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].
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