On the behavior of least energy solutions of a fractional (p,q(p))-Laplacian problem as p goes to infinity

Article type: Research Article

Authors: Ercole, Grey^{a; *} | Medeiros, Aldo H.S.^a | Pereira, Gilberto A.^b

Affiliations: [a] Universidade Federal de Minas Gerais, Belo Horizonte, MG, 30.123-970, Brazil. E-mails: grey@mat.ufmg.br, aldomedeiros@ufmg.br | [b] Universidade Federal de Ouro Preto, Ouro Preto, MG, 35.400-000, Brazil. E-mail: gilberto.pereira@ufop.edu.br

Correspondence: [*] Corresponding author. E-mail: grey@mat.ufmg.br.

Abstract: We study the behavior as p→∞ of up, a positive least energy solution of the problem [(−Δp)α+(−Δq(p))β]u=μp|u(xu)|p−2u(xu)δxuin Ωu=0in RN∖Ω|u(xu)|=‖u‖∞, where Ω⊂RN is a bounded, smooth domain, δxu is the Dirac delta distribution supported at xu, limp→∞q(p)p=Q∈(0,1)if 0<β<α<1(1,∞)if 0<α<β<1 and limp→∞μpp>R−α, with R denoting the inradius of Ω.

Keywords: Asymptotic behavior, Dirac delta distribution, fractional Sobolev spaces, viscosity solutions

DOI: 10.3233/ASY-201632

Journal: Asymptotic Analysis, vol. 123, no. 3-4, pp. 237-262, 2021