研究目的：在薛丁格方程式中，對直立波型態解的分類。並從這樣的分類當中，了解直立波的解會有哪幾種不同的行為。
資料來源：
1、中央大學數學系圖書館期刊室。
2、清華大學數學系圖書館期刊室。
3、MathScinet電子期刊查詢系統。
4、Google搜尋引擎。
研究方法：
研究方法主要是用到以下四種數學工具：
1、 比較定理可用來幫助排除O*及*S解的情形。
2、 可以用來證明靠近原點，解具備開集合的性質
3、 可以讓我們了解，O*這種解的震盪，本身必定有一個上界。
並且可以由這樣子的輔助函數，讓我們觀察到解在無窮遠處，具備有開集合的性質。
4、線性化方程，可以讓我們決定，將兩個開集合隔開的那條平滑曲線，起初從原點出來的行為是如何的。
研究結果：
我們可以得到，當$1<p<frac{n+2}{n-2}$時，所有直立波的解節構就都清礎了。因為在這個情況之下，我們用到一個非常強烈的結果，就是R-G的解是唯一的。並且$S_1$具備開集合的性質。而當$pgeqfrac{n+2}{n-2}$時，我們只能了解$p>p_c$時，解的分佈情形。但$frac{n+2}{n-2}leq pleq p_c$ 的情況下，還無法完全掌握$Gamm_1$這條曲線的行為。

This paper is concerned with the structure of the set of positive radially symmetric solutions for the equation.
$Delta u-u+u^p=0$ on $R^n-{0}$ (1.1)
with $n>2$. Then any radial solution $u=u(r)=u(|x|)$ of the equation is shown to be classified into one of several types according to its behavior as $r
ightarrow 0$ or $r
ightarrowinfty$. Under the assumption that $1<p<frac{n+2}{n-2}$ or $pgeqfrac{n+2}{n-2}$, we clarify the entire structure for the set of solutions of various types. The Pohozaev identity and energy play a crucial role in the investigation of the structure.
We consider equation (1.1) because it came from the following nonlinear Schr"{o}dinger equation
$$iPhi_t=-DeltaPhi-|Phi|^{p-1}Phi$$ where $Phi:R imes R^n
ightarrow C$
. Looking for the standing wave solutions, that is
$Phi(t,x)=e^{it}u(x)$, one is let to the problem:
$$Delta u(x)-u(x)+|u(x)|^{p-1}u(x)=0$$ in $R^{n}-{0}$$(1.2)
Since we consider positive solution, (1.2) becomes (1.1).

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