Multiplicity of positive solutions for eigenvalue problems of \((p,2)\)-equations
type: Item
instance of: scholarly article
title: Multiplicity of positive solutions for eigenvalue problems of \((p,2)\)-equations
zbMATH Open document ID: 1297.35161
DOI: 10.1186/1687-2770-2012-152
author: Leszek Gasiński
author: Nikolaos S. Papageorgiou
published in: Boundary Value Problems
publication date: 2013-11-28T00:00:00Z
review text: Let \(\Omega\subseteq\mathbb R^N\) be a bounded domain with a \(C^2\)-boundary \(\partial\Omega\). This article is devoted to a nonlinear parametric Dirichlet eigenvalue problem \[ \left\{ {-\Delta_p u(z)-\Delta u(z) = \lambda f(z,u(z)) \qquad \text{in} \; \Omega},\atop {u|_{\partial\Omega}=0,\quad u>0,\quad \lambda>0, \quad 2<p<+\infty.} \right. \eqno(P)_{\lambda} \] Here, \(-\Delta_p\) denotes the \(p\)-Laplacian differential operator defined by \[ -\Delta_p u(z)=\text{div}(\|\nabla u(z)\|^{p-2}\nabla u(z))\quad \forall u\in W_0^{1,p}(\Omega). \] The aim of this paper is to determine the precise dependence of the set of positive solutions on the parameter \(\lambda>0\). The hypotheses H on the reaction \(f\) are the following: \(f:\Omega\times\mathbb R\longrightarrow\mathbb R\;\) is a Carathéodory function such that \(\,f(z,0)=0\;\) for almost all \,\(z\in\Omega\), \(f(z,\zeta)\geq 0\;\) for almost all \(z\in\Omega\) and all \(\zeta\geq 0\), and (i) \,\, for every \(\varrho>0\), there exists \(a_{\varrho}\in L^{\infty}(\Omega)\), such that \[ f(z,\zeta)\leq a_{\varrho}(z)\quad \text{for almost all } z\in\Omega, \text{ all } \zeta\in [0,\varrho]; \] (ii) \, \(\lim_{\zeta\to+\infty}{f(z,\zeta) \over \zeta^{p-1}}=0\) uniformly for almost all \(z\in\Omega\); (iii) \(\lim_{\zeta\to 0^+}{f(z,\zeta) \over \zeta^{p-1}}=0\) uniformly for almost all \(z\in\Omega\); (iv) for every \(\varrho>0\), there exists \(\;\xi_{\varrho}>0\;\) such that for almost all \(z\in\Omega\), the map \(\zeta\mapsto f(z,\zeta)+\xi_{\varrho}\zeta^{p-1}\;\) is nondecreasing on \([0,\varrho]; \) (v) \, \, if \[ F(z,\zeta)= \int_0^{\zeta} f(z,s)\, ds, \] \(\qquad \quad \) then there exists \(c\in R\) such that \[ F(z,c)>0\quad \text{for almost all } z\in\Omega . \] Let \[ C_+=\{ u\in C_0^1(\overline{\Omega}): u(z)\geq 0 \text{ for all } z\in \overline{\Omega} \}, \] \[ {\mathcal Y}=\{\lambda>0 : \text{problem \((P)_{\lambda}\) has a nontrivial positive solution} \} \] and let \(S(\lambda)\) be the se…
reviewed by: Petr Tomiczek
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35B51
zbMATH DE Number: 6234139
zbMATH Keywords: nonlinear parametric Dirichlet eigenvalue problem
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: positive solutions
zbMATH Keywords: variational method
zbMATH Keywords: tangency principle
zbMATH Keywords: bifurcation-type theorem
Wikidata QID: Q59288160
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1186/1687-2770-2012-152
OpenAlex ID: W2030506947
cites work: Multiple positive solutions for a class of semilinear elliptic boundary value problems
cites work: Q5680495
cites work: Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems
cites work: Multiple positive solutions for nonlinear eigenvalue problems with the \(p\)-Laplacian
cites work: Solitons in several space dimensions: Derrick's problem and infinitely many solutions
cites work: Nontrivial Solutions forp-Laplace Equations with Right-Hand Side Havingp-Linear Growth at Infinity
cites work: Multiplicity of solutions for a class of the quasilinear elliptic equations at resonance
cites work: Origin and evolution of the Palais-Smale condition in critical point theory
cites work: Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential
cites work: The Ambrosetti–Prodi Problem for thep-Laplace Operator
cites work: Boundary regularity for solutions of degenerate elliptic equations
cites work: Nodal and multiple constant sign solutions for resonant \(p\)-Laplacian equations with a nonsmooth potential
description: scientific article; zbMATH DE number 6234139