Volume 10, Issue 3

Let Ω ⊂ ℝ N , N ≥ 3, be a bounded domain with C 2 boundary, , the critical exponent for the Sobolev imbedding. In this work, we are interested in the following problem: where λ > 0, 0 ≤ q < p - 1. We show that there exists 0 < Λ < ∞ such that for suitable ranges of p and q, (P λ ) admits at least two solutions in W 1,p (Ω) if λ ∈ (0, Λ) and no solution if λ > Λ. The proof of these assertions is done by first finding the local minimum for the variational functional associated to (P λ ) and then applying mountain pass arguments to obtain a saddle point type solution. In the critical case we are considering, there are technical reasons which make the mountain pass argument work for only certain ranges of p and q.

In this paper we consider the following nonlinear elliptic equation with Dirichlet boundary conditions: -Δu = K(x)u p , u > 0 in Ω, u = 0 on ∂Ω, where Ω is a smooth domain in ℝ n , n ≥ 4 and is the critical Sobolev exponent. Using dynamical and topological methods involving the study of critical points at infinity we establish, under generic conditions on K, some existence and multiplicity results.

This article is devoted to the study of the existence of a solution to the periodic nonlinear second order ordinary differential equation with damping uʺ (x) + cuʹ (x) + g(x,u(x)) = f(x), x ∈ [0, T], u(0) = u(T), uʹ(0) = uʹ(T), where c ∈ ℝ, g is a Caratheodory function, f ∈ L 1 ([0, T), a quotient lies between 0 and and a nonlinearity g satisfies Landesman Lazer type condition near the zero. The techniques we use are a variational method and a critical point theorem.

We prove new results for linearized stable and finite Morse index solutions on ℝ N and give applications to half space and bounded domain problems.

Using Melnikov functions at any order, we provide upper bounds for the maximum number of limit cycles bifurcating from the period annulus of the degenerate center ẋ = −y((x 2 + y 2 )/2) m and ẏ = x((x 2 + y 2 )/2) m with m ≥ 1, when we perturb it inside the whole class of polynomial vector fields of degree n. The positive integers m and n are arbitrary. As far as we know there is only one paper that provide a similar result working with Melnikov functions at any order and perturbing the linear center ẋ = −y, ẏ = x.

The paper deals with the initial value problem for the degenerate reaction-diffusion-convection equation u t + h(u)u x = (u m ) xx + f(u), x ∈ ℝ, t > 0, where h is continuous, m > 1, and f is of Fisher-type. By means of comparison type techniques, we prove that the equilibrium u ≡ 1 is an attractor for all solutions with a continuous, bounded, non-negative initial condition u 0 (x) = u(x, 0) ≢ 0. When u 0 is also compactly supported and satisfies 0 ≤ u 0 ≤ 1, the convergence is such that an asymptotic estimate of the interface can be obtained. The employed techniques involve the theory of travelling-wave solutions that we improve in this context. The assumptions on f and h guarantee that the threshold speed wavefront is not stationary and we show that the asymptotic speed of the interface equals this minimal speed.

We consider nonlinear elliptic problems driven by a nonhomogeneous nonlinear differential operator. Using variational methods combined with Morse theory (critical groups), we prove two multiplicity results establishing three nontrivial smooth solutions. For the semilinear problem (linear differential operator), we produce four nontrivial smooth solutions. In the special case of the p-Laplacian differential operator, our framework of analysis incorporates equations which are resonant at infinity with respect to the principal eigenvalue.

We study the possibility of defining a nontrivial continuation after the blow-up time for a system of two heat equations with a nonlinear coupling at the boundary. It turns out that any possible continuation that verifies a maximum principle is identically infinity everywhere after the blow-up time; that is, both components blow up completely. We also analyze the propagation of the singularity to the whole space, the avalanche, when blow-up is non-simultaneous.

Orbital stability property for weakly coupled nonlinear Schrödinger equations is investigated. Different families of orbitally stable standing waves solutions will be found, generated by different classes of solutions of the associated elliptic problem. In particular, orbitally stable standing waves can be generated by least action solutions, but also by solutions with one trivial component whether or not they are ground states. Moreover, standing waves with components propagating with the same frequencies are orbitally stable if generated by vector solutions of a suitable Schrödinger weakly coupled system, even if they are not ground states.

In this article we study the positive solutions of the parabolic semilinear system of competitive type in Ω × (0, T), where Ω is a domain of ℝ N , and p,q > 0, pq ≠ 1, Despite of the lack of comparison principles, we prove local upper estimates in the superlinear case pq > 1 of the form u(x, t) ≦ Ct -(p+1)/(pq-1) , v(x, t) ≦ Ct -(q+1)/(pq-1) in ω × (0, T 1 ), for any domain ω ⊂⊂ Ω, T 1 ∈ (0, T), and C = C(N, p, q, T 1 , ω). For p, q > 1, we prove the existence of an initial trace at time 0, which is a Borel measure on Ω. Finally we prove that the punctual singularities at time 0 are removable when p, q ≧ 1 + 2/N.

We study some Phragmén-Lindelöf type results, which were considered as extensions of maximal modulus theorem. We shall consider them from the viewpoint of Fourier analysis. Our analysis shows that the Phragmén-Lindelöf theorems can be regarded as the convexity of microglobal wave front sets. We prove such theorems for elliptic systems of equations with constant coefficients.