We present instances of the following phenomenon: if a product of topological spaces satisfies some given compactness property then the factors satisfy a stronger compactness property, except possibly for a small number of factors. The first known result of this kind, a consequence of a theorem by A. H. Stone, asserts that if a product is regular and Lindelöf then all but at most countably many...

We introduce compactness theorems for generalized colorings and derive several particular compactness theorems from them. It is proved that the theorems and many of their consequences are equivalent in ZF set theory to BPI, the Prime Ideal Theorem for Boolean algebras. keywords: generalized graph colorings, compactness, prime ideal theorem MSC: 05C15; 03E25 .

let $t$ be a bounded operator on the banach space $x$ and $ph$ be an analytic self-map of the unit disk $bbb{d}$‎. ‎we investigate some operator theoretic properties of‎ ‎bilateral composition operator $c_{ph‎, ‎t}‎: ‎f ri t circ f circ ph$ on the vector-valued hardy space $h^p(x)$ for $1 leq p leq‎ ‎+infty$.‎ ‎compactness and weak compactness of $c_{ph‎, ‎t}$ on $h^p(x)$‎ ‎are characterized an...

Compactness in the space L(0, T ;B), B being a separable Banach space, has been deeply investigated by J.P. Aubin (1963), J.L. Lions (1961,1969), J. Simon (1987), and, more recently, by J.M. Rakotoson and R. Temam (2001), who have provided various criteria for relative compactness, which turn out to be crucial tools in the existence proof of solutions to many abstract time dependent problems re...

A. We consider the semi-linear elliptic PDEs driven by the fractional Laplacian: { (−∆)su = f (x, u), in Ω, u = 0, in Rn\Ω. By the Mountain Pass Theorem and some other nonlinear analysis methods, the existence and multiplicity of non-trivial solutions for the above equation are established. The validity of the Palais-Smale condition without AmbrosettiRabinowitz condition for non-local el...

We construct local minimum solutions for the semilinear bistable equation by minimizing the corresponding functional near some approximate solutions, under the hypothesis that certain global minimum solutions are isolated. The key is a certain characterization of Palais-Smale sequences and a proof that the functional takes higher values away from the approximate solutions.

In this paper, we study the bubbling phenomena of weak solution sequences of a class of degenerate quasilinear elliptic systems of m-harmonic type. We prove that, under appropriate conditions, the energy is preserved during the bubbling process. The results apply to m-harmonic maps from a manifold Ω to a homogeneous space, and to m-harmonic maps with constant volumes, and also to certain Palais...

It is shown that the nonhomogeneous Dirichlet and Neuman problems for the 2nd-order Seiberg-Witten equation on a compact 4-manifold X admit a regular solution once the nonhomogeneous Palais-Smale condition is satisfied. The approach consists in applying the elliptic techniques to the variational setting of the Seiberg-Witten equation. The gauge invariance of the functional allows to restrict th...

We consider a class of variational systems in R N of the form where a; b : R N ! R are continuous functions which are coercive; i.e., a(x) and b(x) approach plus innnity as x approaches plus innnity. Under appropriate growth and regularity conditions on the nonlinearities Fu(:) and Fv (:), the (weak) solutions are precisely the critical points of a related functional deened on a Hilbert space o...

Based on the shaking table test, liquefaction of silt under earthquake load and damage after are studied. The test results show that lower is relatively dense, pore pressure increases slowly, ratio can not reach initial state. However, stronger action, dense may also produce liquefaction, which indicates increase water has a great relationship with compactness intensity load. Because permeabili...