臺灣博碩士論文加值系統

討論在二維上之偏微方程式: .DELTA.u + K(x)exp(2u) = 0 。當 K(X)
是球對稱, 且欲求之解亦為球對稱時, 可改寫上逑偏微分程式為常微方程
式: u''(r) + 1/r u'(r) + K(r) exp (2u) = 0 。此時我們要討論是當
K(r) > 0 時, 微方程式有下列漸近狀態之奇異解: u(r) = a log(r) +
.alpha. + o(1) as r .arrr. 0 u(r) = b log(r) + o(log(r)) as r
.arrr. .ing.其中 b 與 a 和 .aopha. 之間的關係。在本文中我們給幾
個例子是可以把常微方程式的確實解表示出來, 並從這些例子求出 b 與
a 和 .alpha. 之間的關係。再則特別地令 a=0, 並給一些數值圖, 以了
解 b 與 .alpha. 之間的關係。

In this paper we investigate the conformal Gaussian curvature
equation (P.D.E.) : .DELTA.u + K(x)exp(2u) = 0 in .R^2. When K
is radially symmetric and radial solutions u are seeked, above
equation can be reduced to an ordinary differential (O.D.E.) :
u''(r) + 1/r u'(r) + K(r) exp(2u) =0, r>0. In theis paper we
are interested in the singular solutions u of (O.D.E.) having
following asymptotic behaviors: u(r) = a log(r) + .alpha. +
o(1) as r .arrr. 0 u(r) = b log(r) + o(log(r)) as r .arrr.
.inf. We shall give some examples which the exact solutions can
be written down explicitly. These examples indicate us theof b
upon a and .alpha. . And we shall plot numberically the
dependence of b on .alpha. for a=0.17