On a fourth-order quasilinear elliptic equation of concave-convex type
type: Item
instance of: scholarly article
title: On a fourth-order quasilinear elliptic equation of concave-convex type
zbMATH Open document ID: 1179.35130
author: Ederson Moreira dos Santos
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2009-09-02T00:00:00Z
review text: Let \(\Omega\subset\mathbb R^N\), \(N\geq1\), be a bounded domain with a smooth boundary. This paper deals with the following system: \[ \begin{cases} -\Delta\big((-\Delta u)^{\frac{1}{p}}\big)=u^q+\lambda g(x)u^r &\text{in }\Omega,\\ u,-\Delta u=0 &\text{ on }\partial \Omega, \end{cases}\tag{1} \] where \(u^q=|u|^{q-1}u,u^r=|u|^{r-1}u\), and \((-\Delta u)^{\frac{1}{p}}=|-\Delta u| ^{\frac{1}{p}-1}(-\Delta u)\), \(\lambda\geq0\) is a parameter and \(g\in C^1(\overline{\Omega})\) is positive inside \(\Omega\), under the hypotheses \(H_1: 0<r<\frac{1}{p}<q\) and \(H_2:\frac{1}{p+1}+\frac{1}{q+1}\geq 1-\frac{2}{N}\). When \(\lambda\) is small enough, the author studies, by variational methods, the existence of at least two positive and infinitely many solutions to (1). The critical case, when the identity occurs in \(H_2\), and the subcritical case, which corresponds to the strict inequality in \(H_2\), are investigated separately. Some open problems are mentioned at the end of the paper.
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 35B09
zbMATH DE Number: 5601462
zbMATH Keywords: fourth-order quasilinear equation
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: variational methods
reviewed by: Denise Huet
MaRDI profile type: Publication
full work available at URL: s00030-009-0024-y
OpenAlex ID: W1995312638
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DOI: 10.1007/S00030-009-0024-Y
description: scientific article; zbMATH DE number 5601462