A critical Kirchhoff-type problem driven by a p(.)-fractional Laplace operator with variable s(.)-order

Author(s)

Zuo, Jiabin ^{[1, 2]} ; An, Tianqing ^[1] ; Fiscella, Alessio ^[3]

Affiliation

^[1] Hohai Univ, Coll Sci, Nanjing 210098 - Peoples R China

^[2] Jilin Engn Normal Univ, Fac Appl Sci, Changchun 130052 - Peoples R China

^[3] Univ Estadual Campinas, Dept Matemat, IMECC, Rua Sergio Buarque de Holanda 651, BR-13083859 Campinas, SP - Brazil

• Document type: Journal article
• Source: MATHEMATICAL METHODS IN THE APPLIED SCIENCES; v. 44, n. 1 AUG 2020.
• Web of Science Citations: 3

Abstract

The paper deals with the following Kirchhoff-type problem integral M (integral integral(R2N) 1/p(x,y)vertical bar v(x)-v(y)vertical bar(p(x,y))/vertical bar x-y vertical bar(N+p(x,y)s(x,y))dxdy(-Delta)(p(.))(s(.))v(x)=mu g(x,v)+vertical bar v vertical bar(r(x)-2)v in Omega, v = 0 in R-N\textbackslash{}Omega, where M models a Kirchhoff coefficient,(-Delta)(p(.))(s(.)) is a variable s(.)-order p(.)-fractional Laplace operator, with s(.):Double-struck capital R-2N -> (0,1)and p(.) : R-2N -> (1,infinity). Here,Omega subset of R-N is a bounded smooth domain with N > p(x,y)s(x,y) for any (x,y)is an element of(Omega) over bar x (Omega) over bar ,mu is a positive parameter,gis a continuous and subcritical function, while variable exponent r(x)could be close to the critical exponent p(s){*}(x)=N (p) over bar (x)/(N-(s) over bar (x)(p) over bar (x)), given with (p) over bar (x)=p(x,x) and (s) over bar (x)=s(x,x)for x is an element of(Omega) over bar. We prove the existence and asymptotic behavior of at least one non-trivial solution. For this, we exploit a suitable tricky step analysis of the critical mountain pass level, combined with a Brezis and Lieb-type lemma for fractional Sobolev spaces with variable order and variable exponent. (AU)

• FAPESP's process: 17/19752-3 - Fractional problems with lack of compactness
• Grantee: Alessio Fiscella
• Support Opportunities: Regular Research Grants
• FAPESP's process: 19/02512-5 - Systems and partial differential equations
• Grantee: Marcelo da Silva Montenegro
• Support Opportunities: Research Projects - Thematic Grants