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姓名 陳志有(Zhi-you Chen)  查詢紙本館藏   畢業系所 數學系
論文名稱 非線性橢圓方程及系統中解的唯一性和結構性之探討
(Uniqueness and Structure of Solutions for Nonlinear Elliptic Equations and Systems)
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摘要(中) 在本篇文章中,我們將探討橢圓方程和橢圓系統中解的存在性、唯一性及結構性等問題。首先,在橢圓方程中,我們將利用特徵方程來了解某些奇異解的存在性和唯一性, 接者再使用 Pohozeaev function 和 Energy function 來了解其他所有解的存在性和結構性。 然而在橢圓系統中, 我們將透過線性化方程來學解的結構性,進一步結合隱函數定理來證明某些解的唯一性。
摘要(英) In this article, we consider problems involving the existence, uniqueness, and structure of solutions for (single) elliptic equations and (coupled) elliptic systems. To deal with the elliptic equations, we first employ the characteristic equations to realize the existence and uniqueness of certain singular solutions. Then, specific auxiliary functions, Pohozaev functions and energy functions, will be introduced to conduct the uniqueness for solutions of other types and clarify the complete structure of solutions. On the other hand, for the elliptic systems, we analyze the structure of solutions by means of the corresponding linearized equations. Furthermore, combining the Implicit Function Theorem, the consequences related to the uniqueness of some solutions will be offered as well.
關鍵字(中) ★ 非線性橢圓方程及系統中解的唯一性和結構性之探討 關鍵字(英) ★ Uniqueness and Structure of Solutions for Nonlin
論文目次 Part I. Elliptic Equations
On the Classication of Standing Wave Solutions for the Schrodinger Equation
1. Introduction and Main Results 1
2. Structures and Behaviors of Solutions 7
3. Proofs of the Main Results 12
Uniqueness of Finite Total Curvatures and Structure of Radial Solutions for Nonlinear Elliptic Equations
1. Introduction and Main Results 20
2. Proof of Theorem 1.1 30
3. Zeros and Structure of Entire Solutions 35
4. Proofs of Theorems 1.2-1.5 40
The Structure of Radial Solutions for Elliptic Equations Arising from the Spherical Onsager Vortex
1. Introduction and Main Results 43
2. Preliminaries 52
3. Proofs of Main results 57
On the Blowup and Entire Solutions for a Biharmonic Equation
1. Introduction and Main Results 61
2. Blowup Solutions and Nonexistence Results 65
3. Linearization and Structure of Solution 71
Part II. Elliptic Systems
The Linearization and Uniqueness of Solutions for Elliptic Systems
1. Introduction and Main Results 75
2. Linearization and Uniqueness 77
3. Structure of Solutions 82
The Linearization and Uniqueness of Solutions for Elliptic Systems
1. Introduction and Main Results 86
2. The Non-Degeneracy of Linearized Equations 93
3. Uniqueness of Topological Solution 100
4. Asymptotic Behaviors of All Entire Solutions 104
5. Uniqueness and Solution Structures of the Dirichlet Problem 108
6. The Structures of All Entire solutions 110
7. Reference 121
參考文獻 [1] R. Bellman, Stability Theory of Differential Equations, McGraw-Hill, New York, 1953.
[2] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal 82(1983), 313-345.
[3] J. Busca and B. Sirakov, Symmetry results for semi-linear elliptic systems in the whole space, J. Diff. Eqns 163(2000), 41-56.
[4] C. V. Coffman, Uniqueness of positive radial solution on an annulus of the Dirichlet problem for ¢u − u + u3 = 0, J. Diff. Eqns. 128(1996), 379-386.
[5] K.-S. Cheng and J.-L. Chern, Existence of positive solutions of some semi-linear elliptic equations, J. Diff. Eqns. 98(1992), 169-180.
[6] S. Chanillo and M. Kiessling, Rational symmetry of solutions of some non-linear problems in statistical mechanics and in geometry, Commun. Math. Phys. 160(1994), 217-238.
[7] K.-S. Cheng and J.-T. Lin, On the elliptic equations ¢u = K(x)uσ and ¢u = K(x)e2u, Trans. Amer. Math. Soc. 304(1987), 639-668.
