A three critical points theorem revisited
type: Item
instance of: scholarly article
title: A three critical points theorem revisited
zbMATH Open document ID: 1214.47079
author: Biagio Ricceri
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-04-02T00:00:00Z
review text: Let \(X\) be a reflexive Banach space, \(I \subset\mathbb R\) an interval; \(\Phi: X \to\mathbb R\) a sequentially weakly lower semicontinuous \(C^1\) functional, bounded on each bounded subset of \(X\), whose derivative admits a continuous inverse on \(X^*\); \(J: X \to\mathbb R\) a \(C^1\) functional with compact derivative. Assume that \(\lim_{\| x\| \to \infty}(\Phi(x)+\lambda J(x))=+\infty\) for all \(\lambda \in I\), and there exists \(\rho \in\mathbb R\) such that \(\sup_{\lambda \in I}\inf_{x \in X}(\Phi(x)+\lambda(J(x)+\rho))<\inf_{x \in X}\sup_{\lambda \in I}(\Phi(x)+\lambda(J(x)+\rho))\). Then there exists a subset \(A \subset I\), \(A \neq \emptyset\), and \(r>0\) with the following property: for every \(\lambda \in A\) and every \(C^1\) functional \(\Psi: X \to\mathbb R\) with compact derivative, there exists \(\delta>0\) such that, for each \(\mu \in [0,\delta]\), the equation \(\Phi'(x)+\lambda J'(x)+\mu \Psi'(x)=0\) has at least three solutions in \(X\) whose norms are less than \(r\).
reviewed by: Mikhail Yu. Kokurin
Mathematics Subject Classification ID: 47J30
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 49J35
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 5539186
zbMATH Keywords: minimax theorems
zbMATH Keywords: critical points
zbMATH Keywords: multiplicity
MaRDI profile type: Publication
full work available at URL: j.na.2008.04.010
OpenAlex ID: W2018377463
cites work: Three solutions for a Dirichlet boundary value problem involving the \(p\)-Laplacian
cites work: Three solutions for a quasilinear boundary value problem
cites work: Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the \((p_1,\ldots ,p_n)\) -Laplacian
cites work: Existence of three positive solutions for some second-order boundary value problems
cites work: Q5699250
cites work: Existence of three solutions for a two point boundary value problem
cites work: A minimax inequality and its applications to ordinary differential equations
cites work: Q4434331
cites work: Some remarks on a three critical points theorem
cites work: Three solutions to a Neumann problem for elliptic equations involving the \(p\)-Laplacian
cites work: Multiplicity theorems for the Dirichlet problem involving the \(p\)-Laplacian.
cites work: On a class of nonlinear variational–hemivariational inequalities
cites work: Multiple solutions for a two-point boundary value problem
cites work: MULTIPLE SOLUTIONS FOR A QUASILINEAR ELLIPTIC VARIATIONAL SYSTEM ON STRIP-LIKE DOMAINS
cites work: MULTIPLICITY RESULTS FOR A PERTURBED NONLINEAR SCHRÖDINGER EQUATION
cites work: Q3509113
cites work: On a minimax problem of Ricceri
cites work: Three periodic solutions to an eigenvalue problem for a class of second-order Hamiltonian systems
cites work: Further results related to a minimax problem of Ricceri
cites work: Existence and multiplicity results for hemivariational inequalities with two parameters
cites work: Three solutions to Dirichlet boundary value problems for \(p\)-Laplacian difference equations
cites work: Multiplicity results for an eigenvalue problem for hemivariational inequalities in strip-like domains
cites work: EXISTENCE OF TWO NON-TRIVIAL SOLUTIONS FOR A CLASS OF QUASILINEAR ELLIPTIC VARIATIONAL SYSTEMS ON STRIP-LIKE DOMAINS
cites work: On a class of quasilinear eigenvalue problems in RN
cites work: Q3431879
cites work: Multiple solutions for elliptic problems with singular and sublinear potentials
cites work: Multiple solutions of a sublinear Schrödinger equation
cites work: Multiple solutions for \(p\)-Laplacian type equations
cites work: Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden–Fowler equations
cites work: Multiplicity results for a class of quasilinear eigenvalue problems on unbounded domains
cites work: Existence and multiplicity of solutions for fourth-order boundary value problems with param\-eters
cites work: Existence of three solutions for \(p(x)\)-Laplacian equations
cites work: Existence of three solutions for a quasilinear two point boundary value problem
cites work: On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems
cites work: Existence and multiplicity of solutions for a Neumann problem involving the \(p(x)\)-Laplace operator
cites work: A mountain pass theorem
cites work: On a three critical points theorem
cites work: Minimax theorems for limits of parametrized functions having at most one local minimum lying in a certain set
cites work: Q3528689
cites work: Q4454453
cites work: Multiplicity of periodic solutions for a boundary eigenvalue problem
cites work: Saddle point characterization and multiplicity of periodic solutions of non-autonomous second-order systems
cites work: On variational methods for a class of damped vibration problems
cites work: Q3706043
cites work: Three symmetric solutions for a class of elliptic equations involving the \(p\)-Laplacian with discontinuous nonlinearities in \(\mathbb R^{N}\)
DOI: 10.1016/J.NA.2008.04.010
description: scientific article; zbMATH DE number 5539186