The Fu?ík spectrum of Schr?dinger operator and the existence of four solutions of Schr?dinger equations with jumping nonlinearities

摘要

Abstract This paper contains the existence of four solutions of Schr?dinger equations with jumping nonlinearities. The proof procedure is supported by a lot of new results. Initially, a consequence is rendered as a minimax principle on H 1 ( R N ) , which allows us to achieve the feasibility verification of the (PS) condition. Furthermore, the constructions of minimal and maximal curves of Fu?ík spectrum in Q l (see the introduction for the definition of Q l ) warrant an intensive investigation. That we encounter some thorny problems is largely due to the absence of compact embedding and the appearance of essential spectrum. Based on a nontrivial argument, we can compute critical groups of homogeneous functional at zero if ( a , b ) is free of Fu?ík spectrum and ( a , b ) ∈ Q l . This together with convexity and concavity offers a detailed description of the two curves by a series of sophisticated tricks. Additionally, we present a new version of Morse theo