[8] C.-C. Chen and C.-S. Lin, Uniqueness of the ground state solutions of ¢u+ f(u) = 0 in Rn , n ≥ 3, Commun. in Partial Diff. Eqns. 16(1991), 1549-1572.
[9] W. Chen and C. Li, Classici¯cation of solutions of some nonlinear elliptic equation, Duke Math. J. 63(1991), 615-622.
[10] K.-S. Cheng and C.-S. Lin, On the asymptotic behavior of solutions of the conformal Gaussian curvature euqation in R2, Math. Ann. 308(1997), 119-139.
[11] K.-S. Cheng and C.-S. Lin, On the conformal Gaussian Curvature Equation in R2, J. Diff. Eqns. 146(1998), 226-250.
[12] L. A. Caffarelli and Y. Yang, Vortex condensation in the Chern-Simons Higgs model: an existence theorem, Comm. Math. Phys. 168(1995), 321-336.
[13] J.-L. Chern and E. Yanagida, Structure of the sets of regular and singular radial solutions for a semilinear elliptic equation, J. Diff. Eqns. 224(2006), 440-463.
[14] J.-L. Chern, Z.-Y. Chen, and C.-S. Lin, Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles, submitted.
[15] J. L. Chern, C. S. Lin, and J. Shi Uniqueness of Solution to a Coupled Cooperative System, preprint.
[16] J. L. Chern, C. S. Lin, and J. Shi Uniqueness of Solution to a Hamiltonian System, preprint.
[17] J. L. Chern, Z. Y. Chen, and Y. L. Tang, Uniqueness of Finite Total Curvatures and Structure of Radial Solutions for Nonlinear Elliptic Equations, submitted.
[18] Z. Y. Chen, J. L. Chern, and Y. L. Tang, The Structure of Radial Solutions for Elliptic Equations Arising from the Spherical Onsager Vortex, submitted.
[19] Z. Y. Chen, J. L. Chern, and Y. L. Tang, On the Blowup and Entire Solutions for a Biharmonic Equation, preprint.
[20] H. Chan, C.-C. Fu, and C.-S. Lin, Non-topological multi-vortex solutions to the self-dual Chern-Simons-Higgs equation, Commun. Math. Phys. 231(2002), 189-221.
[21] Ph. Clément, D.G. de Figueriredo and E. Mitidieri Positive solutions of semilinear elliptic systems, Commun. Partial Diff. Eqns. 17(1992), 923-940.
[22] D. Chae and O. Y. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys. 215(2000), 119-142.
[23] X. Chen, S. Hastings, J. B. Mcleod, and Y. Yang, A nonlinear elliptic equation arising from gauge filed theory and cosmology, Proc. Roy. Soc. London Ser. A 446(1994), 453-478.
[24] J. L. Chern, Z. Y. Chen, J. H. Chen and Y. L. Tang, On the Classication of Standing Wave Solutions for the Schrödinger Equation, submitted.
[25] Z. Y. Chen, J. L. Chern, Y. L. Tang, and C. S. Lin, The Linearization and Uniqueness of Solutions for Elliptic Systems, submitted.
[26] Z. Y. Chen, J. L. Chern, Y. L. Tang, C. S. Lin, and J. Shi Existence, Uniqueness and Stability of Solution to Sublinear Elliptic System, preprint.
[27] R. Dalmasso, Solutions positives globales d'une equation biharmonique surlineaire, Funkcialaj Ekavacioj 33(1990), 475-492.
[28] G. Dunne, Self-Dual Chern-Simons Theories, Lecture Notes in Physics, Vol. 36, Springer, Berlin, 1995.
[29] R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems, Nonlinear Analysis 39(2000), 559-568.
[30] R. Dalmasso, Existence and uniqueness of positive radial solutions for the Lane-Emden systems, Nonlinear Analysis 57(2004), 341-348.
[31] J. Dziarmaga, Low energy dynamics of [U(1)]N Chern-simons solitons, Phys. Rev. D 49(1994), 5469-5479.
[32] D.G. de Figueiredo and P.L. Felmer, A Liouville-Type for elliptic systems, Ann. Scuola Norm. Sup. Pisa, 21(1994), 387-397.
[33] P. L. Felmer, A. Quaas, M. Tang, and J. Yu, Monotonicity properties for ground states of the scalar field equation, Ann. I. H. Poincaré -AN, 25(2008), 105-119.
