Multiple critical points for nondifferentiable functionals involving Hardy potentials
type: Item
instance of: scholarly article
title: Multiple critical points for nondifferentiable functionals involving Hardy potentials
zbMATH Open document ID: 1120.49013
author: Benedetta Pellacci
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-05-12T00:00:00Z
review text: The author studies general functionals of the calculus of variations with the presence of the Hardy potentials, improving many results obtained in the semilinear framework. The functional \(J\) is defined on the Sobolev space \(W^{1,p}_0 (\Omega)\), where \(\Omega\) is a bounded subset of \({\mathbb R}^N\), \(N \geq 2\), and it is defined in the following way: \[ J(v) = \int_\Omega j(x,v,\nabla v) - {\lambda \over p} \int_\Omega {{|v|^p}\over {|x|^p}} - \int_\Omega G(x,v), \] where \(1 < p < N\), \(j(x,s,\xi) \colon \Omega \times {\mathbb R} \times {\mathbb R}^N \rightarrow {\mathbb R}\) is a measurable function with respect to \(x\) for any \((s,\xi) \in {\mathbb R} \times {\mathbb R}^N\), and of class \(C^1\) with respect to \((s,\xi)\) for almost everywhere \(x \in \Omega\), and \(G \colon \Omega \times {\mathbb R} \rightarrow {\mathbb R}\) is a continuous function. The author first proves a general weak lower semicontinuity result for the functional \(J\), which will imply the existence of a critical point whenever the functional \(J\), under suitable assumptions on \(J\) and \(G\), is coercive. Moreover existence and multiplicity results for critical points of \(J\) are obtained, even if the functional is nonsmooth. The author apply a nonsmooth critical point theory developed in \textit{J.-N. Corvellec} et al. [Topol. Methods Nonlinear Anal. 1, No. 1, 151--171 (1993; Zbl 0789.58021)] and \textit{M. Degiovanni} and \textit{M. Marzocchi} [ Ann. Mat. Pura Appl., IV. Ser. 167, 73--100 (1994; Zbl 0828.58006)].
reviewed by: Antonio Masiello
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 2167830
zbMATH Keywords: Hardy inequality
zbMATH Keywords: Nonsmooth critical point theory
MaRDI profile type: Publication
full work available at URL: j.na.2004.11.015
OpenAlex ID: W2039702221
cites work: Existence and nonexistence results for quasilinear elliptic equations involving the \(p\)-Laplacian with a critical potential
cites work: Dual variational methods in critical point theory and applications
cites work: Critical points for multiple integrals of the calculus of variations
cites work: Some remarks on critical point theory for nondifferentiable functionals
cites work: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations
cites work: Q4440497
cites work: Q4538008
cites work: Extremal functions for Hardy's inequality with weight
cites work: Q4378149
cites work: Multiplicity of solutions for quasilinear elliptic equations
cites work: Q4860444
cites work: Deformation properties for continuous functionals and critical point theory
cites work: A critical point theory for nonsmooth functionals
cites work: Multiple solutions of hemivariational inequalities with area-type term
cites work: Hardy inequalities and some critical elliptic and parabolic problems
cites work: Some remarks on bifurcation.
cites work: Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents
cites work: Metric critical point theory. I: Morse regularity and homotopic stability of a minimum
cites work: Mountain pass theorems and global homeomorphism theorems
cites work: Multiple solutions of quasilinear equations involving an area-type term
cites work: A simple approach to Hardy inequalities
cites work: Nonlinear eigenvalue problems for quasilinear operators on unbounded domains
cites work: Q4359608
cites work: Unbounded critical points for a class of lower semicontinuous functionals
cites work: Weak solutions to general Euler's equations via nonsmooth critical point theory
cites work: Q4708871
cites work: The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential
DOI: 10.1016/J.NA.2004.11.015
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description: scientific article; zbMATH DE number 2167830