Critical groups computations on a class of Sobolev Banach spaces via Morse index
type: Item
instance of: scholarly article
title: Critical groups computations on a class of Sobolev Banach spaces via Morse index
zbMATH Open document ID: 1023.58004
DOI: 10.1016/S0294-1449(02)00011-2
author: Silvia Cingolani
author: Giuseppina Vannella
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2003-05-19T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78579
review text: The aim of this study is to estimate the critical groups of some functionals associated to a class of quasilinear elliptic problems, involving \(p\)-Laplacian. So, a class of Banach spaces which are the natural variational setting for several quasilinear elliptic problems is investigated. Precisely, the authors work on the Sobolev space \(W^{1,p}_0(\Omega)\) where \(2< p<\infty\), and \(\Omega\) is a bounded domain of \(\mathbb{R}^N\) \((N\geq 1)\), with sufficiently regular boundary \(\partial\Omega\). The functional \(f: W^{1,p}_0(\Omega)\to \mathbb{R}\) defined by setting \[ f(u)= {1\over p} \int_\Omega|\nabla u|^p dx+{1\over 2} \int_\Omega|\nabla u|^2 dx+\int_\Omega G(u) dx, \] where \(G(t)= \int^t_0 g(s) ds\) and \(g\in C^1(\mathbb{R}, \mathbb{R})\) satisfies the following assumption: \(|g'(t)|\leq c_1|t|^q+ c_2\) with \(c_1,c_2\) positive constants and \(0\leq q< p^*- 2\) if \(N> p\), while \(q\) is any positive number if \(N= p\). Otherwise, if \(N< p\), no restrictive assumption on the growth of \(g\) is required. The critical points of \(f\) in \(W^{1,p}_0(\Omega)\) correspond to the weak solution of the quasilinear elliptic problem \[ \begin{cases} -\Delta_p u-\Delta u+ g(u)= 0\quad &\text{in }\Omega,\\ u= 0\quad &\text{on }\partial\Omega,\end{cases} \] where \(\Delta_p u=\text{div}(|\nabla u|^{p-2}\nabla u)\), \(p> 2\), arising in the mathematical desription of propagation phenomena of solitary waves. In spite of the difficulties related to the fact that any critical point \(u\) of \(f\) is degenerate and \(f''(u)\) is not a Fredholm operator, so that the generalized splitting Morse lemma cannot be applied in order to describe the behavior of \(f\) near the critical points, the authors obtain critical groups estimates for functionals \(f\) in \(u\), in the spirit of the generalized Morse lemma.
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 1912451
zbMATH Keywords: Morse theory
zbMATH Keywords: critical groups estimate
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: weak solution
zbMATH Keywords: quasilinear elliptic equation
MaRDI profile type: Publication
cites work: Critical points for multiple integrals of the calculus of variations
cites work: Solitons in several space dimensions: Derrick's problem and infinitely many solutions
cites work: Soliton Like Solutions of a Lorentz Invariant Equation in Dimension 3
cites work: Morse theory on Branach space and its applications to partial differential equations
cites work: Infinite dimensional Morse theory and multiple solution problems
cites work: Q4497346
cites work: Some results on critical groups for a class of functionals defined on Sobolev Banach spaces
cites work: Nontrivial solutions of quasilinear equations via nonsmooth Morse theory
cites work: C1 + α local regularity of weak solutions of degenerate elliptic equations
cites work: Existence and nonexistence results for m-Laplace equations involving critical Sobolev exponents
cites work: Q4040640
cites work: On Lower Semicontinuity of Integral Functionals. I
cites work: Linear and quasilinear elliptic equations
cites work: Morse index estimates for continuous functionals associated with quasilinear elliptic equations.
cites work: Critical point theory and Hamiltonian systems
cites work: Q4086468
cites work: Morse theory on Hilbert manifolds
cites work: Morse theory and a non-linear generalization of the Dirichlet problem
cites work: Q5522742
cites work: Regularity for a more general class of quasilinear equations
cites work: On The Dirichletproblem for Quasilinear Equations
cites work: A general approach to Morse theory
cites work: Morse theory on Banach manifolds
OpenAlex ID: W2011950750
description: scientific article; zbMATH DE number 1912451