Multiple solutions for a class of double phase problem without the Ambrosetti–Rabinowitz conditions. (November 2019)

• Record Type: Journal Article
• Authors: Ge, Bin; Lv, De-Jing; Lu, Jian-Fang
• Is Part Of: Nonlinear analysis. Volume 188(2019)
• Page Start: 294
• Page End: 315
• Publication Date: 2019-11
• Subjects: 35J60 -- 03H10 -- 35D05; Double phase problem -- Variational method -- Multiple solutions -- Fountain theorem -- Dual Fountain theorem; 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.2019.06.007
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In the present paper, in view of the variational approach, we consider the existence and multiplicity of weak solutions for a class of the double phase problem − div ( | ∇ u | p − 2 ∇ u + a ( x ) | ∇ u | q − 2 ∇ u ) = λ f ( x, u ), in Ω, u = 0, on ∂ Ω, where N ≥ 2 and 1 < p < q < N . Firstly, by the Fountain and Dual Theorem with Cerami condition, we obtain some existence of infinitely many solutions for the above problem under some weaker assumptions on f . Secondly, we prove that this problem has at least one nontrivial solution for any parameter λ > 0 small enough, and also that the solution blows up, in the Sobolev norm, as λ → 0 + . Finally, by imposing additional assumptions on f, we establish the existence of infinitely many solutions by using Krasnoselskii's genus theory for the above equation.