(34.239.150.57) 您好!臺灣時間:2021/04/14 21:39
字體大小: 字級放大   字級縮小   預設字形  
回查詢結果

詳目顯示:::

我願授權國圖
: 
twitterline
研究生:蕭鈞元
研究生(外文):Jiunn-Yean Shiao
論文名稱:某半線性橢圓方程的徑向解結構及其Pohozaev恆等式
論文名稱(外文):The Structure of Radial Solutions to a Semilinear Elliptic Equation and A Pohozaev Identity
指導教授:羅春光羅春光引用關係
指導教授(外文):C.-K. 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:2003
畢業學年度:91
語文別:英文
論文頁數:34
中文關鍵詞:橢圓方程徑向解
外文關鍵詞:semilinear elliptic equationPohozaev identity
相關次數:
  • 被引用被引用:0
  • 點閱點閱:84
  • 評分評分:系統版面圖檔系統版面圖檔系統版面圖檔系統版面圖檔系統版面圖檔
  • 下載下載:4
  • 收藏至我的研究室書目清單書目收藏:0
本文學習橢圓方程$Delta u+K(|x|)|u|^{p-1}u=0 ,xinmathbf{R}^{n}$ ,其中p>1,n>2,K(r) 是 上的平滑恆正函數。如多數人所了解的,此方程的徑向解可能振盪激烈,或者滿足 (快速遞降),或者滿足 (緩慢遞降)。在此論文中,我們將初值為 的解記做 ,並將之分成下列三個類型:
R(i) 型:u 在$(0,infty)$ 上經i個零點後其振幅便快速遞降。
S(i) 型:u 在$(0,infty)$ 上經i個零點後其振幅便緩慢遞降。
O 型:u 在$(0,infty)$ 上具有無窮多個零點。
當 滿足某些條件時,方程的徑向解結構是完全被確知的。特別地,此時存在著一系列的初始值 使得 是R(i) 型的解,而對所有$al in (al_{i-1},al_{i})$ , 是S(i) 型的解,其中 。這些工作主要是Yanagida 和Yotsutani 所完成的。他們的主要工具是Kelvin轉換、Prüfer轉換、和一個Pohozaev恆等式。這裡我做了一些整理。此外,我引入了一個稱為 的觀念,並對Pohozaev恆等式給出了兩個証明。
The elliptic equation $Delta u+K(|x|)|u|^{p-1}u=0,xin
mathbf{R}^{n}$ is studied, where $p>1$, $n>2$, $K(r)$ is
smooth and positive on $(0,infty)$, and $rK(r)in L^{1}(0,1)$. It
is known that the radial solution either oscillates infinitely, or
$lim_{r
ightarrow
infty}r^{n-2}u(r;al) in Rsetminus
{0}$ (rapidly decaying), or $lim_{r
ightarrow infty}r^{n-2}u(r;al) = infty (or
-infty)$ (slowly decaying). Let $u=u(r;al)$ is a solution
satisfying $u(0)=al$. In this thesis, we classify all the
radial solutions into three types:
Type R($i$): $u$ has exactly $i$ zeros on $(0,infty)$, and is
rapidly decaying at $r=infty$.
Type S($i$): $u$ has exactly $i$ zeros on $(0,infty)$, and is
slowly decaying at $r=infty$.
Type O: $u$ has infinitely many zeros on $(0,infty)$.

If $rK_{r}(r)/K(r)$ satisfies some conditions, then the structure
of radial solutions is determined completely. In particular, there
exists $0<al_{0}<al_{1}<al_{2}<cdots<infty$ such that
$u(r;al_{i})$ is of Type R($i$), and $u(r;al)$ is of Type S($i$)
for all $al in (al_{i-1},al_{i})$, where $al_{-1}:=0$. These
works are due to Yanagida and Yotsutani. Their main tools are
Kelvin transformation, Pr"{u}fer transformation, and a Pohozaev
identity. Here we give a concise account. Also, I impose a
concept so called $r-mu graph$, and give two proofs of the
Pohozaev identity.
1 Introduction 2
2 Notations and Some Basic Propositions 6
3 Prufer Transformation 13
3.1 Lemmas for Theorem 1 13
3.2 Lemmas for Theorem 2 16
4 Proofs of Theorems 25
5 Appendix 29
5.1 Proof of Proposition 2.1 29
5.2 Proof of Eq.(3.5) 32
[1] G. Birkhoff and G. C. Rota , Ordinary
Differential Equations, 4th ed. New York: Wiley, 1959.

[2] K.-S. Cheng and J.-L. Chern , Existence of positive
entire solutions of some semilinear elliptic equations, J. Diff.
Eqns , $mathbf{98}$ (1992), 169-180.

[3] E. A. Coddington and N. Levinson , Theory of Ordinary
Differential Equations, New York: McGraw-Hill, 1955.

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

[5] 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.

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

[7] E. Yanagida , Structure of radial solutions to $Delta u+K(|x|)|u|^{p-1}u=0 $in$mathbf{R}^{n}$, SIAM J. Math. Anal., $mathbf{27}$ (1996),
997-1014.

[8] 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.

[9] E. Yanagida and S. Yotsutani , Existence of nodal fast-decay solutions to
$Delta u+K(|x|)|u|^{p-1}u=0$ in $mathbf{R}^{n}$, Nonlinear
Anal., $mathbf{22}$ (1994), 1005-1015.

[10] S. Yotsutani , Positive radial solutions to
nonlinear elliptic boundary value problems, Lecture Note, NCTS,
(2000).
QRCODE
 
 
 
 
 
                                                                                                                                                                                                                                                                                                                                                                                                               
第一頁 上一頁 下一頁 最後一頁 top
系統版面圖檔 系統版面圖檔