Imperfect bifurcations in nonlinearelliptic equations on spherical caps
Imperfect bifurcations in nonlinearelliptic equations on spherical caps
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球冠非线性椭圆方程中的不完美分岔
DOI：
10.3934/cpaa.2010.9.1189
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发表时间：
2010
影响因子：
1
通讯作者：
H. Ninomiya
中科院分区：
文献类型：
--
作者：
C. Bandle;Yoshitsugu Kabeya;H. Ninomiya
本文考虑大球面上的椭圆边值问题，
电容具有参数依赖的功率非线性。本文证明
这种不完美的分叉发生在工作中[13]。当域是
在整个球体中，有一个恒定的解。如果域是一个
球冠，但是，常数解消失，由于边界
条件对于大的球冠，我们构造接近于
在整个n维球面上的常数解，使用
在全球和不动点参数的基础上的线性化问题
Lyapunov-Schmidt型还原从数字上看，
在这个问题的图表和[18]中获得的图表之间，也是[5]，
关于球冠的Brezis-Nirenberg型问题。
We consider elliptic boundary value problems on large spherical
caps with parameter dependent power nonlinearities. In this paper we show
that imperfect bifurcation occurs as in the work [13]. When the domain is the
whole sphere, there is a constant solution. In the case where the domain is a
spherical cap, however, the constant solution disappears due to the boundary
condition. For large spherical caps we construct solutions which are close to the
constant solution in the whole n-dimensional sphere, using the eigenvalues of
the linearized problem in the whole sphere and fixed point arguments based on a
Lyapunov-Schmidt type reduction. Numerically there is a surprising similarity
between the diagrams of this problem and the ones obtained in [18], also [5],
for a Brezis-Nirenberg type problem on spherical caps.