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Complete classification of ground state solutions with different Morse index for critical fractional Laplacian system
Author(s) -
Zhen Maoding,
Zhang Binlin
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.6862
Subject(s) - morse code , ground state , mathematics , morse theory , index (typography) , laplace operator , state (computer science) , fractional laplacian , type (biology) , mathematical analysis , pure mathematics , mathematical physics , physics , quantum mechanics , algorithm , world wide web , computer science , electrical engineering , engineering , ecology , biology
In this paper, we first obtain the existence of positive ground state solutions for the following critical fractional Laplacian system:( − Δ ) s u = μ 1 | u |2 s ∗ − 2 u + α γ2 s ∗| u | α − 2 u | v | βinℝ n ,( − Δ ) s v = μ 2 | v |2 s ∗ − 2 v + β γ2 s ∗| u | α | v | β − 2 vinℝ n ,then we give a complete classification of positive ground state solutions with different Morse index. More precisely, we show that if ( u , v ) be any positive ground state solution of system (1.1), then ( u , v ) must be ( C 1 U ϵ , y , C 2 U ϵ , y ) type with Morse index 1 and Morse index 2, where U ϵ , y is a positive ground state solution for a given equation.