Multiple solutions with sign information for a \(( p, 2)\)-equation with combined nonlinearities
type: Item
instance of: scholarly article
title: Multiple solutions with sign information for a \(( p, 2)\)-equation with combined nonlinearities
zbMATH Open document ID: 1436.35117
author: Nikolaos S. Papageorgiou
author: Calogero Vetro
author: Francesca Vetro
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2020-04-07T00:00:00Z
review text: The authors study a parametric \((p,2)\)-equation of the form \[ \begin{cases} -\Delta_p u(z) -\Delta u(z)=\lambda f(z,u(z))+g(z,u(z))\quad & \text{in }\Omega,\\ u(z)=0 & \text{on }\partial\Omega,\\ \lambda>0, \ 2<p<+\infty,\tag{P\(_{\lambda}\)} \end{cases} \] where \(\Omega \subseteq \mathbb{R}^N\) is a bounded domain with a \(C^2\)-boundary \(\partial\Omega\), \(\Delta_r\) denotes the \(r\)-Laplacian for \((1,+\infty)\) and the reaction has the competing effects of two distinct nonlinearities, namely, a term \(f\colon \Omega \times\mathbb{R}\to\mathbb{R}\) which is \((p-1)\)-superlinear (convex term) in the second variable and a perturbation \(g\colon \Omega \times\mathbb{R}\to\mathbb{R}\) which is \((p-1)\)-sublinear (concave term) in the second variable. In the first part it is shown that problem P\(_{\lambda}\) has at least five nontrivial smooth solutions (all with sign information) provided the parameter \(\lambda>0\) is small enough. In the second part, by strengthening the regularity of the two nonlinearities, the authors prove the existence of two more nodal solutions, for a total of seven nontrivial smooth solutions all with sign informations.
reviewed by: Patrick Winkert
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 7187813
zbMATH Keywords: convex-concave problem
zbMATH Keywords: nonlinear regularity
zbMATH Keywords: strong comparison principle
zbMATH Keywords: constant sign and nodal solutions
zbMATH Keywords: critical groups
zbMATH Keywords: flow invariance
MaRDI profile type: Publication
full work available at URL: j.na.2019.111716
OpenAlex ID: W2992468615
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DOI: 10.1016/J.NA.2019.111716
description: scientific article; zbMATH DE number 7187813