A new and simpler proof is given of the result of P. Rabinowitz for nontrivial time periodic solutions of a vibrating string equation utt-ux, + g(u) = 0 and Dirichlet boundary conditions on a finite interval. We assume essentially that g is nondecreasing, and g(u)/u+m as Iu(+w. The proof uses a modified form (PS), of the Palais-Smale condition (PS). Let g : R-+R be a continuous nondecreasing fu...

Let f be a positive smooth function on a close Riemann surface (M,g). The f − energy of a map u from M to a Riemannian manifold (N, h) is defined as E f (u) = M f |∇u| 2 dV g. In this paper, we will study the blow-up properties of Palais-Smale sequences for E f. We will show that, if a Palais-Smale sequence is not compact, then it must blows up at some critical points of f. As a sequence, if an...

In this paper, the notion of βS∗−compactness is introduced in L−fuzzy topological spaces based on S∗−compactness. A βS∗−compactness L-fuzzy set is S∗−compactness and also β−compactness. Some of its properties are discussed. We give some characterizations of βS∗−compactness in terms of pre-open, regular open and semi-open L−fuzzy set. It is proved that βS∗−compactness is a good extension of β−co...

Under generic assumptions, we prove boundedness of Palais-Smale sequences relative to some geometric functional defined on H(M), where M is a four-dimensional manifold. Our analysis is useful to find critical points (via minimax arguments) of this functional, which give rise to conformal metrics of constant Q-curvature. The proof is based on a refined bubbling analysis, for which the main estim...