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On a systems involving fractional Kirchhoff‐type equations and Krasnoselskii's genus
Author(s) -
Costa Augusto,
Pereira Fabio
Publication year - 2018
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.5295
Subject(s) - mathematics , type (biology) , bounded function , fractional laplacian , hilbert space , operator (biology) , genus , pure mathematics , class (philosophy) , space (punctuation) , domain (mathematical analysis) , mathematical analysis , combinatorics , ecology , biochemistry , chemistry , botany , linguistics , philosophy , repressor , artificial intelligence , gene , transcription factor , computer science , biology
We consider a class of variational systems involving fractional Kirchhoff‐type equations of the form:M 1 ( ‖ u‖ X 2 ) ( − Δ ) s u =F u ( x , u , v )inΩ ,M 2 ( ‖ v‖ X 2 ) ( − Δ ) s v =F v ( x , u , v )inΩ ,u = v = 0inR N \ Ω ,where s ∈ (0,1), N > 2 s , Ω ⊂ R N a smooth and bounded domain, the functions F u , F v , M 1 and M 2 are continuous and ( − Δ) s is the fractional Laplacian operator. In this paper, we show that, under appropriate growth conditions on the nonlinearities F u and F v and on the nonnegative functions M 1 and M 2 , the (weak) solutions are precisely the critical points of a related functional defined on a fractional Hilbert space Y (Ω) = X (Ω) × X (Ω) and the existence infinitely many solutions can be obtained by the use of the Krasnoselskii's genus. Besides, a regularity result can also be obtained by using specific results for systems in conjunction with the growth assumptions of these functions.