论文标题
半星式微分方程中的次要分叉
Secondary bifurcations in semilinear ordinary differential equations
论文作者
论文摘要
我们考虑了方程式$ u_ {xx}+λf（u）= 0 $的neumann问题，在刺穿的间隔$（-1,1）\ setMinus \ {0 \} $中，其中$λ> 0 $是Bifurcation参数，$ f（u-u-f（$ f（u-u-u-f（u-u-f（u-u-f（u-u-u-f（u-u-u-u-u-u-u-u-u-u-u-u-u-u-e），在$ x = 0 $时，我们强加条件$ u（-0）+au_x（-0）= u（+0）-au_x（+0）$和$ u_x（-0）= u_x（-0）= u_x（+0）$ for常数$ a> 0 $（符号$+0 $+0 $+0 $ and $ -0 $ and $ -0 $ and $ -0 $ and-$ -0 $ and-$ -0 $ and-$ -0 $ and-$ -0 $ and-$ -0 $支架）。该问题似乎是在较高维度域中缩小到间隔$（-1,1）$的较高维域中半线性椭圆方程的限制方程。首先，我们证明奇数解决方案甚至解决方案构成了分支的家族$ \ {\ Mathcal {c}^o_k \ \} _ {k \ in \ Mathbb {n}} $和$ \ {\ {\ Mathcal {c} $ \ MATHCAL {C}^O_K $和$ \ MATHCAL {C}^E_K $ BIFURCATE来自Trivial解决方案$ U = 0 $。然后，我们表明$ \ MATHCAL {C}^e_k $不包含其他分叉点，而$ \ Mathcal {c}^o_k $包含次要分叉发生的两个点。最后，我们确定分支上解决方案的摩尔斯索引指数。还给出了$ f（u）$的一般条件，以供同一主张持有。
We consider the Neumann problem for the equation $u_{xx}+λf(u)=0$ in the punctured interval $(-1,1) \setminus \{0\}$, where $λ>0$ is a bifurcation parameter and $f(u)=u-u^3$. At $x=0$, we impose the conditions $u(-0)+au_x(-0)=u(+0)-au_x(+0)$ and $u_x(-0)=u_x(+0)$ for a constant $a>0$ (the symbols $+0$ and $-0$ stand 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 $(-1,1)$. First we prove that odd solutions and even solutions form families of branches $\{ \mathcal{C}^o_k\}_{k \in \mathbb{N}}$ and $\{ \mathcal{C}^e_k\}_{k \in \mathbb{N}}$, respectively. Both $\mathcal{C}^o_k$ and $\mathcal{C}^e_k$ bifurcate from the trivial solution $u=0$. We then show that $\mathcal{C}^e_k$ contains no other bifurcation point, while $\mathcal{C}^o_k$ contains two points where secondary bifurcations occur. Finally we determine the Morse index of solutions on the branches. General conditions on $f(u)$ for the same assertions to hold are also given.
