The Existence and Multiplicity of Normalized Solutions for Kirchhoff Equations in Defocusing Case

In this paper, we study the existence of solutions for Kirchhoff equationwith mass constraint conditionwhere a, b, c > 0, µ ∈ R and 2 < q < p < 6.The λ ∈ R appears as a Lagrange multiplier.For the range of p and q, the Sobolev critical exponent 6 and mass critical exponent 14 3 are involved which corresponding energy functional is unbounded from below on S c .We consider the defocusing case, i.e. µ < 0 when (p, q) belongs to a certain domain in R 2 .We prove the existence and multiplicity of normalized solutions by using constraint minimization, concentration compactness principle and Minimax methods.We partially extend the results have been studied.