Volume 2, Issue 3

We extend to the CR framework an equivalent problem to the well known Kazdan- Warner problem on Riemannian manifolds. Let (M, θ) be a three-dimensional pseudo Hermitian CR compact manifold, locally conformally CR equivalent to the sphere S 3 of C 4 . We give suitable conditions on a C 3 positive function K, defined on M, to be the Webster scalar curvature for a contact form on M conformal to θ.

This work is devoted to study the existence of solutions to equations of the p Laplacian type in unbounded domains. We prove the existence of at least one solution, and under further assumptions, the existence of infinitely many solutions. We apply the mountain pass theorem in weighted Sobolev spaces.

The paper is concerned with the existence of nontrivial solutions of the obstacle problem: where K = {υ ∈ H 1 0 (Ω) : υ ≤ ψ a.e. on Ω}. By using a generalized mountain pass theorem for inequalities, we prove, subject to some restrictions on the obstacle ψ, the existence of nontrivial solutions of the above inequality.

We deal with the Dirichlet boundary value problem associated to a parameter-dependent second order vector differential equation. Using the method of lower and upper solutions together with degree theory, we provide existence and multiplicity of positive solutions.

The paper deals with the existence of twist solutions of a Hill's equation with singular term (often called Brillouin equation in the related literature) for a given region of parameters involved on the equation. Solutions of twist type are in particular Lyapunov stable and present interesting dynamical features around them. The techniques of proof include upper and lower solutions, topological degree and classical tools on normal forms of area preserving maps.

In this paper we consider the planar system x' = f (x, y), y' = -g(x, y), where f -1 (0) is the graph of a function F and g(x, y)x > 0 for x ≠ 0, and we study the intersection of the trajectories of the system with the curves y = F(x) and x = 0. We improve some results in the line of [I], [4] and [3].

Using some recent extensions of upper and lower solutions techniques and continuation theorems to the periodic solutions of quasilinear equations of p-Laplacian type, we prove the existence of positive periodic solutions of equations of the form (|xʹ| p-2 xʹ)ʹ + f(x)xʹ + g(x) = h(t) with p > 1, f arbitrary and g singular at 0. This extends results of Lazer and Solimini for the undamped ordinary differential case.

We study uniqueness for nonlinear ordinary differential equations arising in constructing blow-up and extinction self-similar solutions of various reaction-diffusion- absorption equations. Such particular similarity solutions describe the asymptotic singular behaviour of wide classes of general solutions of nonlinear heat equations. We prove that under some monotonicity assumptions, such similarity profiles are unique.