Uniqueness of global positive solution branches of nonlinear elliptic problems
Uniqueness of global positive solution branches of nonlinear elliptic problems
复制标题
非线性椭圆问题全局正解分支的唯一性
DOI：
10.1007/bf01450485
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发表时间：
1994
期刊：
影响因子：
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通讯作者：
H. Kielhöfer
中科院分区：
文献类型：
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作者：
M. Holzmann;H. Kielhöfer
(1.1)Au+2f(U)=0 in OcIR n，2Ep-.，u=0 on of 2，u>；0 inf2已在许多文献中被研究过(第一个全局描述在[19]中)。尽管如此，仍然存在一些悬而未决的问题，其中一些问题已在本文件中得到解答。我们现在提出其中的一些问题：对于一类合适的(光滑)函数，f：IR+~IR问题(1.1)对于2=1有无穷多个解。这在[15]中通过上下解的方法显示出来，它还给出了这些解的顺序：0<；ul<；u2<；……此外，如果f(0)-0和f(0)>；0(不排除在[15]中)，则解(2，u)的全局分支(=连续)C+从平凡解线{(2，0)}在2=21处分叉，其中#1=21 f‘(0)是线性化的主特征值
(1.1) Au+ 2f (u)= 0 in OcIR n, 2EP-., u= 0 on Of 2, u> 0 inf2, has been studied in many publications (the first global description was given in [19]). Nonetheless there were open questions some of which are answered in this paper. We motivate some of these questions now: For a suitable class of (smooth) fimctions f: IR+~ IR problem (1.1) has infinitely many solutions for 2= 1. This was shown in [15] via the method of sub-and supersolutions which also gives an order of these solutions: 0< ul< u2<.... If moreover f (0)--0 and f (0)> 0 (which is not excluded in [15]) there is also a global branch (= continuum) C+ of solutions (2, u) bifurcating from the trivial solution line {(2, 0)} at 2= 21 where# 1= 21 f'(0) is the principal eigenvalue of the linearization