Dual variational methods and nonvanishing for the nonlinear Helmholtz equation
Dual variational methods and nonvanishing for the nonlinear Helmholtz equation
复制标题
DOI：
10.1016/j.aim.2015.04.017
复制
发表时间：
2014-02
影响因子：
1.7
通讯作者：
G. Evéquoz;T. Weth
中科院分区：
文献类型：
--
作者：
G. Evéquoz;T. Weth
我们建立了一个双变分框架来检测非线性亥姆霍兹方程的实驻波解− Δ u− k 2 u= Q (x)|你| p− 2 u, u∈ W 2, p (R N) 且 N≥ 3, 2 (N+ 1)(N− 1)< p< 2 N N− 2 且非负 Q∈ L∞(R N)。我们证明了周期 Q 以及 Q (x)→ 0 as| 的情况下非平凡解的存在性。 x|→∞。在周期性情况下，该方法的关键要素是与相关积分方程相关的新的非零定理。我们研究的解决方案是出射波和入射波的叠加，其特征是非线性远场关系。
We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation− Δ u− k 2 u= Q (x)| u| p− 2 u, u∈ W 2, p (R N) with N≥ 3, 2 (N+ 1)(N− 1)< p< 2 N N− 2 and nonnegative Q∈ L∞(R N). We prove the existence of nontrivial solutions for periodic Q as well as in the case where Q (x)→ 0 as| x|→∞. In the periodic case, a key ingredient of the approach is a new nonvanishing theorem related to an associated integral equation. The solutions we study are superpositions of outgoing and incoming waves and are characterized by a nonlinear far field relation.