Real Solutions to the Nonlinear Helmholtz Equation with Local Nonlinearity
Real Solutions to the Nonlinear Helmholtz Equation with Local Nonlinearity
复制标题
具有局部非线性的非线性亥姆霍兹方程的实数解
DOI：
10.1007/s00205-013-0664-2
复制
发表时间：
2013
影响因子：
2.5
通讯作者：
T. Weth
中科院分区：
文献类型：
--
作者：
G. Evéquoz;T. Weth
本文研究了非线性Helmholtz方程-Δu-k2u=f(x，u)，x∈RN\DocumentClass[12pt]{Minimum}\usepackage{amsath}\usepackage{amssym}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{matrsbsy}\usepackage{upgreek}\setlong{oddsidemargin}{-69pt}\Begin{Document}$-\Delta u-k^2u=f(x，u)，满足渐近条件的四边形x\in{R}^N$\end{Document}u(x)=O|x|1-N2and∂2u∂r2(x)+k2u(x)=o|x|1-N2asr=|x|→∞.\documentclass[12pt]{minimal}\usepackage{amsath}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{matrsfs}\usepackage{upgreek}\setLength{\oddsidemargin}{-69pt}\Begin{Document}$$u(X)=O\Left(|x|^{\FRAC{1-N}{2}}\右)\FRAD{\Rm和}\FRAC{\Partial^2 u}{\Partial r^2}(x)+k^2u(x)=o\left(|x|^{\frac{1-N}{2}}\right)\Quad{\rm As}\，R=|x|\to\inty.$$\end{文档}我们发展了变分框架，在没有任何对称假设的情况下证明了紧支集非线性方程非平凡解的存在性。此外，我们还考虑了径向情形，在这种情形下，对于更大的一类非线性，无穷多个解被证明存在。我们的结果给出了相应的具有任意大频率的非线性Klein-Gordon方程驻波解的存在性。
In this paper, we study real solutions of the nonlinear Helmholtz equation -Δu-k2u=f(x,u),x∈RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- \Delta u - k^2 u = f(x,u),\quad x\in \mathbb{R}^N$$\end{document}satisfying the asymptotic conditions u(x)=O|x|1-N2and∂2u∂r2(x)+k2u(x)=o|x|1-N2asr=|x|→∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(x)=O\left(|x|^{\frac{1-N}{2}}\right) \quad {\rm and} \quad \frac{\partial^2 u}{\partial r^2}(x)+k^2u(x)=o\left(|x|^{\frac{1-N}{2}}\right) \quad {\rm as}\, r=|x| \to\infty.$$\end{document}We develop the variational framework to prove the existence of nontrivial solutions for compactly supported nonlinearities without any symmetry assumptions. In addition, we consider the radial case in which, for a larger class of nonlinearities, infinitely many solutions are shown to exist. Our results give rise to the existence of standing wave solutions of corresponding nonlinear Klein–Gordon equations with arbitrarily large frequency.