Volume 15, Issue 3

Let α be a contact form on S 3 , let ξ be its Reeb vector-field and let v be a non-singular vector-field in ker α. Let C β be the space of curves x on S 3 such ẋ = aξ + bv, ȧ = 0, a ≩ 0. Let L + , respectively L − , be the set of curves in C β such that b ≥ 0, respectively b ≤ 0. Let, for x ∈ C β , J(x) = ∫ 0 1 α x (ẋ)dt. The framework of the present paper has been introduced previously in eg [3]. We establish in this paper that some cycles (an infinite number of them, indexed by odd integers, tending to ∞) in the S 1 - equivariant homology of C β , relative to L + ∪ L − and to some specially designed ”bottom set”, see section 4, are achieved in the Morse complex of (J,C β ) by unions of unstable manifolds of critical points (at infinity)which must include periodic orbits of ξ; ie unions of unstable manifolds of critical points at infinity alone cannot achieve these cycles. At the odd indexes (2k−1) = 1+(2k−2), 1 for the linking, (2k−2) for the S 1 -equivariance, we find that the equivariant contributions of a critical point at infinity to L + and to L − are fundamentally asymmetric when compared to those of a periodic orbit [5]. The topological argument of existence of a periodic orbit for ξ turns out therefore to be surprisingly close, in spirit, to the linking/equivariant argument of P. Rabinowitz in [12]; e.g. the definition of the ”bottom sets” of section 4 can be related in part to the linking part in the argument of [12]. The objects and the frameworks are strikingly different, but the original proof of [12] can be recognized in our proof, which uses degree theory, the Fadell-Rabinowitz index [8] and the fact that π n+1 (S n ) = Z 2 , n ≥ 3. We need of course to prove, in our framework, that these topological classes cannot be achieved by critical points at infinity only, periodic orbits of ξ excluded, and this is the fundamental difficulty. The arguments hold under the basic assumption that no periodic orbit of index 1 connects L + and L − . It therefore follows from the present work that either a periodic orbit of index 1 connects L + and L − (as is probably the case for all three dimensional over-twisted [8] contact forms, see the work of H. Hofer [10], the periodic orbit found in [10] should be of index 1 in the present framework); or (with a flavor of exclusion in either/or) a linking/ equivariant variational argument a la P. Rabinowitz [12] can be put to work. Existence of (possibly multiple) periodic orbits of ξ, maybe of high Morse index, follows then. Therefore, to a certain extent, the present result runs, especially in the case of threedimensional over-twisted [8] contact forms, against the existence of non-trivial algebraic invariants defined by the periodic orbits of ξ and independent of what ker α and/or α are.

We consider the problem of attainability of the best constant C > 0 in the following critical fractional Hardy-Sobolev inequality: For all , , where and γ ∈ ℝ. This allows us to establish the existence of nontrivial weak solutions for the following doubly critical problem on ℝ n , , where is the critical α-fractional Sobolev exponent, and , the latter being the best fractional Hardy constant on ℝ n .

In this paper we characterize the dynamics of a generalized version of the model of Belgacem and Cosner [5] within the range of values of the parameters where the trivial steady state is unstable and the problem cannot admit a positive steady state. In such regimes, the dynamics is governed by the metasolutions.

In this paper, we shall prove even symmetry of monotone entire solutions to the balanced Allen-Cahn equation with one spatial variable. Related results for the unbalanced Allen- Cahn equation are also discussed.

We shall be concerned with the buckling of a thin circular elastic plate simply supported along a boundary, subjected to a radial compressive load uniformly distributed along its boundary. One of the main engineering concerns is to reduce deformations of plate structures. It is well known that von Kármán equations provide an established model that describes nonlinear deformations of elastic plates. Our approach to study plate deformations is based on bifurcation theory. We will find critical values of the compressive load parameter by reducing von Kármán equations to an operator equation in Hölder spaces with a nonlinear Fredholm map of index zero. We will prove a sufficient condition for bifurcation by the use of a gradient version of the Crandall-Rabinowitz theorem due to A.Yu. Borisovich and basic notions of representation theory. Moreover, applying the key function method by Yu.I. Sapronov we will investigate the shape of bifurcation branches.

We consider Dirichlet p-Laplacian equations which may be resonant with respect to the principal eigenvalue at ±∞. We show the existence of extremal nontrivial constant sign solutions and of nodal solutions. In the semilinear case (p = 2) we produce additional nodal solutions. We show that certain parametric equations (eigenvalue problems) studied in the past, are a special case of our multiplicity theorem. Finally, we establish the stability of the extremal solutions.

Following the constructive method of Wei and Yan [29], with new ingredients to take care of high-space dimensions, we prove the existence of infinitely many solutions of the nonlinear biharmonic equation in the unit ball of ℝ n (n ≥ 6, α > 0) with the Navier conditions v = Δv = 0 on the boundary.

In this paper we study the existence of weak positive solutions for the following class of quasilinear Schrödinger equations −Δu + V(x)u − [Δ(u 2 )]u = h(u) in ℝ N , where h satisfies some “mountain-pass” type assumptions and V is a nonnegative continuous function. We are interested specially in the case where the potential V is neither bounded away from zero, nor bounded from above. We give a special attention to the case when V may eventually vanish at infinity. Our arguments are based on penalization techniques, variational methods and Moser iteration scheme.

We consider a nonlinear parametric Robin problem driven by the p-Laplacian. We assume that the reaction exhibits a concave term near the origin. First we prove a multiplicity theorem producing three solutions with sign information (positive, negative and nodal) without imposing any growth condition near ±∞ on the reaction. Then, for problems with subcritical reaction, we produce two more solutions of constant sign, for a total of five solutions. For the semilinear problem (that is, for p = 2), we generate a sixth solution but without any sign information. Our approach is variational, coupled with truncation, perturbation and comparison techniques and with Morse theory.