Secondary bifurcations in semilinear ordinary differential equations
Secondary bifurcations in semilinear ordinary differential equations
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半线性常微分方程的二次分岔
DOI：
10.1007/s42985-022-00180-5
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
菅徹
中科院分区：
文献类型：
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作者：
Y. Satake;M. Oozawa;T. Sogabe;Y. Miyatake;T. Kemmochi;S.-L. Zhang;剱持智哉;剱持智哉;剱持智哉;剱持智哉;剱持智哉;剱持智哉;Tomoya Kemmochi;Tomoya Kemmochi;剱持智哉;剱持智哉;剱持智哉;剱持智哉;剱持智哉;菅徹
我们考虑该方程在打孔区间上的Neumann问题，其中有一个分支参数和。在，我们对常数施加条件u（-0）+aux（-0）=u（+0）-aux（+0）和（符号和代表单侧极限）。该问题表现为收缩到区间的高维域中半线性椭圆方程的极限方程。首先证明了奇解和偶解分别构成分支族和.都是从平凡解分支出来的.然后，我们证明，不包含其他分歧点，而包含两个点，二次分歧发生。最后确定了分支上解的莫尔斯指数。还给出了相同断言成立的一般条件onf（u）。
We consider the Neumann problem for the equationin the punctured interval, whereis a bifurcation parameter and. At, we impose the conditions u(-0)+aux(-0)=u(+0)-aux(+0) andfor a constant(the symbolsandstand for one-sided limits). The problem appears as a limiting equation for a semilinear elliptic equation in a higher dimensional domain shrinking to the interval. First we prove that odd solutions and even solutions form families of branchesand, respectively. Bothandbifurcate from the trivial solution. We then show thatcontains no other bifurcation point, whilecontains two points where secondary bifurcations occur. Finally we determine the Morse index of solutions on the branches. General conditions onf(u) for the same assertions to hold are also given.