在環上混合型邊界條件的半線性方程

Abstract

此篇學位論文中，我們考慮這個方程式的正半徑對稱解的存在性 .DELTA. u+g(.lgvert.x.lgvert.)f(u)=0在R<.lgvert.x.lgvert.<.Rbar. x.in.歐 幾里得N-空間 N.gtoreq.2有下列的邊界條件集合之一個：u=0在.lgvert. x.lgvert.=R，和u=0在.lgvert.x.lgvert.=.Rbar.，u=0在 .lgvert.x. lgvert.=R，和u對半徑之微分等於零，在.lgvert.x.lgvert.= .Rbar.，u 對半徑之微分等於零，在.lgvert.x.lgvert.=R，和u=0在 .lgvert.x. lgvert.=.Rbar.。此篇論文，是由參考文獻[1]，[2]，和[3]所引發的。 我們使用射擊法及估計，證明了解的存在性，得到以下的結果：在給定區 間[a,b]，0<a<b<.inf.後，假設(A-1)'，(A-2)'，(A-3)'都成立，那麼我 們所要探討的方程式，都至少有一個正半徑對稱解。從內容中也可輕易看 出此篇也再次確証了一些已知的好結果。 In this thesis we consider the existence of the positive radial solutions of the equation.DELTA.u+g(.lgvert.x.lgvert.)f(u)=0 in R <.lgvert.x.lgvert.<.Rbar.x.in. Euclidean N-space,N.gtoreq.2, with one of following sets of boundary conditions: u=0 on .lgvert.x .lgvert.=R and u=0 on .lgvert.x.lgvert.=.Rbar., u=0 on .lgvert.x .lgvert.=R and the differentiation in the radial direction=0 on .lgvert.x.lgvert.=.Rbar., the differentiation in the radial dire- ction = 0 on .lgvert.x.lgvert.=R and u=0 on .lgvert.x.lgvert.= .Rbar.. This thesis was motivated by reference [1],[2],and[3]. We used shooting methods and estimation to prove the existence of t- he positive radial solutions, and got the following results: giv- en [a,b],0<a<b<. inf., assume(A-1)'(A-2)',and(A-3)'hold,then thses equations we consider have positeve radial solutions. It is easy to see that this thesis verifies some good results.