Topic
Palais–Smale compactness condition
About: Palais–Smale compactness condition is a research topic. Over the lifetime, 46 publications have been published within this topic receiving 8677 citations.
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TL;DR: In this paper, general existence theorems for critical points of a continuously differentiable functional I on a real Banach space are given for the case in which I is even.
4,081 citations
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01 Jul 1986TL;DR: The mountain pass theorem and its application in Hamiltonian systems can be found in this paper, where the saddle point theorem is extended to the case of symmetric functionals with symmetries and index theorems.
Abstract: An overview The mountain pass theorem and some applications Some variants of the mountain pass theorem The saddle point theorem Some generalizations of the mountain pass theorem Applications to Hamiltonian systems Functionals with symmetries and index theorems Multiple critical points of symmetric functionals: problems with constraints Multiple critical points of symmetric functionals: the unconstrained case Pertubations from symmetry Variational methods in bifurcation theory.
3,685 citations
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TL;DR: In this paper, the authors derive a generic theorem for a wide class of functionals, having a mountain pass geometry, and show how to obtain, for a given functional, a special Palais-Smale sequence possessing extra properties that help to ensure its convergence.
Abstract: Using the ‘monotonicity trick’ introduced by Struwe, we derive a generic theorem. It says that for a wide class of functionals, having a mountain-pass (MP) geometry, almost every functional in this class has a bounded Palais-Smale sequence at the MP level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais–Smale sequence possessing extra properties that help to ensure its convergence. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the formWe assume that the functional associated to (P) has an MP geometry. Our results cover the case where the nonlinearity f satisfies (i) f(x, s)s−1 → a ∈)0, ∞) as s →+∞; and (ii) f(x, s)s–1 is non decreasing as a function of s ≥ 0, a.e. x → ℝN.
815 citations
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TL;DR: In this article, the nature of critical points generated by Ghoussoub's type min-max principle for locally Lipschitz continuous functionals fulfilling a weak Palais-Smale assumption, which contains the so-called non-smooth Cerami condition, is investigated.
Abstract: The nature of critical points generated by a Ghoussoub's type min–max principle for locally Lipschitz continuous functionals fulfilling a weak Palais–Smale assumption, which contains the so-called non-smooth Cerami condition, is investigated. Two meaningful special cases are then pointed out; see Theorems 3.5 and 3.6.
205 citations
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TL;DR: In this article, the existence of a local minimum for a continuously Gâteaux differentiable function, possibly unbounded from below, without requiring any weak continuity assumption, is established and a novel definition of Palais-Smale condition is presented.
Abstract: The aim of this paper is to establish the existence of a local minimum for a continuously Gâteaux differentiable function, possibly unbounded from below, without requiring any weak continuity assumption. Several special cases are also emphasized. Moreover, a novel definition of Palais–Smale condition, which is more general than the usual one, is presented and a mountain pass theorem is pointed out. As a consequence, multiple critical points theorems are then established. Finally, as an example of applications, an elliptic Dirichlet problem with critical exponent is investigated.
167 citations