Nontrivial solutions for critical potential elliptic systems
type: Item
instance of: scholarly article
title: Nontrivial solutions for critical potential elliptic systems
zbMATH Open document ID: 1218.35096
author: Ezequiel R. Barbosa
author: Marcos Montenegro
published in: Journal of Differential Equations
publication date: 2011-03-31T00:00:00Z
review text: Extensions of some results on Brezis-Nirenberg type problems are obtained. More precisely, the following potential elliptic system involving the \(p\)-Laplace operator is studied: \[ \begin{aligned} -\Delta_pu = f(u) + g(u) &\quad \text{in } \Omega\subset \mathbb{R}^n,\\ u= 0 &\quad \text{on } \partial\Omega, \end{aligned} \] where \(f=\frac{1}{P^*}\nabla F\), \(g = \frac{1}{p}\nabla G\), for \(F, G : \mathbb R^k \to \mathbb R\) of class \(C^1\) with \(F\) being positively homogeneous of degree \(p^*\) and \(G\) being homogeneous of degree \(p\). Necessary and sufficient conditions for the existence of a nontrivial solution to this system are presented. More precisely: {\parindent=6,5mm \begin{itemize}\item[(1)] If \(\Omega\) is star-shaped with smooth boundary and \(G\) is negative on \(\mathbb{R}^k\backslash\{0\}\), then the system has no nontrivial solution. \item[(2)] If \(n \geq p^2 > 1\), \(M_G = \max_{|t|_p=1}G(t) < \lambda_1\) and \(G\) is positive at a suitable point, then the system has a nontrivial solution. \item[(3)] If \(p^2>n>p>1\), \(M_G < \lambda_1\), and \(F, G\) satisfy suitable conditions at some point, then the system has a nontrivial solution. \end{itemize}}
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 35J91
Mathematics Subject Classification ID: 35J57
zbMATH DE Number: 5872522
zbMATH Keywords: Critical Sobolev exponents
zbMATH Keywords: potential systems
zbMATH Keywords: nonlinear eigenvalues
zbMATH Keywords: coercivity
reviewed by: Marlène Frigon
MaRDI profile type: Publication
full work available at URL: j.jde.2011.01.001
OpenAlex ID: W2088924260
cites work: Q4530288
cites work: Some results on \(p\)-Laplace equations with a critical growth term.
cites work: Existence and nonexistence results for critical growth polyharmonic elliptic systems
cites work: Existence of multiple solutions for quasilinear systems via fibering method
cites work: A Relation Between Pointwise Convergence of Functions and Convergence of Functionals
cites work: Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
cites work: An existence result for nonliner elliptic problems involving critical Sobolev exponent
cites work: Q4363056
cites work: Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results
cites work: Semilinear elliptic systems: A survey of superlinear problems
cites work: On Superquadratic Elliptic Systems
cites work: Systems ofp-laplacean equations involving homogeneous nonlinearities with critical sobolev exponent degrees
cites work: Linking solutions for \(p\)-Laplace equations with nonlinearity at critical growth
cites work: Semilinear elliptic equations involving critical Sobolev exponents
cites work: Q3778373
cites work: Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations
cites work: Q4040640
cites work: Q4223568
cites work: Quasilinear elliptic equations involving critical Sobolev exponents
cites work: Strongly indefinite systems with critical Sobolev exponents
cites work: Boundary regularity for solutions of degenerate elliptic equations
cites work: On nontrivial solutions of critical polyharmonic elliptic systems
cites work: Q3027501
cites work: Q3816412
cites work: Q3997570
cites work: The best sobolev constant
cites work: On The Dirichletproblem for Quasilinear Equations
cites work: Regularity for a more general class of quasilinear equations
cites work: Q5542608
cites work: A strong maximum principle for some quasilinear elliptic equations
DOI: 10.1016/J.JDE.2011.01.001
description: scientific article; zbMATH DE number 5872522