Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations in R3

In this paper, we study the multiplicity of solutions with a prescribed L2-norm for a class of nonlinear Kirchhoff type problems in R3−(a+b∫R3|∇u|2)Δu−λu=|u|p−2u, where a,b>0 are constants, λ∈R, p∈(143,6). To get such solutions we look for critical points of the energy functional Ib(u)=a2∫R3|∇u|2+b4(∫R3|∇u|2)2−1p∫R3|u|p restricted on the following set Sr(c)={u∈Hr1(R3):‖u‖L2(R3)2=c},c>0. For the value p∈(143,6) considered, the functional Ib is unbounded from below on Sr(c). By using a minimax procedure, we prove that for any c>0, there are infinitely many critical points {unb}n∈N+ of Ib restricted on Sr(c) with the energy Ib(unb)→+∞(n→+∞). Moreover, we regard b as a parameter and give a convergence property of unb as b→0+.