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).