Nodal properties and variational characterizations of solutions to nonlinear Sturm-Liouville problems
Nodal properties and variational characterizations of solutions to nonlinear Sturm-Liouville problems
复制标题
非线性 Sturm-Liouville 问题解的节点性质和变分特征
DOI：
10.1016/0022-0396(86)90089-6
复制
发表时间：
1986
影响因子：
2.4
通讯作者：
H. Heinz
中科院分区：
文献类型：
--
作者：
H. Heinz
我们考虑了形式为-(p (t) u ') ' + q (t) u+ f (t, u)= dr (t) u（1.1）的常微分方程的非线性特征值问题，并考虑了Dirichlet边界条件。将式（1.1）的左手边看作一个合适的泛函Y的欧拉-拉格朗日算子，我们可以辨识出这样一个非线性特征值问题的解，该问题具有Y对水平曲面S的限制的临界点，该约束形式为S am r (t) dt= R2，其中r >为任意的。为了描述和激励本论文的结果，让我们简要回顾一些众所周知的事实，其中fz0和假设是这样的，我们正在处理一个常规的Sturm-Liouville问题。设（,4,，A,，…）为问题特征值的递增序列，设(cpl,（p2，…）为归一化特征函数的关联序列。任意K > 0,把u =中移动,和c,: = $描述:* (n = 1,2,…)。则c = Y (u,)（即c是与An相关的临界能级），S上临界值c对应的特征函数恰好是u，和u，。数字c，= Y (u,)（因此函数u，）享有著名的变分特征c，= inf sup u ' (u) YEAn ucS, qnV
We consider nonlinear eigenvalue problems for ordinary differential equations of the form-(p (t) u’)‘+ q (t) u+ f (t, u)= dr (t) u(1.1) together with Dirichlet boundary conditions. Viewing the left-hand side of (1.1) as the Euler-Lagrange operator of an appropriate functional Y, we may identify the solutions of such a nonlinear eigenvalue problem with the critical points of the restriction of Y to the level surface S, given by a constraint of the form s am r (t) dt= R2 with R> 0 arbitrary. In order to describe and motivate the results of the present paper, let us briefly recall some of the well-known facts pertaining to the case where f z 0 and the assumptions are such that we are dealing with a regular Sturm-Liouville problem. Let (, 4,, A,,...) be the increasing sequence of eigenvalues of the problem, and let (cpl,(p2,...) be the associated sequence of normalized eigenfunctions. For an arbitrary K> 0, put u,:= Rq, and c,:= $ &R*(n= 1, 2,...). Then c,= Y (u,)(ie, c, is the critical level associated with An), and the eigenfunctions corresponding to the critical value c, on S, are precisely u, and-u,. The numbers c,= Y (u,)(and hence the functions u,) enjoy the famous variational characterizations c,= inf sup u’(u) YEAn ucS, qnV