Multiple solutions with sign information for a (p, 2)-equation with combined nonlinearities. (March 2020)

• Record Type: Journal Article
• Authors: Papageorgiou, Nikolaos S.; Vetro, Calogero; Vetro, Francesca
• Is Part Of: Nonlinear analysis. Volume 192(2020)
• Publication Date: 2020-03
• Subjects: 35J20 -- 35J60 -- 58E05; Convex–concave problem -- Nonlinear regularity -- Strong comparison principle -- Constant sign and nodal solutions -- Critical groups -- Flow invariance; 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.2019.111716
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider a parametric nonlinear Dirichlet problem driven by the sum of a p -Laplacian and of a Laplacian (a ( p, 2 ) -equation) and with a reaction which has the competing effects of two distinct nonlinearities. A parametric term which is ( p − 1 ) -superlinear (convex term) and a perturbation which is ( p − 1 ) -sublinear (concave term). First we show that for all small values of the parameter the problem has at least five nontrivial smooth solutions, all with sign information. Then by strengthening the regularity of the two nonlinearities we produce two more nodal solutions, for a total of seven nontrivial smooth solutions all with sign informations. Our proofs use critical point theory, critical groups and flow invariance arguments.