Standing and Outward Radiating Wave Solutions of Nonlinear Helmholtz Equations

在非线性分析和偏微分方程领域中，波在非线性介质中的传播已成为最引人注目的研究课题之一。虽然驻波、色散和波散射的发生现在在线性环境中已经相当好地理解了，但在非线性环境中却产生了大量的开放性问题。本课题致力于研究高频驻波或散射向外辐射波非线性Klein-Gordon方程的时间周期解。时间周期分析导致非线性亥姆霍兹方程，近年来在各种问题中引起越来越多的关注，因为本质谱的存在阻碍了对非线性效应的理解。本建议建立在第一个资助期取得的进展的基础上，其中通过对偶变分和拓扑不动点方法，推导了非临界增长非线性的驻波解和连续解的存在性结果。现在，我们将解决一些具有挑战性的开放性问题，包括临界指数的影响、双基态的形状、浓度现象的发生、溶液分支的持续以及平稳散射的性质。

Wave propagation in nonlinear media has emerged as one of the most intriguing research topics in the field of nonlinear analysis and partial differential equations. Whereas the occurence of standing waves, dispersion, and wave scattering is by now reasonably well understood in a linear context, it gives rise to an abundance of open questions in the nonlinear setting. The present project is devoted to the study of time periodic solutions of the nonlinear Klein-Gordon equation representing standing or scattered outward radiating waves with high frequency. The time periodic ansatz leads to the nonlinear Helmholtz equation, which has attracted growing attention recently as it arises in various problems where the presence of essential spectrum hampers the understanding of nonlinear effects. The present proposal builds on the progress made in the first funding period, in which results on the existence of standing wave solutions and solution continua have been derived, for non-critically growing nonlinearities, by means of dual variational and topological fixed point methods. We shall now tackle challenging open questions concerning the impact of critical exponents, the shape of dual ground states, the occurence of concentration phenomena, the contination of solution branches, and the nature of stationary scattering.