Volume 13, Issue 2

We prove a global continuation result for T-periodic solutions of a T-periodic parametrized second order retarded functional differential equation on a boundaryless compact manifold with nonzero Euler-Poincaré characteristic. The approach is based on the fixed point index theory for locally compact maps on ANRs. As an application, we prove the existence of forced oscillations of retarded functional motion equations defined on topologically nontrivial compact constraints. This existence result is obtained under the assumption that the frictional coefficient is nonzero, and we conjecture that it is still true in the frictionless case.

The aim of this note is to present a generalization of the topological method for detecting chaotic dynamics in a periodic local processes generated by non-autonomous ordinary differential equations, based on the notion of periodic segments.

Let Ω be a bounded domain in ℝ N . In this paper, we consider the following nonlinear elliptic equation of N-Laplacian type: when f is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to (0.1) without the Ambrosetti-Rabinowitz (AR) condition. Earlier works in the literature on the existence of nontrivial solutions to N−Laplacian in ℝ N when the nonlinear term f has the exponential growth only deal with the case when f satisfies the (AR) condition. Our approach is based on a suitable version of the Mountain Pass Theorem introduced by G. Cerami [9, 10, 21]. Examples of f and comparison with earlier assumptions in the literature are given.

A Morse decomposition of a global attractor describes its internal dynamics, i.e., the dynamics on invariant compact sets in the attractor and the connections between them. When we deal with non-autonomous dynamical systems, the concept of pullback attractor for the associated skew-product flow appears as a powerful tool to analyze the asymptotic behaviour of these systems. In this paper we develop a Morse decomposition theory for pullback attractors of non-autonomous dynamical systems in Banach spaces with compact base space which, in particular, defines a (non-autonomous) Lyapunov functional on the attractor describing a decaying energy level on the evolution of trajectories.

In this paper we obtain some results on the existence of solution, and of pullback attractors, for a 2D Navier-Stokes model with finite delay studied in [4] and [6]. Actually, we prove a result of existence and uniqueness of solution under less restrictive assumptions than in [4]. More precisely, we remove a condition on square integrable control of the memory terms, which allows us to consider a bigger class of delay terms (for instance, just under a measurability condition on the delay function leading the delayed time). After that, we deal with dynamical systems in suitable phase spaces within two metrics, the L 2 norm and the H 1 norm. Moreover, we prove that under these assumptions, pullback attractors not only of fixed bounded sets but also of a set of tempered universes do exist. Finally, from comparison results of attractors we establish relations among them, and under suitable additional assumptions we conclude that these families of attractors are in fact the same object.

We are interested in the existence of multiple symmetric solutions to the Lane-Emden system with Hénon type weights Here B stands for the open unit ball in ℝ N centered at zero, N ≥ 3, α, β ≥ 0, p, q > 0. In this paper we present a complete picture of the quantitative study of solutions to the system above.

We establish existence results and energy estimates of solutions for a homogeneous Neumann problem involving the p-Laplace operator. The case of large dimensions, corresponding to the lack of compactness of W 1,p (Ω) in C 0 (Ω̅) is also considered. In particular, for a precise localization of the parameter, the existence of a non-zero solution is established, without requiring any asymptotic condition at zero or at infinity of the nonlinear term. In the case of (p − 1)-sublinear terms at the origin, we deduce the existence of solutions for small positive values of the parameter and we obtain that the corresponding solutions have smaller and smaller energies as the parameter goes to zero. Finally, a multiplicity result is obtained and concrete examples of applications are provided. A basic ingredient in our arguments is a recent local minimum theorem for differentiable functionals.

This paper studies the long-time behavior of solutions for p-Laplacian equations with a polynomial growth nonlinearity of arbitrary order and dynamic flux boundary conditions in d-dimensional bounded smooth domains. We first prove the existence of global attractors in L 2 (Ω)×L 2 (Γ) and L q (Ω)×Lq(Γ) for the p-Laplacian evolution equations subject to dynamic nonlinear boundary conditions by using the Sobolev compactness embedding theory, and then the existence of (L 2 (Ω) × L 2 (Γ), L 2(q−1) (Ω) × L 2(q−1) (Γ))-global attractor is obtained by asymptotical regularity method. Finally, we prove the existence of global attractor in W 1,p (Ω) ∩ L q (Ω)) × L q (Γ) by asymptotic a priori estimate.

When the plane is pie-sliced in n ≤ 4 parts (with nonempty interior and common vertex at the origin) our main result provides sufficient conditions for any map L, that is continuous and piecewise linear relatively to this slicing, to be invertible. This result cannot be plainly extended to a greater number of slices. Also, some examples show that the assumptions cannot be relaxed too much. Our result is proved by a combination of linear algebra and topological arguments.

The homogeneous Dirichlet problem for a partial differential inclusion involving the p- Laplace operator and depending on a parameter λ > 0 is investigated. The existence of three smooth solutions, a smallest positive, a biggest negative, and a nodal one, is obtained for any λ sufficiently large by combining variational methods with truncation techniques.

We study existence and stability for solutions of −Lu + g(x, u) = ω where L is a second order elliptic operator, g a Caratheodory function and ω a measure in Ω. We present a unified theory of the Dirichlet problem and the Poisson equation. We prove the stability of the problem with respect to weak convergence of the data.

The theory of distributions provides generalized solutions for problems which do not have a classical solution. However, there are problems which do not have solutions, not even in the space of distributions. As model problem you may think of with Dirichlet boundary conditions in a bounded open star-shaped set. Having this problem in mind, we construct a new class of functions called ultrafunctions in which the above problem has a (generalized) solution. In this construction, we apply the general ideas of Non Archimedean Mathematics (NAM) and some techniques of Non Standard Analysis. Also, some possible applications of ultrafunctions are discussed.

In this note we prove that if u is a negative solution to a nonlinear elliptic equation involving a Hessian operator, and u is zero on the boundary of a ball, then u is radially symmetric and increasing along the radii.

In this paper we consider the following elliptic system in ℝ 3 where K(x), α(x) are non-negative real functions defined on ℝ 3 so that . When K(x) ≡ K ∞ and α(x) ≡ α ∞ we have already proved the existence of a radial ground state of the above system. Here, by using a new version of the moving plane method, we show that all positive solutions of the above system with K(x) ≡ K ∞ and a(x) ≡ α ∞ are radially symmetric and the linearized operator around a radial ground state is also non-degenerate. Using these results we further prove, under additional assumptions on K(x) and α(x), but not requiring any symmetry property on them, the existence of a positive solution for the system.

In this paper we study a class of quasilinear problems, in particular we deal with multiple sign-changing solutions of quasilinear elliptic equations. We further develop an approach used in our earlier work by exploring elliptic regularization. The method works well in studying multiplicity and nodal property of solutions.