A note on a nonlinear elliptic problem with a nonlocal coefficient
type: Item
instance of: scholarly article
title: A note on a nonlinear elliptic problem with a nonlocal coefficient
zbMATH Open document ID: 1338.35172
author: Daisuke Naimen
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-12-21T00:00:00Z
review text: The author studies a nonlocal and nonlinear elliptic problem \[ \begin{cases} -a(\int_\Omega| \nabla u|^2 dx) \Delta u=\lambda u+u^p & \text{in } \Omega,\\ u=0 & \text{on }\partial\Omega \end{cases} \tag{P} \] where \(\Omega\subset \mathbb R^N\), \(N\geq 3\) is a bounded domain with smooth boundary \(\partial \Omega\), \(a\) is a nondegenerate continuous function, \(p>1\) and \(\lambda\in \mathbb R\). There are obtained several effects of the nonlocal coefficient \(a\) on the structure of the solution set of (P). Introducing a scaling observation and describing the solution set by using that of an associated semilinear problem the author shows unbounded continua of solutions \((\lambda, u)\). A rich variety of new bifurcation and multiplicity results are observed. It is also proved that the nonlocal coefficient can induce up to uncountably many solutions in a convenient way. There are given some remarks from the variational point of view.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 6522740
zbMATH Keywords: nonlocal problem
zbMATH Keywords: nonlinear elliptic problem
zbMATH Keywords: bifurcation
zbMATH Keywords: variational method
zbMATH Keywords: Kirchhoff type
reviewed by: Lubomira Softova
MaRDI profile type: Publication
full work available at URL: j.jmaa.2015.11.030
OpenAlex ID: W2219167160
cites work: An elementary proof of the uniqueness of positive radial solutions of a quasilinear Dirichlet problem
cites work: Uniqueness and nondegeneracy for some nonlinear elliptic problems in a ball.
cites work: On semilinear elliptic equations with indefinite nonlinearities
cites work: Positive solutions for a quasilinear elliptic equation of Kirchhoff type
cites work: Combined effects of concave and convex nonlinearities in some elliptic problems
cites work: Dual variational methods in critical point theory and applications
cites work: Global nonexistence for nonlinear Kirchhoff systems
cites work: Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
cites work: Q4735483
cites work: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions
cites work: Boundary layer solutions to functional elliptic equations
cites work: Some remarks on non local elliptic and parabolic problems
cites work: Remarks on a nonlocal problem involving the Dirichlet energy
cites work: On an elliptic equation of p-Kirchhoff type via variational methods
cites work: Global bifurcation and positive solution for a class of fully nonlinear problems
cites work: Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument
cites work: Study of a nonlinear Kirchhoff equation with non-homogeneous material
cites work: A critical Kirchhoff type problem involving a nonlocal operator
cites work: Symmetry and related properties via the maximum principle
cites work: On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN
cites work: Positive solutions to some equations with homogeneous operator
cites work: Existence of a positive solution to Kirchhoff-type problems without compactness conditions
cites work: Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior
cites work: The concentration-compactness principle in the calculus of variations. The limit case. I
cites work: On doubly nonlocal fractional elliptic equations
cites work: Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent
cites work: The critical problem of Kirchhoff type elliptic equations in dimension four
cites work: On the Brezis-Nirenberg Problem with a Kirchhoff Type Perturbation
cites work: Q3992017
cites work: Uniqueness of positive solutions of semilinear elliptic equations and related eigenvalue problems
cites work: Nontrivial solutions of Kirchhoff-type problems via the Yang index
cites work: Some global results for nonlinear eigenvalue problems
cites work: Uniqueness of solutions of nonlinear Dirichlet problems
cites work: Variational Methods
cites work: Existence and multiplicity of solutions for Kirchhoff type equations
cites work: On nonhomogeneous elliptic equations involving critical Sobolev exponent
DOI: 10.1016/J.JMAA.2015.11.030
description: scientific article; zbMATH DE number 6522740