The Quantitative Analysis of Singular Solutions for Semilinear Elliptic Equations with Nonlinear Critical and Supercritical Potential

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姓名

陳志和(Chih-Her Chen) 查詢紙本館藏

畢業系所

數學系

論文名稱

(The Quantitative Analysis of Singular Solutions for Semilinear Elliptic Equations with Nonlinear Critical and Supercritical Potential)

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摘要(中)

在第一部份，我們首先考慮具備哈帝位勢及臨界非線性項的橢圓
方程，並且在位勢項做了相當一般性的假設，探討其奇異解的節
構。一般而言，我們可以證明存在唯一一個特殊奇異解及無窮多個
奇異解在此特殊奇界解附近震盪。我們也學習這些圍繞在此唯一奇
異解的特殊解的極限行為。我們的結果可以應用在不同的問題上
面，例如純量場方程、細胞重製模型以及卡法芮利-科恩-尼倫柏格不
等式。另外，我們也個別討論了三個橢圓方程，並依據每個方程的
特徵考慮其在超臨界的情況下解在無窮遠處的行為或分類所有解節
構。在第二部份，我們證明了來自於乘積阿貝爾規範場論的橢圓系
統其非拓樸解的存在性。

摘要(英)

For the first part, we consider the structure of singular solutions for
elliptic equations with the Hardy potential and critical nonlinearity under
quite general conditions on the potential terms. In general, it is shown that
there exists a unique special singular solution, and other infinitely many
singular solutions are oscillatory around the special singular solution. We
also study the asymptotic behavior of the solutions around the singular
point. Our results can be applied to various problems such as the
scalar field equation, a self-replication model and the Cafarelli-Kohn-
Nirenberg inequality. In particular, we discuss the three elliptic equations
separately and to consider the asymptotic behavior of the solutions at
infinity under supercritical case or classify all the solutions structure
according to the characteristic of each equation.
For the second part, we prove the existence of Non-Topological solutions
for the elliptic system arising from a product Abelian Gauge Field theory.

關鍵字(中)

★ 半線性橢圓方程
★ 非線性臨界
★ 超線性位勢
★ 奇異解
★ 定量分析

關鍵字(英)

★ Semilinear Elliptic Equations
★ Nonlinear Critical
★ Supercritical Potential
★ Singular solutions
★ The Quantitative Analysis

論文目次

Part I
On the study of singular solutions for semilinear elliptic equations with nonlinear critical and supercritical potential
1 Classication of singular solutions of Nonlinear Schrodinger Equation with potential term 1
1.1 Introduction . . . . . . . . . . . . . . . . . . 1
1.2 Fundamental properties of solutions . . . . . . . 5
1.3 Existence of Singular solutions . . . . . . . . . 9
2 Classication of radial solutions for the minimizer of Cafarelli-Kohn-Nirenberg inequality 17
2.1 Introduction . . . . . . . . 17
2.2 Existence and uniqueness of special singular solutions . . . . . . . . . . . . . . 22
2.3 Solutions structure of CKN . . . . . . . . . . . 23
2.4 Neumann Problem for CKN inequality . . . . . . . 27
3 A PDE for self-replication model 29
3.1 Introduction . . . . . . . . . . . . . . . . . . 29
3.2 Fundamental properties and Nonexistence of Solutions . . . . . . . . . . . . . . 33
3.3 Neumann problem for self-replication model . . . .39
3.4 Existence of Oscillatory Singular Solutions . . . 40
3.5 Existence and Uniqueness of Special Singular Solution . . . . . . . . . . . . . . 43
3.6 Structures of the set of Singular Solutions . . . 47
3.7 Uniqueness of Singular Solutions for the supercritical case . . . . . . . . . . . . 47
3.8 Layer Property Of Regular Solutions . . . . . . .52
3.9 Existence of special singular solutions under supercritical case . . . . . . . . . . 56

Part II
The family of non-topological solutions for the elliptic system arising from a product Abelian gauge feld theory

4 The Existence of Non-Topological Solutions for The Elliptic System Arising from a Product Abelian Gauge Field theory 58
4.1 Introduction . . . . . . . . . . . . . . . . . . 58
4.2 Reduction to a single equation . . . . . . . . . 63
4.3 Solution sets with bN 1 . . . . . . . .. . . . 68
4.4 Approximation of non-topological solutions .. . 75
Bibliography -------------------------------------83

