Título:	The asymptotic behavior of constant sign and nodal solutions of (p,q)-Laplacian problems as p goes to 1
Autor(es):	Figueiredo, Giovany de Jesus Malcher Pimenta, Marcos Tadeu de Oliveira Winkert, Patrick
ORCID:	https://orcid.org/0000-0003-1697-1592 https://orcid.org/0000-0003-4961-3038 https://orcid.org/0000-0003-0320-7026
Afiliação do autor:	Universidade de Brasília, Departamento de Matemática Universidade Estadual Paulista, Departamento de Matemática e Computação Institut für Mathematik, Technische Universität Berlin
Assunto:	Comportamento assintótico Funções de variação limitada p-Laplaciano
Data de publicação:	1-out-2024
Editora:	Elsevier
Referência:	FIGUEIREDO, Giovany M.; PIMENTA, Marcos T. O.; WINKERT, Patrick. The asymptotic behavior of constant sign and nodal solutions of (p,q)-Laplacian problems as p goes to 1. Nonlinear Analysis, [S.l.], v. 251, e113677, 2025. DOI: https://doi.org/10.1016/j.na.2024.113677. Disponível em: https://www.sciencedirect.com/science/article/pii/S0362546X24001962?via%3Dihub. Acesso em: 28 jan. 2026.
Abstract:	In this paper we study the asymptotic behavior of solutions to the (𝑝, 𝑞)-equation −𝛥𝑝 𝑢 − 𝛥𝑞 𝑢 = 𝑓(𝑥, 𝑢) in 𝛺, 𝑢 = 0 on 𝜕𝛺, as 𝑝 → 1 +, where 𝑁 ≥ 2, 1 < 𝑝 < 𝑞 < 1 ∗ ∶= 𝑁∕(𝑁 − 1) and 𝑓 is a Carathéodory function that grows superlinearly and subcritically. Based on a Nehari manifold treatment, we are able to prove that the (1, 𝑞)-Laplace problem given by − div ( ∇𝑢 |∇𝑢| ) − 𝛥𝑞 𝑢 = 𝑓(𝑥, 𝑢) in 𝛺, 𝑢 = 0 on 𝜕𝛺, has at least two constant sign solutions and one sign-changing solution, whereby the signchanging solution has least energy among all sign-changing solutions. Furthermore, the solutions belong to the usual Sobolev space 𝑊 1,𝑞 0 (𝛺) which is in contrast with the case of 1-Laplacian problems, where the solutions just belong to the space BV(𝛺) of all functions of bounded variation. As far as we know this is the first work dealing with (1, 𝑞)-Laplace problems even in the direction of constant sign solutions.
Unidade Acadêmica:	Instituto de Ciências Exatas (IE) Departamento de Matemática (IE MAT)
DOI:	https://doi.org/10.1016/j.na.2024.113677
Versão da editora:	https://www.sciencedirect.com/science/article/pii/S0362546X24001962?via%3Dihub
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