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

In this paper, we study the existence of solutions for Kirchhoff equation (-a+b integral(R3 )|del u|(2) dx) triangle u = lambda u + mu|u|(q-2)u + |u|(p-2)u in R-3 with mass constraint condition S-c := {u is an element of H-1(R-3) : integral(R3 )|u|(2)dx = c}, where a, b, c > 0, mu is an element of R and 2 < q < p < 6. The lambda is an element of R appears as a Lagrange multiplier. For the range of p and q, the Sobolev critical exponent 66 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. mu < 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 that have been studied.