[34] Y. Furusho and K. Taka^si, Supersolution-subsolution method for nonlinear biharmonic equations in Rn, Czechoslovak Math. J. 47(122)(1997), 749-768.
[35] F. Gazzola and H.-C. Grunau Radial entire solutions for supercritical bi-harmonic equations, Math. Ann. 334(2006), 905-936.
[36] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in RN , Mathematical Analysis and Applica-tions, Part A, 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London(1981).
[37] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964 (2nd ed. Barkhauser, Boston Basel Stattgart, 1982).
[38] J. Hulshof and R. Van der Vorst, Differential systems with strongly indefinite variational structure, J. Funct. Anal. 114(1993), 32-58.
[39] J. Hong, Y. Kim, and P. Y. Pac, Multivortex solutions of the Abelian Chern-Simons-Higgs theory, Phys. Rev. Lett. 64(1990), 2230-2233.
[40] R. Jackiw and S.-Y. Pi, Soliton solutions to the gauged nonlinear Schrödinger equation on the plane, Phys. Rev. Lett. 64(1990), 2969-2972.
[41] R. Jackiw and E. J. Weinberg, Self-dual Chern-Simons vortices, Phys. Rev. Lett. 64(1990), 2234-2237.
[42] A. Jaffe and C. Taubes, Vortices and Monopoles, Progress in Physics Vol. 2, BirkhÄauser Boston. Mass., 1980.
[43] R. Johnson, X.-B. Pan, and Y.-F. Yi, Singular solutions of the elliptic equation ¢u − u + up = 0, Ann. Mat. Pura Appl. 166(1994), 203-225.
[44] M. K. Kwong, Uniqueness of positive solutions of ¢u − u + up = 0 in Rn , Arch. Rational Mech. Anal. 105(1989), 243-266.
[45] C. N. Kumar and A. Khare, Charged vortex of ¯nite energy in nonabelian gauge theories with Chren-Simons term, Phys. Lett. B 178(1986), 395-399.
[46] K. Taka^si and C. A. Swanson, Positive entire solutions of semilinear biharmonic equations, Hiroshima Math. J. 17(1987), 13-28.
[47] N. Kawano, E. Yanagida, and S. Yotsutani, Structure theorems for positive radial solutions to ¢u + K(jxj)up = 0 in Rn, Funkcial. Ekvac. 36(1993), 557-579.
[48] C. Kim, C. Lee, P. Ko, B. H. Lee, and H. Min, Schrödinger fields on the plane with [U(1)]N Chern-Simons interactions and generalized self-dual solitons, Phys. Rev. 48(1993), 1821-1840.
[49] P.-L. Lions, Isolated singularities in semilinear problems, J. Diff. Eqns. 38(1980), 441-450.
[50] C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in Rn, Comment. Math. Helv.73(1998), 206-231.
[51] C.-S. Lin, Uniqueness of solutions to the mean field equations for the spherical Onsager vortex, Arch. Rational Mech. Anal. 153(2000), 153-176.
[52] E. H. Lieb and J. P. Solovej, Ground state energy of the two-component charged Bose gas, Commun. Math. Phys. 252(2004), 485-534.
[53] Y. Liu, Y Li, and Y. Deng, Separation property of solutions for a semilinear elliptic equation, J. Diff. Eqns. 163(2000), 381-406.
[54] C.-S. Lin, A. C. Ponce, and Y. Yang, A system of elliptic equations arising in Chern-Simons ¯eld theory, J. Funct. Anal. 247(2007), 289-350.
[55] G. Lu, J. Wei, and X. Xu, On Conformally invariant equation (¡¢)pu − K(x)u N+2p N¡2p = 0 and its generalizations, Annali Di Math. Pura ed appl.(IV) CLXXIX(2001), 309-329.
[56] K. Mcleod, Uniqueness of positive radial solutions of ¢u + f(u) = 0 in Rn . II. Ameri. Math. Soc. 339(1993), 495-505.
[57] E. Mitidieri, A Rellich type identity and applications , Commun. Partial Diff. Eqns. 18 (1993), 125-151.
[58] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in RN . , Differential and Integral Equations 9 No.3 (1996), 465-479.
