Normalized solutions to nonlinear scalar field equations with doubly nonlocal terms and critical exponent

Abstract

In this paper, we study the existence and regularity of normalized solutions to the following doubly nonlocal equation−Δu=λu+μ1(Iα⁎|u|p)|u|p−2u+μ2(Iβ⁎|u|p)|u|p−2uinR3, having a prescribed mass ∫R3u2=a>0, where λ∈R will arise as a Lagrange multiplier, α,β∈(0,3) with α⩽β, p∈[2⁎β,2α⁎], Iα and Iβ are Riesz potentials, μ1,μ2>0 are constants. In particular, we estimate the energy level ingeniously and consider the existence of normalized solutions to the above equation with Hardy–Littlewood–Sobolev lower critical exponent p=2⁎β or upper critical exponent p=2α⁎. The results are supplements to the works of Cao et al. (2021) [5].