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

In present paper, we study the normalized solutions (λ c , u c) ∈ R × H 1 (R N) to the following Kirchhoff problem − (a + b ∫ R N | ∇ u | 2 d x) Δ u + λ u = g (u) in R N , 1 ≤ N ≤ 3 satisfying the normalization constraint ∫ R N u 2 = 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 → + ∞. [ABSTRACT FROM AUTHOR]