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arXiv 1509.06221math.CA
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具有Sturm-Liouville型多点边界条件的线性和非线性二阶问题

Linear and nonlinear, second-order problems with Sturm-Liouville-type, multi-point boundary conditions

Bryan P. Rynne

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摘要：

考虑非线性方程$$-u'' = f(u) + h , \quad \text{on} \quad (-1,1),$$，其中$f : {\mathbb R} \to {\mathbb R}$和$h : [-1,1] \to {\mathbb R}$是连续的，以及一般的Sturm-Liouville型，在$\pm 1$上的多点边界条件.我们将得到这个边值问题在一定的“非共振”条件下的解的存在性，以及Rabinowitz型全局分歧的结果，从而产生问题的节点解。这些结果依赖于由方程$$-u'' = λu, \quad \text{on} \quad (-1,1),$$和多点边界条件组成的本征值问题的谱性质。在以前的文件中，它表明，在某些“最佳”条件下，这个本征值问题的基本谱性质是类似的标准Sturm-Liouville问题与单点边界条件。特别地，对于每个整数$k \geq 0$，存在唯一的简单本征值$λ_k$，其本征函数的“振荡计数”等于$k$，其中“振荡计数”是根据复杂的普吕弗角构造来定义的。不幸的是，似乎很难将Prüfer角构造应用于非线性问题。因此，在本文中，我们使用替代的，非最优的，振荡计数方法，以获得所需的线性问题的谱特性，然后将这些应用到非线性问题，以产生上述结果。

英文摘要：

We consider the nonlinear equation $$-u'' = f(u) + h , \quad \text{on} \quad (-1,1),$$ where $f : {\mathbb R} \to {\mathbb R}$ and $h : [-1,1] \to {\mathbb R}$ are continuous, together with general Sturm-Liouville type, multi-point boundary conditions at $\pm 1$. We will obtain existence of solutions of this boundary value problem under certain `nonresonance' conditions, and also Rabinowitz-type global bifurcation results, which yield nodal solutions of the problem. These results rely on the spectral properties of the eigenvalue problem consisting of the equation $$-u'' = λu, \quad \text{on} \quad (-1,1),$$ together with the multi-point boundary conditions. In a previous paper it was shown that, under certain `optimal' conditions, the basic spectral properties of this eigenvalue problem are similar to those of the standard Sturm-Liouville problem with single-point boundary conditions. In particular, for each integer $k \geq 0$ there exists a unique, simple eigenvalue $λ_k$, whose eigenfunctions have `oscillation count' equal to $k$, where the `oscillation count' was defined in terms of a complicated Prüfer angle construction. Unfortunately, it seems to be difficult to apply the Prüfer angle construction to the nonlinear problem. Accordingly, in this paper we use alternative, non-optimal, oscillation counting methods to obtain the required spectral properties of the linear problem, and these are then applied to the nonlinear problem to yield the results mentioned above.

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