Limit behaviors of pseudo-relativistic Hartree equation with power-type perturbations

Abstract

We consider the following pseudo-relativistic Hartree equations with power-type perturbation,i∂tψ=−Δ+m2ψ−(1|x|⁎|ψ|2)ψ+ε|ψ|p−2ψ, with (t,x)∈R×R3 where 2<p<3, ε>0 and m>0, p=83 can be viewed as a Slater modification. We mainly focus on the normalized ground state solitary waves φε, where ‖φε‖22=N. Firstly, we prove the existence and nonexistence of normalized ground states under L2-subcritical, L2-critical (p=83) and L2-supercritical perturbations. Secondly, we classify perturbation limit behaviors of ground states when ε→0+, and obtain two different blow-up profiles for N=Nc and N>Nc, where Nc be regard as “Chandrasekhar limiting mass”. We prove that 〈φε,−Δφε〉∼ε−23p−4 for N=Nc and 2<p<3, while 〈φε,−Δφε〉∼ε−23p−8 for N>Nc and 83<p<3. Finally, we study the asymptotic behavior for ε→+∞, and obtain an energy limit limε→+∞⁡eε(N)=12mN and a vanishing rate ∫R3|φε|pdx≲ε−1 when N>Nc and 83<p<3.