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研究生:陳定邦
研究生(外文):Den-bon Chen
論文名稱:某半線性橢圓方程正解結構之分類
論文名稱(外文):Classification of the Structure of Positive Radial Solutions to some Semilinear Elliptic Equation
指導教授:羅春光羅春光引用關係
指導教授(外文):Chun-kong Law
學位類別:碩士
校院名稱:國立中山大學
系所名稱:應用數學系研究所
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
論文頁數:39
中文關鍵詞:橢圓方程
外文關鍵詞:Semilinear Elliptic equation
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我們將討論橢圓方程$Delta u+K(|x|)u^{p}=0 mbox{in} R^{n}$(其中p>1,n>2)正則徑向解的分類,已知此方程的任何徑向解必有零點或滿足快速遞降,或滿足緩慢遞降。在此論文中,我们將初值為alpha的解記做$u(r; alpha)$時,當$K(r)$滿足某些條件時,我們利用參數$r_{G}$跟$r_{H}$(被$K(r)$所決定),將方程式的徑向解結構可以分成下列三型:
Z 型:對任意初值$alphain(0,infty)$時,u在$(0,infty)$ 上有零點。
S 型:u對任意初值$alphain(0,infty)$時,u在$(0,infty)$ 上的振幅便緩慢遞降。
M 型:存在一個初值$alpha_{f}$使得當初值$alphain(alpha_{f},infty)$時,u在$(0,infty)$ 上有零點。當初值$alpha=alpha_{f}$時,u在$(0,infty)$ 上的振幅便緩慢遞降。當初值$alphain(0,alpha_{f})$時,u在$(0,infty)$ 上的振幅便快速遞降。以上乃日本教授Yanagida和Yotsutani好幾篇論文的工作,我在此論文做一個整理報告。
In this thesis, we shall give a concise account for the classification of the structure of positive radial solutions of the semilinear elliptic equation$$Delta u+K(|x|)u^{p}=0 .$$ It is known that a radial solution $u$ is crossing if $u$ has a zero in $(0, infty)$; $u$
is slowly decaying if $u$ is positive but $displaystylelim_{r
ightarrow{infty}}r^{n-2}u=infty$; u is rapidly decaying if $u$ is positive,
$displaystylelim_{r
ightarrow{infty}}r^{n-2}u$ exists and is positive. Using some Pohozaev identities, we show that under certain condition on $K$, by comparing some parameters $r_{G}$ and $r_{H}$, the structure of positive radial solutions for various initial conditions can be classified as Type Z ($u(r; alpha)$ is crossing for all $r>0$ ), Type S ($u(r; alpha)$ is slowly decaying for all $r>0$), and Type M (there is some $alpha_{f}$ such that
$u(r; alpha)$ is crossing for $alphain(alpha_{f},
infty)$, $u(r; alpha)$ is slowly decaying for
$alpha=alpha_{f}$, and $u(r; alpha)$ is rapidly decaying for $alphain(0, alpha_{f})$). The above work is due to Yanagida and Yotsutani.
1.Introduction
2.Properties of solutions
3.Kelvin Transformation
4.Proof of Theorem A
5.Proof of Key Proposition
[1] K.-S. Cheng and J.-L. Chern , Existence of positive
entire solutions of some semilinear elliptic equations, J.
Differential Equations, $mathbf{98}$ (1992), 169-180.

[2] N. Kawano , W.-M. Ni , and S. Yotsutani , A
generalized Pohozaev identity and its applications, J. Math. Soc.
Japan, $mathbf{42}$(1990), 541-564.

[3] N. Kawano , E. Yanagida , and S. Yotsutani , Structure
theorems for positive radial solutions to $Delta
u+K(|x|)u^{p}=0$ in $mathbf{R}^{n}$, Funkcial. Ekvac.
$mathbf{36}$ (1993), 557-579.

[4] W.-M. Ni and S. Yotsutani , Semilinear elliptic
equations of Matukuma-type and related topics , Japan J. Appl.
Math. , $mathbf{5}$(1988), 1-32.

[5] E. Yanagida and S. Yotsutani , Classifications of
the structure of positive radial solutions to $Delta
u+K(|x|)u^{p}=0$ in $mathbf{R}^{n}$, Arch. Rational Mech. Anal.,
$mathbf{124}$ (1993), 239-259.

[6] E. Yanagida and S. Yotsutani , Existence of
positive radial solutions to $Delta u+K(|x|)u^{p}=0$ in
$mathbf{R}^{n}$ , J. Differential Equations,
$mathbf{115}$(1995), 477-502.

[7] S. Yotsutani , Positive radial solutions to
nonlinear elliptic boundary value problems, Lecture
Notes(Gidas-Ni-Nirenberg), NCTS, (2000).
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