Complete classification of ground state solutions with different Morse index for critical fractional Laplacian system. (7th September 2020)

• Record Type: Journal Article
• Authors: Zhen, Maoding; Zhang, Binlin
• Is Part Of: Mathematical methods in the applied sciences. Volume 44:Number 2(2021)
• Page Start: 1601
• Page End: 1614
• Publication Date: 2020-09-07
• Subjects: classification of solutions -- fractional Laplacian system -- ground state solution -- morse index -- variational method; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.6862
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : 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 γ 2 s ∗ | u | α − 2 u | v | β in ℝ n, ( − Δ ) s v = μ 2 | v | 2 s ∗ − 2 v + β γ 2 s ∗ | u | α | v | β − 2 v in ℝ 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.