Standing and Outward Radiating Wave Solutions of Nonlinear Helmholtz Equations

波在非线性介质中的传播已成为非线性分析和偏微分方程组领域中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背景下已经被很好地理解了，但它在非线性背景下引起了大量的未决问题。本项目致力于研究非线性Klein-Gordon方程的时间周期解，该方程代表高频向外辐射的驻波或散射波。时间周期的ansatz引出了非线性Helmholtz方程，它最近引起了越来越多的关注，因为它出现在各种问题中，其中本质谱的存在阻碍了对非线性效应的理解。本提案建立在第一个资助期取得的进展的基础上，在第一个资助期，利用对偶变分不动点方法和拓扑不动点方法，对于非临界增长的非线性项，导出了驻波解的存在性和解的连续性。我们现在将解决一些具有挑战性的公开问题，如临界指数的影响、双基态的形状、集中现象的发生、解分支的连续性以及静止散射的性质。

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.