On solutions for a class of Kirchhoff systems involving critical growth in R2

Article type: Research Article

Authors: de Albuquerque, J.C.^a | do Ó, J.M.^b | dos Santos, E.O.^a | Severo, U.B.^{b; *}

Affiliations: [a] Departamento de Matemática, Universidade Federal de Pernambuco, 50670-901 Recife–PE, Brazil. E-mails: joserre@gmail.com, jc@dmat.ufpe.br, edjane.mat@gmail.com | [b] Departamento de Matemática, Universidade Federal da Paraíba, 58051-900, João Pessoa–PB, Brazil. E-mails: jmbo@pq.cnpq.br, uberlandio@mat.ufpb.br

Correspondence: [*] Corresponding author: Uberlandio B. Severo. E-mail: uberlandio@mat.ufpb.br.

Abstract: In this work we study the existence of solutions for the following class of elliptic systems involving Kirchhoff equations in the plane: m(‖u‖2)[−Δu+u]=λf(u,v),x∈R2,ℓ(‖v‖2)[−Δv+v]=λg(u,v),x∈R2, where λ>0 is a parameter, m,ℓ:[0,+∞)→[0,+∞) are Kirchhoff-type functions, ‖·‖ denotes the usual norm of the Sobolev space H1(R2) and the nonlinear terms f and g have exponential critical growth of Trudinger–Moser type. Moreover, when f and g are odd functions, we prove that the number of solutions increases when the parameter λ becomes large.

Keywords: Kirchhoff systems, exponential critical growth, Trudinger–Moser inequality

DOI: 10.3233/ASY-201610

Journal: Asymptotic Analysis, vol. 122, no. 1-2, pp. 69-85, 2021