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A variable‐order fractional p (·) ‐Kirchhoff type problem in ℝ N
Author(s) -
Zuo Jiabin,
Yang Libo,
Liang Sihua
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.6995
Subject(s) - mathematics , multiplicity (mathematics) , type (biology) , variable (mathematics) , embedding , mathematical analysis , order (exchange) , fixed point theorem , laplace transform , compact space , operator (biology) , pure mathematics , ecology , finance , economics , biology , biochemistry , chemistry , repressor , artificial intelligence , computer science , transcription factor , gene
This paper is concerned with the existence and multiplicity of solutions for the following variable s (·) ‐order fractional p (·) ‐Kirchhoff type problemM∬ℝ 2 N1 p ( x , y )| v ( x ) − v ( y ) | p ( x , y )| x − y | N + p ( x , y ) s ( x , y )d x d y( − Δ ) p ( · ) s ( · ) v ( x ) + | v ( x ) |p ‾ ( x ) − 2 v ( x ) = μ g ( x , v )inℝ N ,v ∈ W s ( · ) , p ( · ) ( ℝ N ) ,where N > p ( x , y ) s ( x , y ) for any ( x , y ) ∈ ℝ N × ℝ N ,( − Δ ) p ( · ) s ( · )is a variable s (·) ‐order p (·) ‐fractional Laplace operator with s ( · ) : ℝ 2 N → ( 0,1 ) and p ( · ) : ℝ 2 N → ( 1 , ∞ ) ,p ‾ ( x ) = p ( x , x ) for x ∈ ℝ N , and M is a continuous Kirchhoff‐type function, g ( x , v ) is a Carathéodory function, and μ > 0 is a parameter. Under the weaker conditions, we obtain that there are at least two distinct solutions for the above problem by applying the generalized abstract critical point theorem. Moreover, we also show the existence of one solution and infinitely many solutions by using the mountain pass lemma and fountain theorem, respectively. In particular, the new compact embedding result of the spaceW s ( · ) , p ( · ) ( ℝ N ) intoL a ( x ) q ( · ) ( ℝ N ) will be used to overcome the lack of compactness inℝ N . The main feature and difficulty of this paper is the presence of a double non‐local term involving two variable parameters.