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Schrödinger‐Poisson system with Hardy‐Littlewood‐Sobolev critical exponent
Author(s) -
Su Yu,
Wang Li,
Han Tao
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5694
Subject(s) - mathematics , sobolev space , exponent , mathematical analysis , critical exponent , schrödinger's cat , poisson distribution , perturbation (astronomy) , transformation (genetics) , initial value problem , infinity , sobolev inequality , mathematical physics , quantum mechanics , geometry , physics , philosophy , linguistics , statistics , scaling , biochemistry , chemistry , gene
In this paper, we consider the following Schrödinger‐Poisson system:− Δ u + λ ϕ | u |2 α ∗ − 2 u =∫ R 3| u |2 β ∗| x − y | 3 − βd y| u |2 β ∗ − 2 u , inR 3 ,( − Δ )α 2ϕ = A α − 1| u |2 α ∗, inR 3 ,where parameters α , β ∈(0,3), λ >0, A α = Γ ( 3 − α 2 )2 α π 3 2Γ ( α 2 ) ,2 α ∗ = 3 + α , and2 β ∗ = 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.