Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equation in the plane
Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equation in the plane
复制标题
平面上非线性亥姆霍兹方程驻波的存在性及其渐近行为
DOI：
10.1515/anly-2016-0023
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发表时间：
2016
期刊：
影响因子：
1.6
通讯作者：
G. Evéquoz
中科院分区：
文献类型：
--
作者：
G. Evéquoz
本文研究半线性椭圆问题-Δu-k 2u=q|u|p-2u inℝ2，-\≥u-k^{2}u=q|u|^{p-2}u\quad\Text{in}\mathbb{R}^{2}，其中k>；0{k>；0}，p-gt；6{p\geq6}，q是有界函数。证明了衰减系数Q和周期系数Q的实值W2，p{W^{2，p}-解的存在性，并导出了这些解的一个非线性远场关系.
Abstract In this paper we study the semilinear elliptic problem - Δ u - k 2 u = Q | u | p - 2 u in ℝ 2 , -\Delta u-k^{2}u=Q|u|^{p-2}u\quad\text{in }\mathbb{R}^{2}, where k > 0 {k>0} , p ≥ 6 {p\geq 6} and Q is a bounded function. We prove the existence of real-valued W 2 , p {W^{2,p}} -solutions, both for decaying and for periodic coefficient Q. In addition, a nonlinear far-field relation is derived for these solutions.