Existence and blow up behavior of positive normalized solution to the Kirchhoff equation with general nonlinearities: Mass super-critical case

Abstract

In present paper, we study the normalized solutions (λc,uc)∈R×H1(RN) to the following Kirchhoff problem −(a+b∫RN|∇u|2dx)Δu+λu=g(u)inRN,1≤N≤3 satisfying the normalization constraint ∫RNu2=c, which appears in free vibrations of elastic strings. The parameters a,b>0 are prescribed as is the mass c>0. The nonlinearities g(s) considered here are very general and of mass super-critical. Under some suitable assumptions, we can prove the existence of ground state normalized solutions for any given c>0. After a detailed analysis via the blow up method, we also make clear the asymptotic behavior of these solutions as c→0+ as well as c→+∞.