On finding sign-changing solutions
On finding sign-changing solutions
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DOI：
10.1016/j.jfa.2005.09.004
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发表时间：
2006-05
影响因子：
1.7
通讯作者：
W. Zou
中科院分区：
文献类型：
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作者：
W. Zou
建立了一些依赖于参数的链接定理，这些定理允许产生有界且符号变化的 Palais-Smale 序列。对于偶泛函，获得依赖于参数的喷泉定理，该定理提供了无限多个有界且符号变化的 Palais-Smale 序列。一个变体山口定理是建立在圆锥体中的，它产生有界的、正的和负的 Palais-Smale 序列。通常的 Palais-Smale 型紧性条件及其变体对于这些理论来说完全不是必要的。可以确定关键序列的更准确位置。将抽象结果应用于具有（或不具有）临界 Sobolev 指数的薛定谔方程：其中 2* 是临界 Sobolev 指数。获得（多个）变号解的存在性。正解和负解也作为副产品获得。我们还将证明具有跳跃非线性的薛定谔问题与 Fučík 谱无关。
Some parameter-depending linking theorems are established, which allow to produce a bounded and sign-changing Palais–Smale sequence. For even functionals, a parameter-depending fountain theorem is obtained which provides infinitely many bounded and sign-changing Palais–Smale sequences. A variant mountain pass theorem is built in cones which yields bounded, positive and negative Palais–Smale sequences. The usual Palais–Smale type compactness condition and its variants are completely not necessary for these theories. More exact locations of the critical sequences can be determined. The abstract results are applied to the Schrödinger equation with (or without) critical Sobolev exponents: where 2*is the critical Sobolev exponent. The existence of (multiple) sign-changing solutions is obtained. The positive and negative solutions are also gained as by-products. We will also show that this Schrödinger problem with jumping nonlinearity is independent of the Fučík spectrum.