Existence and multiplicity of normalized solutions for the nonlinear Kirchhoff type problems

In this paper, we prove the existence and multiplicity results of solutions with prescribed L2-norm for a class of Kirchhoff type problems −a+b∫R3|∇u|2dxΔu−λu=f(u)inR3,where a,b>0 are constants, λ∈R andf∈C(R,R). To obtain such solutions, we look into critical points of the energy functional Eb(u)=a2∫R3|∇u|2+b4∫R3|∇u|22−∫R3F(u)constrained on the L2-spheres S(c)=u∈H1(R3):||u||22=c. Here, c>0 and F(s)≔∫0sf(t)dt. Under some mild assumptions on f, we show that critical points of Eb unbounded from below on S(c) exist for c>0. In addition, we establish the existence of infinitely many radial critical points {unb} of Eb on S(c) provided that f is odd. Finally, the asymptotic behavior of unb as b↘0 is analyzed. These conclusions extend some known ones in previous papers.