參考文獻

[1] Weiwei Ao, Chang-Shou Lin and Juncheng Wei: On non-topological solutions of the A2
and B2 Chern-Simons system, Mem. Amer. Math. Soc. 239 (2016), no. 1132
[2] Fursikov, A. V., Imanuvilov, O. Yu.: Local Exact Boundary Controllability of the Boussinesq
Equation. SIAM Journal of Control and Optimization 36, 391-421 (1998).
[3] H.Berestycki, P.-L.Lions. Nonlinear scalar eld equations. I. Existence of a ground state.
Arch.Ration.Mech.Anal(1983). 82, pp313-345
[4] H.Berestycki and P.-L. Lions; Nonlinear scalar eld equations. I. Existence of a ground
state. Arch. Rational Mech. Anal. 82 (1983),313-345, 379-386
[5] LUIS A. CAFFARELLI , BASILIS GIDAS and JOEL SPRUCK, Asymptotic symmetry
and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure
Appl. Math. 42 (1989), 271{297.
[6] Chih-Her Chern, Jann-Long Chern, Eiji Yanagida Structure of Singular Solutions for Nonlinear
Elliptic Equations with the Critical Potential. Preprint
[7] Chiun-Chuan Chen and Theodore Kolokolnikov Simple PDE model of spot replication in
any dimension SIAM J. MATH. ANAL. Vol44, No5, (2012), pp. 3364-3593.
[8] C.-C. Chen and C.-S. Lin, On the asymptotic symmetry of singular solutions of the scalar
curvature equations, Math. Ann. 313 (1999), 229{245.
[9] Chiun-Chuan Chen, Chih-Chiang Huang, Theodore Kolokolnikov Critical exponent of a
simple model of spot replication J.Dierential Equations 263 (2017) 5507-5520
[10] Chih-Her Chen, Gyeongha Hwang and Jann-Long Chern Classication of radial solutions
for the minimizer of the best constant in Cafarelli-Kohn-Nirenberg inequality. preprint
[11] Wenxiong,Chen, Congming,Li.,Wenxiong: Qualitative properties of solutions to some nonlinear
elliptic equations in R2, Duke Math. J. 71 427-439 (1993).
[12] Dongho Chae, Oleg Yu. Imanuvilov: The existence of non-topological multivortex solutions
in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys. 215 (2000) 119-142.
[13] Dongho Chae: On the elliptic system arising from a self-gravitating Born-Infeld Abelian
Higgs theory, Nonlinearity 18 (2005) 1823-1833.
[14] Jann-Long Chern, Chang-Shou Lin; Minimizers of Caarelli-Kohn-Nirenberg Inequalities
with the Singularity on the Boundary Arch.Rational Mech.Anal.197(2010)401-432 Digital
Object Identier(DOI) 10.1007/s00205-009-0269-y
[15] Janne-Long Chern and Eiji Yanagida; Singular Solutions of the Scalar Field Equation
with a Critical Exponent Geometric Properties for Parabolic and Elliptic PDE′s, Springer
Proceedings in Mathematics and Statistics 176 (2016), pp277-288
[16] Janne-Long Chern, Zhi-You Chern, Jhih-He Chern, Yong-Li Tang On the Classication of
Standing Wave Solutions for the Schrodinger equation Communications in Partial Dieren-
tial Equation (2010), pp275-301
[17] Jann-Long Chern, Eiji Yanagida Structure of the sets of regular and singular radial solutions
for a semilinear elliptic equation. J.Dierential Equations 224(2006) 440-463
[18] J.-L. Chern, Z.-Y. Chern and Y.-L. Tang, Uniqueness of nite total curvatures and the
structure of radial solutions for nonlinear elliptic equations, Transactions of the American
Mathematical Society 363 (2011), 3211-3231
[19] K.S.CHOU and C.W.CHU On the best constant for a weighted sobolev-Hardy inequality
J-London Math.Soc. (2)48(1993), no,1,137-151, http://dx.doi.,org/jlms/s2-48.1.137
MR1223899(94h:46052) Arch.Rational.Mech.Anal 105 (1989), pp243-266
[20] Kuo-Shung Cheng, Chang-ShouLin: On the conformal Gaussian curvature equation in R2,
J. Di. Eqns 146, 226-250 (1998).
[21] Kwangseok Choe. (2005): Uniqueness of the topological multivortex solution in the selfdual
in the Chern-Simons Theorem, J. Math. Phys, 46:012305.
[22] Kwangseok Choe, Namkwon Kim and Chang-Shou Lin: Existence of self-dual nontopological
solutions in the Chern-Simons Higgs model, Ann.Inst. H. Poincare Anal. Non
Lineaire 28 (2011) 837-852.
[23] Kwangseok Choe, Namkwon Kim and Chang-Shou Lin: Self-dual symmetric nontopological
solutions in the SUp3q model in R2, Commun. Math. Phys. 33 (2015), 1-37.
[24] Kwangseok Choe, Namkwon Kim and Chang-Shou Lin.: Existence of mixed type solutions
