A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of X = OP1(−1,−1) to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero, one boundary component open Gromov-Witten invariants for Lagrangian submanifolds LK ⊂ X obtained from the conormal bundles of knots K ⊂ S3. This computation is then performed for two...

Abstract We show continuity under equivariant Gromov–Hausdorff convergence of the critical exponent discrete, non-elementary, torsion-free, quasiconvex-cocompact groups with uniformly bounded codiameter acting on Gromov-hyperbolic metric spaces.

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...

Let X be a real Banach space and Φ ∈ C 1 (X, R) a function with a mountain pass geometry. This ensures the existence of a Palais-Smale, and even a Cerami, sequence {u n } of approximate critical points for the mountain pass level. We obtain information about the location of such a sequence by estimating the distance of u n from S for certain types of set S as n → ∞. Under our hypotheses we can ...

The main purpose of this paper is to establish a three critical points result without assuming the coercivity of the involved functional. To this end, a mountain-pass theorem, where the usual Palais-Smale condition is not requested, is presented. These results are then applied to prove the existence of three solutions for a two-point boundary value problem with no asymptotic conditions.

where Ω ⊂ R is a smooth domain with smooth boundary ∂Ω such that 0 Î Ω, Δpu = div(|∇u|∇u), 1 < p < N, μ < μ̄ = ( N−p p ), l >0, 1 < q < p, sign-changing weight functions f and g are continuous functions on ̄, μ̄ = ( N−p p ) p is the best Hardy constant and p∗ = Np N−p is the critical Sobolev exponent. By extracting the Palais-Smale sequence in the Nehari manifold, the multiplicity of positive solu...

We show the existence of the nontrivial periodic solution for a class of the system of the nonlinear suspension bridge equations with Dirichlet boundary condition and periodic condition by critical point theory and linking arguments. We investigate the geometry of the sublevel sets of the corresponding functional of the system, the topology of the sublevel sets and linking construction between ...

We prove the Morse relations for all geodesics connecting two non-conjugate points on a class of globally hyperbolic Lorentzian manifolds. We overcome the difficulties coming from the fact that the Morse index of every geodesic is infinite, and from the lack of the Palais-Smale condition, by using the Morse complex approach. Introduction Let M be a smooth connected manifold without boundary of ...