z.-Q.Wang,On the Morse indices of sign changing solution of nonlinear elliptic problem,Math.Z.,233 (2000) 655-677.
[8]T.Bartsch and Y.Ding,Deformation theorems on non-metrizable vector spaces and applications to critical point theory,Math.Nachr.,279(2006)1267-1288.
[9]V.Benci and P.H.Rabinowitz,Critical point theorems for indefinite functionals,Invent.Math.,52(1979)241-273.
[10]O.V.Besov,V.P.Il'in and S.M.Nikol'ski(?),Integral representations of functions and imbedding theorems,Vol.Ⅰ,V.H.Winston &;Sons,Washington,D.C,1978.
[11]Jo(?)o Marcos Bezerra do (?),Existence of solutions for Quasilinear elliptic equations,J.Math.Anal.Appl.,207(1997)104-126.
[12]J.Bourgain,On the Cauchy problem for the Kadomtsev-Petviashvili equation,Geom.Funct.Anal.,4(1993) 315-341.
[13] H. Br(?)zis and E. Lieb, A relation between pointwise convergence of functions and convergence of functional,Proc.Amer. Math. Soc, 88(1983)486-490.
[14] H. Br(?)zis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm.Pure Appl.Math., 36(1983)437-477.
[15] H. Br(?)zis and L. Nirenberg, H~1 versus C~1 local minimizers, C. R. Acad. Sci. Paris, 317(1993) 465-475.
[16] H. Br(?)zis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math.,64(1991)939-963.
[17] G. Cerami, Un criterio di esistenza per i punti critici su variet(?) illimitate, Rend. Acad. Sci.Lett.Ist. Lombardo, 112(1978)332-336.
[18] K.C.Chang, Infinite dimensional Morse theory and multiple solution problems, Birkh(a|¨)user,Boston, MA, 1993.
[19] K.C. Chang, S.J. Li and J. Q. Liu, Remarks on multiple solutions for asymptotically linear elliptic boundary value problems, Topo. Meth. Nonl.Anal., 3(1994)179-187.
[20] D. C. Clark, A variant of the Ljusternik-Schnirelmann theory, Indiana Univ. Math. J.,22(1972)65-74.
[21] Ph.Cl(?)ment, M. Garc(?)a-Huidobro, R. Man(?)sevich and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc.Var.,11(2000)33-62.
[22] P. Clement, B. Pagter, G. Sweers and F. Thelin, Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces, Mediterranean Journal of Mathematics, 1(2004)241-267.
[23] P. Concus and R. Finn, The shape of a pendent liquid drop, Philos. Trans. Roy. Soc. London Ser. A, 292(1979)307-340.
[24] A. De Bouard and J. C. Saut, Solitary waves of generalized Kadomtsev-Petviashvili equations, Ann. I. H. Poincare, 14(1997) 211-236.
[25]W.Y.Ding and W.M.Ni,On the existence of positive entire solutions of a semilinear elliptic equation,Arch.Rational Mech.Anal.,91(1986) 283-308.
[26]T.Donaldson,Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces,J.Differential Equations,10(1971)507-528.
[27]I.Ekeland and N.Ghoussoub,Selected new aspects of the the calculus of variations in the large,Bull.Amer.Math.Soc,39(2002)207-265.
[28]R.Finn,Equilibrium capillary surfaces,Springer-Verlag,New York-Berlm-Heiderberg-Tokyo,1986.
[29]N.Pukagai,M.Ito and K.Narukawa,Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on R~N,Funkcialaj Ekvacioj,49(2006)235-267.
[30]N.Fukagai and K.Narukawa,On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems,Annali di Matematica,186(2007)539-564.
[31]M.Garcfa-Huidobro,V.K.Le,R.Manasevich and K.Schmitt,On principal eigenvalues for quasilinear elliptic differential operators:an Orlicz-Sobolev space setting,Non.Diff.Equ.Appl.,6(1999)207-225.
[32]J.Gossez,Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients,Trans.Amer.Math.Soc,190(1974)163-205.
[33]J.Gossez,A strongly nonlinear elliptic problem in Orlicz-Sobolev space,Proc.Symp.Pure Math.,45(1986)455-462.
[34]D.Gromoll and W.Meyer,On differential functions with isolated point,Topology,8(1969)361-369.
[35]H.P.Heinz,Free Ljusternik-Schnirelmann theory and the bifurcation diagrams of certain singular nonlinear systems,J.Differential Equations,66(1987)263-300.
[36]N.Hirano,S.J.Li and Z.Q.Wang,Morse theory without (PS) condition at isolated values and strong resonance problems,Calc.Var.,10(2000)223-247.
[37] P. Isaza and J. Mejia, Local and global Cauchy problem for the Kadomtsev-Petviashvili equation in Sobolev spaces of negative indices, Comm. Partial Differntial Equations,26(2001)1027-1054.
[38] M. A. Krasnosel'skii and Ya. B. Rutickii, Convex functions and Orlicz spaces, Noordhoff,Groningen, 1961.
[39] A. Kufner, O.John and S. Fucik,Function spaces,Noordhoff, Leyden, 1997.
