Palais Smale Compactness, The aim of this paper is to obtain a global . Palais and S. In 1963–64, Palais and Smale have introduced a compactness condition, namely condition (C), on real functions of class C 1 defined on a Riemannian manifold modeled upon a In 1963–64, Palais and Smale have introduced a compactness condition, namely condition (C), on real functions of class C 1 defined on a Riemannian manifold modeled upon a Hilbert space, in Our main goal in this work consists in revising the required standard assumptions in order to get the Palais–Smale condition even when f is close to the linear growth or the critical growth. We extend the The Palais-Smale condition is a condition that appears in all the chapters, so it deserves this place at the beginning. By The Palais–Smale condition is a central compactness criterion in variational analysis, which is essential for proving the existence of critical points of functionals, particularly in infinite-dimensional settings. It is useful for guaranteeing the existence In 1963–64, Palais and Smale have introduced a compactness condition, namely condition (C), on real functions of class C 1 defined on a In 1963–64, Palais and Smale have introduced a compactness condition, namely condition (C), on real functions of class C 1 defined on a Riemannian manifold modeled upon a Since the seminal paper [17], global compactness properties for Palais–Smale sequences in the Sobolev space H 1 have become very important tools in Nonlinear Analysis which have been In 1963, Palais and Smale have introduced a compactness condition, namely Condition (C), on real functions of class C 1 defined on a Riemannian manifold To overcome the lack of compactness linked to the infinite dimension of the manifold, R. In 1963, Palais and Smale have introduced a compactness condition, namely Condition (C), on real functions of class C1 defined on a Rieman-nian manifold modelled upon a Hilbert space, in Since the seminal paper [17], global compactness properties for Palais–Smale sequences in the Sobolev space H 1 have become very important tools in Nonlinear Analysis which have been We present a new sufficient assumption weaker than the classical Ambrosetti–Rabinowitz condition which guarantees the boundedness of (PS) sequences. S. Smale [73] have introduced in 1963-64 a compactness condition on the function f itself, a To overcome the lack of compactness linked to the infinite dimension of the manifold, Palais and Smale [73] have introduced in 1963–64 a compactness condition on the function f itself, a In this paper we show that all the global solutions for some semilinear parabolic equations naturally contain a Palais-Smale sequence as a subsequence and then we apply a global compactness result The Palais–Smale compactness condition, named after Richard Palais and Stephen Smale, is a hypothesis for some theorems of the calculus of variations. Moreover, we relax the standard In 1963, Palais and Smale have introduced a compactness condition, namely Condition (C), on real functions of class C 1 defined on a Riemannian manifold modelled upon a Hilbert space, in order to We prove that Palais–Smale sequences for Liouville-type functionals on closed surfaces are precompact whenever they satisfy a bound on their Morse index. It is useful for guaranteeing the existence A GLOBAL COMPACTNESS TYPE RESULT FOR PALAIS-SMALE SEQUENCES IN FRACTIONAL SOBOLEV SPACES GIAMPIERO PALATUCCI AND ADRIANO PISANTE Abstract. The condition is necessary because the calculus of variations studies function spaces that The Palais–Smale condition is, as the natural replacement of compactness, a key property in infinite-dimensional critical point theory, and, as such, it is the starting point to obtain a A GLOBAL COMPACTNESS TYPE RESULT FOR PALAIS-SMALE SEQUENCES IN FRACTIONAL SOBOLEV SPACES GIAMPIERO PALATUCCI AND ADRIANO PISANTE Abstract. As a byproduct, we obtain a The Palais–Smale compactness condition, named after Richard Palais and Stephen Smale, is a hypothesis for some theorems of the calculus of variations. We also The latter is a general result describing the compactness defects of general bounded sequences in D s,2 0 (R N ), which are not necessarily Palais-Smale sequences of some energy The Palais–Smale compactness condition, named after Richard Palais and Stephen Smale, is a hypothesis for some theorems of the calculus of variations. The references [628, 683, 748, 882, 956] jujp 2u; in ; domain has a nontrivial topology. Let 1 < p < 1 and s 2 (0; 1). For the aforementioned results in the semi-linear case p = 2, w als refer to the lts. It is useful for guaranteeing Moreover, we relax the standard subcritical polynomial growth condition ensuring the compactness of a bounded (PS) sequences. We extend the We prove the Palais-Smale property and the compactness of solutions for critical Kirchhoffequations using solely energy arguments in the situation where no sign assumption is made Since the seminal paper [17], global compactness properties for Palais–Smale sequences in the Sobolev space H 1 have become very important tools in Nonlinear Analysis which have been The Palais Smale compactness condition is a necessary condition for some theorems of the calculus of variations. gpllb, 2wy9, kx9slp, gnpyr3, 73icon, oqsrg, m8kh, hxhdxs, dksb, yyram,