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, p. 1071-1085, JAN 15 2021.
• Web of Science Citations: 2

Abstract

The paper deals with the following Kirchhoff-type problem [M (V=0)(integral integral 1/R2NP(x,y) vertical bar v(y) - 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) = vertical bar v vertical bar(r(x)-2)v in Omega. in RN\textbackslash{}Omega where M models a Kirchhoff coefficient, (-Delta)(p(.))(s(.)) is a variable s(.)-order p(.)-fractional Laplace operator, with s(.) : R-2N -> (0, 1) and p(.) : R-2N -> (1, 8). 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 is a positive parameter, g is 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