Premium
On a class of coupled critical Hartree system with deepening potential
Author(s) -
Zheng Yu,
Gao Fashun,
Shen Zifei,
Yang Minbo
Publication year - 2020
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.6785
Subject(s) - mathematics , sobolev space , hartree , infinity , function (biology) , class (philosophy) , critical exponent , variational inequality , state (computer science) , pure mathematics , mathematical analysis , geometry , quantum mechanics , physics , computer science , biology , algorithm , evolutionary biology , artificial intelligence , scaling
In this paper, we are interested in the following coupled critical Hartree system− Δ u + λ V 1 ( x ) u = α 1 u + β v + | x | − μ * | v |2 μ *| u |2 μ * − 2 u inℝ N ,− Δ v + λ V 2 ( x ) v = β u + α 2 v + | x | − μ * | u |2 μ *| v |2 μ * − 2 v inℝ N ,where λ > 0, β ≥ 0,α 1 , α 2 ∈ ℝ , 0 < μ < N , N ≥ 4,2 μ * = ( 2 N − μ ) / ( N − 2 ) are the upper critical exponents due to the Hardy‐Littlewood‐Sobolev inequality and the nonnegative potential functionV 1 , V 2 ∈ C ( ℝ N , ℝ ) has potential as well. By using variational methods, we prove the existence of ground state solutions and also characterize the asymptotic behavior of the solutions as the parameter λ goes to infinity. Furthermore, we are able to find the existence of multiple semitrivial weak solutions by the Lusternik‐Schnirelmann category theory.