Asymptotic profiles for a nonlinear Kirchhoff equation with combined powers nonlinearity

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation −(a+b∫RN|∇u|2)Δu+λu=uq−1+up−1inRN,(Pλ)as λ→0 and λ→+∞, where N=3 or N=4, 20, b≥0 are constants and λ>0 is a parameter. In particular, we prove that in the case 20 is sufficiently large or sufficiently small. Our results also show that in the space dimension N=3, there is a striking difference between the cases b=0 and b≠0. More precisely, if b≠0, then both p0≔10/3 and pb≔14/3 play a role in the existence, non-existence, the exact number and asymptotic behavior of the normalized solutions of the mass constrained problem, which is completely different from those for the corresponding nonlinear Schrödinger equation and which reveals the special influence of the nonlocal term.