Standing and Outward Radiating Wave Solutions of Nonlinear Helmholtz Equations

波在非线性介质中的传播问题是非线性分析和偏微分方程领域的一个重要研究课题。而驻波，色散和波散射的发生是现在合理地理解在线性的背景下，它引起了大量的开放性问题，在非线性设置。本项目致力于研究代表高频驻波或散射向外辐射波的非线性Klein-Gordon方程的时间周期解。非线性Helmholtz方程是由时间周期近似得到的，由于本质谱的存在阻碍了人们对非线性效应的理解，使得非线性Helmholtz方程引起了越来越多的关注。本提案建立在第一个供资期间取得的进展的基础上，在第一个供资期间，通过对偶变分和拓扑不动点方法，针对非临界增长的非线性问题，得出了驻波解和解连续体存在的结果。我们现在将解决具有挑战性的开放问题的影响，临界指数，双基态的形状，浓度现象的发生，解决方案分支的延续，和静态散射的性质。

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.