in the SUp3q Chern-Simons theory in R2, Calc. Var. Partial Dierential Equations 56 (2017)
no. 2, 56:17.
[25] Kwangseok Choe, Namkwon Kim and Chang-Shou Lin: Existence of mixed type solutions
in the Chern-Simons gauge theory of rank two in R2, Journal of Functional Analysis 273
(2017) 1734-1761.
[26] Arjen Doelman,Tasso J.Kaper,and Lambertus A.Peletier. Homoclinic furcations at the
onset of pulse replication, J.Dierential Equations, 231(2006),pp359-423
[27] Francois Genoud and Charles A.Stuart Schrodinger equations with a spatially decaying
nonlinearity:existence and stability of standing waves Ecole Polytechnique Federale deLau-
sanne CH-1015 Lausanne, Switzerland
[28] B.Gidas, Wei-Ming Ni, Nirenberg.L. Symmetry of Positive Solutions of Nonlinear Elliptic
Equations in Rn MATHEMATICAL ANALYSIS AND APPLICATIONS, PART A AD-
VANCES IN MATHEMATICS SUPPLEMENTARY STUDIES, VOL. 7A
[29] Chun-Hsiung Hsia,Chang-Shou Lin, ZHI-Qiang Wang. Asymptotic Symmetry and local
Behaviors of Solutions to a class of Anisotropic Elliptic Equations. Indiana University Math-
ematics Journal, Vol.60, No.5(2011)
[30] Genggeng Huang and Chang-Shou Lin: The existence of non-topological solutions for a
skew-symmetric Chern-Simons system, Indiana Univ. Math. J. 65 (2016), no. 2, 453-491.
[31] M.K.Kwong Uniqueness of positive solutions of u u ? up 0 in Rn.
Arch.Rational.Mech.Anal 105 (1989), pp243-266
[32] C.-S. Lin and J. V. Prajapat, Asymptotic symmetry of singular solutions of semilinear
elliptic equations, J. Dierential Equations 245 (2008), 2534{2550.
[33] C.-S. Lin Interpolation inequalities with weighs Comm.Partial Di. Equ. 11(14) 1515-
1538(1986)
[34] Caarelli,L., Kohn, R.,Nirenberg.L: First order interpolation inequalities with weighs
Composit.Math.53(3), 259-275 (1984)
[35] C.MURATOV AND V.V.OSIPOV Static spike autosolitons in the Gray-Scott model
J.Phys.A, 33(2000), pp.8893-8916
[36] Johnson,R.,Pan, X.,Yi, Y. Singular solutions of the elliptic equation u u ? up 0
Ann.Mat.Pura Appl. 166(4) (1994), pp203-225
[37] John E. Pearson Complex patterns in a simple system, Science,216(1993), pp189-192
[38] M. Hoshino and E. Yanagida, Convergence rate to singular steady states in a semilinear
parabolic equation, Nonlinear Anal. 131 (2016), 98{111.
[39] Nirenberg, L.: Topics in Nonlinear Analysis, Courant Lecture Notes in Math. 6, American
Mathematical Society, 2001.
[40] W.-M. Ni,Serrin,J. Nonexistence theorems for singular solutions of quasilinear partial differential
equations. Comm.Pure Appl.Math.39 (1986), pp379-399
[41] P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence
and Steady States, Birkhauser Advanced Texts, Birkhauser, Basel, 2007.
[42] S.EI,Y.NISHIURA,AND K.UEDA 2n splitting or edge eplitting?:A manner of splitting in
dissipative systems, Japan.J.Indust. Appl.Math., 18(2001),pp181-205
[43] S. Sato and E. Yanagida, Asymptotic behavior of singular solutions of a semilinear
parabolic equation, Discrete Contin. Dyn. Syst. 32 (2012), 4027{4043.
[44] Sze-Guang Yang.;Zhi-You Chen;Chih-Her Chen ; Jann-Long Chern : The Non-Topological
Solutions for The Elliptic System Arising from A Product Abelian Gauge Field Theory, In
the reviewing at Journal Nonlinearity.
[45] Tong, D., Wong, K.: J. High Energy Phys. 1401 (2014) 090.
[46] T.KOLOKOLNIKOV,M.J.WARD,AND J.WEI, The existence and stability of spike
equilibria in the one-dimensional Gray-Scott model: The pulse-splitting regime Phys.
D,202(2005), pp258-293
[47] W.-Y.Ding, W.-M.Ni. On the existence of positive entire solutions of a semilinear elliptic
equation Arch.Ration.Mech.Anal.91(1)(1986)283-308
[48] Y.NISHIURA AND D.UEYAMA A skeleton structure of self-replicating dynamics
Phys.D,130(1999), pp73-104
[49] Yang, Y.: Cosmic strings in a product Abelian gauge eld theory, Nucl. Phys. B 885 (2014)
25-33.
[50] Yang, Y.: Prescribing zeros and poles on a compact Riemann surface for a gravitationally
coupled Abelian gauge eld theory, Comm. Math. Phys. 249, 579-609 (2004).
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指導教授

陳建隆(Jann-Long Chern)

審核日期

2018-7-10

推文