Normalized solutions to the Kirchhoff Equation with triple critical exponents in [formula omitted

Abstract

In this paper, we investigate the normalized solutions for the nonlinear critical Kirchhoff equations with combined nonlinearities: −a+b∫RN|∇u|2dxΔu=λu+μ|u|u+|u|2u,∫RNu2dx=c2,x∈RN,where N=4, λ, μ∈R and a, b, c>0 are constants. In R4, some interesting phenomena occur, which are, the L2-critical exponent for Δu is 2+4N=3, while the L2-critical exponent for (∫R4|∇u|2dx)Δu is equal to the Sobolev critical exponent, i.e., 2+8N=2NN−2=4. This paper investigates the case that the nonlinearity with triple critical term and proves the existence of a positive mountain-pass type normalized solution through variational methods and energy estimations.