Existence of a nontrivial solution for the (p, q)-Laplacian in RN without the Ambrosetti–Rabinowitz condition. (February 2015)

• Record Type: Journal Article
• Authors: Chaves, Marcio Fialho; Ercole, Grey; Miyagaki, Olimpio Hiroshi
• Is Part Of: Nonlinear analysis. Volume 114(2015)
• Page Start: 133
• Page End: 141
• Publication Date: 2015-02
• Subjects: 35J92 -- 47J30; Ambrosetti–Rabinowitz condition -- Cerami condition -- Nontrivial weak solution -- (p, q)-Laplacian; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2014.11.010
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we prove the existence of at least one nonnegative nontrivial weak solution in D 1, p ( R N ) ∩ D 1, q ( R N ) for the equation − Δ p u − Δ q u + a ( x ) | u | p − 2 u + b ( x ) | u | q − 2 u = f ( x, u ), x ∈ R N, where 1 < q < p < q ⋆ : = N q N − q, p < N, Δ m u : = div ( | ∇ u | m − 2 ∇ u ) is the m -Laplacian operator, the coefficients a and b are continuous, coercive and positive functions, and the nonlinearity f is a Carathéodory function satisfying some hypotheses which do not include the Ambrosetti–Rabinowitz condition.