[59] H. Morishita, E. Yanagida, ans S. Yotsutani, Structure of positive radial solutions including all singular solutions to Matukuma's equation, Commun. Pure Appl. Anal. 4(2005), 871-888.
[60] W.-M. Ni, On the elliptic equation ¢u(x) + k(x)u (n+2) (n¡2) (x) = 0 its generalizations, and Applications in Geometry, Indiana Univ. Math. 31(1982), 41-56.
[61] M. Naito, A note on bounded positive entire solutions of semilinear elliptic equations, Hiroshima Math. J. 14(1984), 211-214.
[62] W.-M. Ni and J. Serrin, Nonexistence theorems for singular solutions of quasilinear partial differential equations, Comm. Pure Appl. Math., 39(1986), 379-399.
[63] E. S. Noussair, C. A. Swanson, and J. Yang, Transcritical biharmonic in Rn, Annali Di Math. Pura ed appl.(IV) 35(1992), 533-543.
[64] W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma-type and related topics, Japan J. Appl. Math. 5(1988), 1-32.
[65] L. Polvani and D. Dritschel, Wave and vortex dynamics on the surface of a sphere, J. Fluid Mech. 255(1993), 35-64.
[66] L.A. Peletier, and R. Van der Vorst, Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equation, J. Differential and Integral Equations 5(1992), 747-767.
[67] J. Qi and Y. Lu, The slowly decaying solutions of ¢u + f(u) = 0 in Rn , Funkcial. Ekvac. 41(1998), 317-326.
[68] B. Sirakov, Standing waves solutions of the nonlinear Schrödinger equation in RN , Ann. Mat. Pura Appl. 181(2002), 73-83.
[69] J. Spruck and Y. Yang, The existence non-topological solutions in the self-dual Chern-Simons theory, Comm. Math. Phys. 149(1992), 361-376.
[70] C. A. Swanson and L. S. Yu, Radial polyharmonic problems in Rn, J. Math. Anal. Appl. 174(1993), 461-466.
[71] J. Spruck and Y. Yang, Topological solutions in the self-dual Chren-Simons theory: existence and approximation, Ann. Inst. H. Poincaré Anal. Non Linéaire 12(1995), 75-97.
[72] J. Serrin and H. Zou, Nonexistence of positive solutions of Lane-Emden systems, Differential and Integral equations 9 No.4 (1996), 635-653.
[73] G. Tarantello, Uniqueness of selfdual periodic Chern-Simons vortices of topological type, Calc. Var. Partial Diff. Eqns 29(2007), 191-217.
[74] R. Van der Vorst, Variational identites and applications to differential systems , Arch. Rational Mech. Anal. 116(1991), 375-398.
[75] H. J. de Vega and F. A. Schaponsnilk, Electrically charged vortices in non-abelian gauge theories with Chren-Simons term, Phys. Rev. Lett. 56(1986), 2564-2566.
[76] E. Yanagida, Mini-maximizers for reaction-diffusion systems with skewgradient structure, J. Diff. Eqns 179(2002), 311-335.
[77] E. Yanagida, Structure of radial solutions to ¢u + K(jxj)jujp¡1u = 0 in Rn, SIAM J. Math.Anal. 27(1996), 997-1014.
[78] E. Yanagida, Uniqueness of positive radial solutions of ¢u + f(u, jxj) = 0, Nonlinear Anal. 19(1992), 1143-1154.
[79] Y. Yang, The relativistic non-abelian Chren-Simons equations, Comm. Math. Phys. 186(1997), 199-218.
[80] Y. Yang, Solitons in Filed Theory and Nonlinear Analysis, Springer Monographs in Mathematics, Springer-Verlag, New York, 2001.
[81] E. Yanagida and S. Yotsutani, Existence of positive radial solutions to ¢u+ K(jxj)up = 0 in Rn, J. Diff. Eqns 115(1995), 477-502.
[82] E. Yanagida and S. Yotsutani, Global structure of positive solutions to equations of Matukuma Type, Arch. Rational Mech. Anal. 134(1996), 199-226.
[83] Z. Y. Chen, J. L. Chern, Y. L. Tang, and J. Shi On the Uniqueness and Structure of Solutions for a Nonlinear Elliptic System via Linearization Approaches, preprint.
指導教授 陳建隆(Jann-long Chern) 審核日期 2008-7-1
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