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 系統識別號 U0002-0506201310231100 中文論文名稱 一些橢圓偏微分方程解的存在性與唯一性之探討 英文論文名稱 On the Existence and Uniqueness of Solutions for Some Elliptic Partial Differential Equations 校院名稱 淡江大學 系所名稱(中) 數學學系博士班 系所名稱(英) Department of Mathematics 學年度 101 學期 2 出版年 102 研究生中文姓名 陳宜榮 研究生英文姓名 Yi-Jung Chen 學號 894150068 學位類別 博士 語文別 英文 口試日期 2013-05-21 論文頁數 66頁 口試委員 指導教授-錢傳仁委員-錢傳仁委員-郭滄海委員-陳建隆委員-郭忠勝委員-張茂盛委員-陳功宇 中文關鍵字 擬線性橢圓方程  Hardy-Sobolev方程  Dirichlet 條件  Robin條件  奇異解 英文關鍵字 Quasilinear elliptic equation  Hardy-Sobolev equation  Dirichlet condition  Robin condition  singular solution 學科別分類 中文摘要 在本論文中我們將探討擬線性橢圓方程在Dirichlet 和 Robin 條件下，解的存在性問題；更進一步，我們在Robin 條件下得到解的唯一性。同時，我們對Hardy-Sobolev方程作了徑對稱解的研究，得到在上臨界的條件下最多僅有一個奇異解。 英文摘要 In this thesis, we study the existence of solutions to quasilinear elliptic equations with Dirichlet and Robin conditions, and the uniqueness of solutions under Robin conditions. Also, we investigate the radial symmetric solutions for a Hardy-Sobolev equation and derive the result that there exists at most one positive radial symmetric singular solution for the supercritical case. 論文目次 Contents Chapter 1 Introduction………………………………………………1 Chapter 2 Quasilinear Elliptic Equations with Dirichlet Conditions………………………………………………………………5 2.1 The Maximum Principl……………………………………………9 2.2 The Estimate……………………………………………………13 2.3 The Existence of Strong Solutions…………………………15 Chapter 3 Quasilinear Elliptic Equations with Robin Conditions ……………………………………………………………31 3.1 The Comparison Principle ……………………………………34 3.2 and Estimates………………………………………………38 3.3 The Existence and Uniqueness of Solutions in ………41 Chapter 4 Hardy-Sobolev Equations………………………………45 4.1 Asymptotic Behavior s…………………………………………46 4.2 The Uniqueness of Positive Radial Singular Solutions………………………………………………………………56 Appendix A Basic Notations and Fixed Point Theorems………60 A.1 Basic Notations…………………………………………………60 A.2 Fixed Point Theorems……………………………………… …62 Bibliography …………………………………………………………63 參考文獻  L. Boccardo, F. Murat and J. P. Puel, Existence de solutions faibles pour des equations elliptiques quasi-lineaires a croissance quadratique, in: J. L. Lions, H. Brezis (Eds.), it Nonlinear Partial Differential Equations and Their Applications, in: it College de France Seminar, Vol. IV, Research Notes in Math., vol. 84, Pitman, London, 1983, pp. 19-73.  L. Boccardo, F. Murat and J. P. Puel, Resultats d'existence pour certains problemes elliptiques quasi-lineaires, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 11(1984), 213-235.  J.-M. Bony, P. Courrege et P. Priouret, Semi-groupes de Feller sur une variete a bord compacte et problemes aux limites integro-differentiels du second ordre donnant lieu au principe du maximum, it Ann. Inst. Fourier (Grenoble) 18(1968), 369-521.  H. Brezis, Analyse Fonctionnelle Theorie et Applications, Masson, Paris (1983).  H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36(1983), 437-477.  F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequality: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math. 54(2001), 229-258.  J.-L. Chern and C.-S. Lin, Minimizers of Caffarelli-Kohn-Nirenberg inequalities on domains with the singularity on the boundary, Arch. Ration. Mech. Anal. 197(2010), 401-432.  J.-L. Chern, Y.-L. Tang, C.-J. Chyan and Y.-J. Chen, On the Uniqueness of Singular Solutions for a Hardy-Sobolev Equation, accepted for publication in Discrete and Continuous Dynamical Systems, Supplement 2013.  M. Delgado and A. Suarez, Weak solutions for some quasilinear elliptic equations by the sub-supersolution method, Nonlinear Analysis 42(2000), 995-1002.  A. Friedman, Partial differential equations, Holt, Rinehart and Winston(1969).  N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincar?Anal. Non Linaire 21(2004), 767-793.  D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, second edition, Springer-Verlag, New York (1983).  C.-H. Hsia, C.-S. Lin and H. Wadade, Revisiting an idea of Brezis and Nirenberg, J. Funct. Anal. 259(2010), 1816-1849.  T.-H. Kuo, Estimates on solutions to certain quasilinear equations in divergence form, Taiwanese J. Math. 9, No.2 (2005), 237-243.  T.-H. Kuo and Y.-J. Chen, Existence of strong solutions to some quasilinear elliptic problems on bounded smooth domains, Taiwanese J. Math. 6, No.2 (2002), 187-204.  T.-H. Kuo and Y.-J. Chen, The existence of solutions to certain quasilinear elliptic equations, Nonlinear Analysis 74, Issue 4 (2011), 1286-1289.  O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear elliptic equations, Academic Press, New York (1968).  C.-S. Lin and Z.-Q. Wang, Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities, Proc. Amer. Math. Soc. 132(2004), 1685-1691.  K. Taire, Diffusion processes and partial differential equations, Academic Press, San Diego New York London Tokyo(1988).  K. Taire, On the existence of Feller semigroups with boundary conditions, Memoirs Amer. Math. Soc., Vol. 99, No. 475(1992).  K. Taira, Analytic semigroups and semilinear initial boundary value problems, London Mathematical Society Lecture Note Series 223, Cambridge Univ. Press(1995).  K. Taira, Boundary value problems for elliptic integro-differential operators, Math. Z. 222(1996), 305-327.  K. Taira, Existence and uniqueness theorems for semilinear elliptic boundary value problems, Adv. Differential Equation 2(1997), 509-534.  K. Taira, D. K. Palagachev and P. R. Popivanov, A degenerate Neumann problem for quasilinear elliptic equations, Tokyo J. Math. Vol. 23, No. 1(2000), 227-234.  C.-C. Tsai and T.-H. Kuo, {it On the existence of solutions to some quasilinear elliptic problems}, Ph. D. 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