Schrödinger‐Poisson system with Hardy‐Littlewood‐Sobolev critical exponent

다국어 초록 (Multilingual Abstract)

In this paper, we consider the following Schrödinger‐Poisson system:

−Δu+λϕ|u|2α∗−2u=∫R3|u|2β∗|x−y|3−βdy|u|2β∗−2u,inR3,(−Δ)α2ϕ=Aα−1|u|2α∗,inR3,
where parameters α,β∈(0,3),λ>0,
Aα=Γ(3−α2)2απ32Γ(α2),
2α∗=3+α, and
2β∗=3+β are the Hardy‐Littlewood‐Sobolev critical exponents. For α<β and λ>0, we prove the existence of nonnegative groundstate solution to above system. Moreover, applying Moser iteration scheme and Kelvin transformation, we show the behavior of nonnegative groundstate solution at infinity. For β<α and λ>0 small, we apply a perturbation method to study the existence of nonnegative solution. For β<α and λ is a particular value, we show the existence of infinitely many solutions to above system.