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

Abstract

In this paper, we consider the following Schrödinger‐Poisson system: −normalΔu+λϕfalse|ufalse|2α∗−2u=()∫R3false|ufalse|2β∗false|x−yfalse|3−βnormaldyfalse|ufalse|2β∗−2u,in5ptR3,false(−normalΔfalse)α2ϕ=Aα−1false|ufalse|2α∗,in5ptR3, where parameters α,β∈(0,3),λ>0, Aα=normalΓfalse(3−α2false)2απ32normalΓfalse(α2false), 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.

Fractional Schrodinger-Poisson systems with weighted Hardy potential and critical exponent

“…where V ∈ C 1 (R 3 , R + ) and f is subcritical. For more details and recent works, we refer to [5,31,37,41,42] and the references therein.…”