[40] E.M. Landesman and A.C.Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19(1970)609-623.
[41] E.M. Landesman, S. Robinson and A. Rumbos, Multiple solutions of semilinear elliptic problems at resonace, Nonlinear Anal., 24(1995)1049-1059.
[42] S.J. Li and M. Willem,Application of local linking to critical point theory, J. Math. Anal.Appl.,189(1995)6-32.
[43] S.J. Li and J.B. Su, Existence of multiple solutions of a two-point boundary value problem at resonance, Topo. Meth. Nonl.Anal., 10(1997)123-135.
[44] S.J. Li and J.Q. Liu, Computations of critical groups at degenerate critical point and application to nonlinear differential equations with resonance, Houston J. Math., 25(1999)563-582.
[45] S. J. Li and Z. Q. Wang, Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, J. Anal. Math., 81(2000)373-396.
[46] S. J. Li and Z. Q. Wang, Dirichlet problem of elliptic equations with strong resonances,Comm. Partial Differntial Equations, 27(2002)2007-2030
[47] S. J. Li, A new Morse theory and strong resonace problems, Topo. Meth. Nonl. Anal.,21(2003)81-100.
[48] P. Lindqvist, On the equation div(|(?)u|~(p-2)(?)u)+λ|u|~(p-2)u=0,Proc.Amer.Math. Soc,109(1990)157-164.
[49] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ann. I. H. Poincare, 1(1984) 109-145 and 223-283.
[50] J. Q. Liu and S. J. Li, Some existence theorems on multiple critical points and their applications, Kexue Tongbao, 17(1984)1025-1027.
[51] J. Q. Liu, A Morse index for a saddle point, Syst. Sci. Math. Sci., 2(1989)32-39.
[52] J. Q. Liu and J. B. Su, Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. Math. Anal. Appl., 258(2001)209-222.
[53] Z. L. Liu and J. X. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172(2001)257-299.
[54] W. Luxemberg,Banach function spaces, PHD thesis, Technische Hogeschool te Delft, The Netherland, 1955.
[55] J. Mawhin and M. Willem,Critical point theory and Hamiltonian systems, Springer, Berlin,1989.
[56] M. Mih(?)ilescu and V. Radulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting, J. Math. Anal. Appl.,330(2007)416-432.
[57] K. Narukawa and T. Suzuki, Nonlinear eigenvalue problem associated to capillary surfaces,Boll. Un. Mat.Ital.,4-B(1990)223-241.
[58] K. Narukawa and T. Suzuki, Nonlinear eigenvalue problem for a modified capillary surface equation, Funkcialaj Ekvacioj, 37(1994)81-100.
[59] D. Passaseo, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Punct. Anal., 114(1993)97-105.
[60] K. Perera, Critical groups of critical points produced by local linking with applications,Abst. Appl. Anal., 3(1998)437-446.
[61] M. del Pino and J. C. Wei, Supercritcal elliptic problems in domains with small holes, Ann.I. H. Poincar(?),24(2007)507-520.
[62] S. P. Pohozaev, On the eigenfunctions of the equation -Δu+λf(u) =0,Dokl.Akad. Nauk SSSR,165(1965)36-39.
[63] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math. 65, AMS, Providence, R.I., 1986.
[64] P.H. Rabinowitz, J.B. Su and Z.Q. Wang, Multiple solutions of superlinear elliptic equations, Rend. Lincei Mat. Appl,18(2007)97-108.
[65] M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Marcel Dekker, New York, 1991.
[66] S. Robinson, Multiple solutions for semilinear elliptic boundary value problem at resonance,Elec. J. Diff.Equ., 1995(1995)1-14.
[67] J. B. Su, Semilinear elliptic resonant problems at higher eigenvalue with unbounded nonlinear terms, Acta Math. Sinica, New Series, 14(1998)411-419.
[68] J.B. Su and C.L. Tang, Multiplicity results for semilinear elliptic equations with resonance at higher eigenvalues, Nonlinear Anal., 44 (2001)311-321.
[69] J.B. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48(2002)881-895.
[70] J.B. Su, Multiplicity results for asymptotically linear elliptic problems at resonance, J.Math. Anal. Appl., 278(2003)397-408.
[71] J.B. Su and L.G. Zhao, An elliptic resonance problem with multiple solutions, J. Math.Anal. Appl., 319(2006)604-616.
[72] Z. Q. Wang, Multile solutions for indefinite functionals and applications to asymptotically linear problems, Acta Math. Sinica, 5 (1989)101-113.
[73] Z. Q. Wang, On a superlinear elliptic equation, Ann. I. H. Poincare, 8(1991)43-57.
[74] Z. Q. Wang and M. Willem, A multiplicity result for the generalized Kadomtsev-Petviashvili equation, Topo. Meth. Nonl. Anal., 7(1996) 261-270.
[75] H. C. Wente, The symmetry of sessile and pendent drops, Pacific J. Math., 88(1980)387-397.
[76] M. Willem, Minimax theorems, progress in nonlinear differential equations and their applications, Birkhauser,1996.
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