Deformation argument under PSP condition and applications

In this paper we introduce a new deformation argument, in which $C^0$-group action and a new type of Palais-Smale condition PSP play important roles. This type of deformation results are studied in [HT],[IT]] and has many different applications. Typically it can be applied to nonlinear scalar field equations. We give a survey in an abstract functional setting. We also present another application to nonlinear elliptic problems in strip-like domains. Under Berestycki-Lions type conditions, we show the existence of infinitely many solutions. This extends the results in [ByT].