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研究生:葉秋萍
研究生(外文):Chiou-Ping Ye
論文名稱:P-拉普拉斯問題正解分枝曲線之研究
論文名稱(外文):On the Bifurcation Curve of Positive Solutions for a P-Laplacian Two Point Boundary Value Problem
指導教授:王信華
指導教授(外文):Shin-Hwa Wang
學位類別:碩士
校院名稱:國立清華大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2004
畢業學年度:92
語文別:英文
中文關鍵詞:P-拉普拉斯分枝曲線
外文關鍵詞:P-LaplacianBifurcation curve
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P-拉普拉斯問題正解分枝曲線之研究. 這份論文分成兩部分, 第一部分是對於 dead core solution 的討論, 第二部分是對於 bifurcation curve 的研究.
Part I:
Explicit Necessary and Sufficient Conditions for the Existence of Dead Core Solutions of a P-Laplacian Steady-State Reaction-Diffusion Problem

1. Introduction……………………… 2
2. Main Results……………………... 3
3. Lemmas………………………….. 4
4. Proofs of Main Results…………... 7
5. One-Dimensional Case…………... 9
6. Appendix………………………… 14
References………………………….. 13

Part II:
On the Bifurcation Curve of Positive Solutions for a P-Laplacian Two Point Boundary Value Problem

1. Introduction……………………… 2
2. Main Results……………………... 4
3. Lemmas………………………….. 9
4. Proofs of Main Results…………... 11
5. Appendix…………….…………... 25
References………………………….. 23
Part I:
1. C. Bandle, T. Nanbu and I. Stakgold, Porous medium equation with absorption. SIAM J. Math. Anal. 29 (1998), 1268--1278.
2. C. Bandle, R. P. Sperb and I. Stakgold, Diffusion and reaction with monotone kinetics. Nonlinear Analysis, TMA 8 (1984), 321--333.
3. C. Bandle and I. Stakgold, The formation of the dead core in parabolic reaction-diffusion problems, Trans. Amer. Math. Soc. 286 (1984), 275--293.
4. C. Bandle and S. Vernier-Piro, Estimates for solutions of quasilinear problems with dead cores, Z. Angew. Math. Phys. 54 (2003), 815--821.
5. J. I. Díaz, Nonlinear partial differential equations and free boundaries. Vol. I. Elliptic Equations. Research Notes in Mathematics, 106, Pitman, Boston, MA, 1985.
6. J. I. Díaz, and J. E. Saa, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 521--524.
7. A. Friedman and D. Phillips, The free boundary of a semilinear elliptic equation. Trans. Amer. Math. Soc. 282 (1984), 153--182.
8. I. S. Gradshtein, I. M. Ryzhik, and A. Jeffrey, Table of integrals, series, and products, Academic Press, Boston, 1994.
9. S. P. Hastings and J. B. McLeod, The number of solutions to an equation from catalysis, Proc. Roy. Soc. Edinburgh 101A (1985), 15--30.
10. T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J. 20 (1970), 1--13.
11. P. Pucci and J. Serrin, The strong maximum principle revisited, J. Differential Equations 196 (2004), 1--66.
Rainville : E. D. Rainville, Special functions, Macmillan, New York, 1960.
12. I. Stakgold, Reaction-diffusion problems in chemical engineering. Nonlinear Diffusion Problems, Lecture Notes in Math., 1224, 119--152, Springer, Berlin, 1986.
13. S.-H. Wang and F.-P. Lee, Bifurcation of an equation from catalysis theory, Nonlinear Analysis, TMA 23 (1994), 1167--1187.
14. R. L. Wheeden and A. Zygmund, Measure and integral: An introduction to real analysis, M. Dekker, New York, 1977.
15. S. Zhang and J. Jin, Computation of special functions, Wiley, New York, 1996.

Part II:

1. C. Bandle, R. P. Sperb, and I. Stakgold, Diffusion and reaction with monotone kinetics. Nonlinear Analysis, TMA 8 (1984), 321--333.
2. E. N. Dancer, On the structure of the solutions of an equation in catalysis theory when a parameter is large, J. Differential Equations 37 (1980), 404--437.
3. J. I. Díaz, Nonlinear partial differential equations and free boundaries. Vol. I. Elliptic Equations. Research Notes in Mathematics, 106, Pitman, Boston, MA, 1985.
4. S. P. Hastings and J. B. McLeod, The number of solutions to an equation from catalysis, Proc. Roy. Soc. Edinburgh 101A (1985), 15--30.
5. T. Laetsch, The number of solutions of a nonlinear two point boundary value problem. Indiana Univ. Math. J. 20 (1970), 1--13.
6. S.-Y. Lee, S.-H. Wang and C.-P. Ye, Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-Laplacian steady-state reaction-diffusion problem, preprint.
7. P. L. Lions, On the existence of positive solutions of semilinear elliptic equations. SIAM Review 24 (1982), 441--467.
8. H. O. Peitgen, D. Saupe, and K. Schmitt, Nonlinear elliptic boundary value problems versus their finite difference approximations: numerically irrelevant solutions. J. Reine Angew. Math. 322 (1981), 74--117.
9. P. Pucci and J. Serrin, The strong maximum principle revisited, J. Differential Equations 196 (2004), 1--66.
10. J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations 39 (1981), 269--290.
11. I. Stakgold, Reaction-diffusion problems in chemical engineering. Nonlinear Diffusion Problems, Lecture Notes in Math., 1224, 119--152, Springer, Berlin, 1986.
12. S.-H. Wang and F.-P. Lee, Bifurcation of an equation from catalysis theory, Nonlinear Analysis, TMA 23 (1994), 1167--1187.
13. S.-H. Wang and T.-S. Yeh, Exact multiplicity and ordering properties of positive solutions of a p-Laplacian Dirichlet problem and their applications, J. Math. Anal. Appl. 287 (2003), 380--39.
14. R. L. Wheeden and A. Zygmund, Measure and integral: An introduction to real analysi. M. Dekker, New York, 1977.
15. S. Zhang and J. Jin, Computation of special functions, Wiley, New York, 1996.
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