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Positive solutions of nonlinear Schrödinger equation with peaks on a Clifford torus
Author(s) -
Santra Sanjiban,
Wei Juncheng
Publication year - 2016
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201400321
Subject(s) - torus , mathematics , nonlinear system , nonlinear schrödinger equation , mathematical analysis , clifford algebra , mathematical physics , schrödinger equation , clifford torus , function (biology) , pure mathematics , algebra over a field , geometry , physics , quantum mechanics , evolutionary biology , biology
We prove the existence of large energy positive solutions for a stationary nonlinear Schrödinger equation Δ u − V ( x ) u + u p = 0inR Nwith peaks on a Clifford type torus. HereV ( x ) = V ( r 1 , r 2 , ⋯ , r s )=1 + 1 ( a 1 r 1 m + a 2 r 2 m + a 3 r 3 m + ⋯ + a s r s m )+ O 1 ( a 1 r 1 m + a 2 r 2 m + a 3 r 3 m + ⋯ + a s r s m ) 1 + τwhereR N = R N 1 × R N 2 × ⋯ × R N s , withN i ≥ 2 for all i = 1 , 2 , ⋯ , s ,m > 1 , τ > 0 , r i = | x i | . Each r i is a function r , ϕ 1 , ⋯ , ϕ i − 1and is defined by the generalized notion of spherical coordinates. The solutions are obtained by a max ( r , ϕ 1 , ⋯ , ϕ s − 1 )or amax r min ( ϕ 1 , ⋯ , ϕ s − 1 )process.