Volume 15, Issue 2

This paper studies the existence of nodal solutions for a class of boundary value problems that arise in population dynamics models. These problems are degenerate in the sense that the nonlinear term vanishes in a subinterval of the underlying region. In contrast to the non-degenerate case, in these degenerate situations the structure of the set of nodal solutions might consist of several components, making the analysis of this problem more complicated. This paper provides an initial step towards the solution of this general problem.

In this article we present a careful survey and critique of 5 landmark results and methods that seem to have played a pivotal role in the development of Mathematics since 1984. This article conveys the slow development of our understanding over the years and could not have been possible without the numerous communications with some authors of these articles as well as conversations with other friends and collaborators. The key contribution and the kindness of these colleagues is hereby gratefully acknowledged.

We study the existence of solutions of elliptic systems of the type and we examine how the structure of the system gives rise to a regularizing effect on the summability of the solutions; following the theory of elliptic (single) equations, the regularizing effect is unexpected.

Let G be a free Carnot group (i.e. a connected simply connected nilpotent stratified free Lie group) of step 2. In this paper, we prove that the variational functional generated by “intrinsic” Maxwell’s equations in G is the Γ-limit of a sequence of classical (i.e. Euclidean) variational functionals associated with strongly anisotropic dielectric permittivity and magnetic permeability in the Euclidean space.

We use the variational concept of Γ-convergence to prove existence, stability and exhibit the geometric structure of four families of stationary solutions to the singularly perturbed parabolic equation u t = ε 2 Δu + u[a 2 (x) − u 2 ], for (t, x) ∈ ℝ + × Ω, where Ω ⊂ ℝ 2 and a is positive, supplied with no-flux boundary condition. Let γ ⊂ Ω be a smooth simple closed curve and N(γ) a narrow tubular neighborhood of γ. Roughly speaking, the sufficient condition found for existence of such solutions relates the geometric profile of the function a in N(γ) to the signed curvature of γ .

In this work we show some multiplicity results for the anisotropic equation where Ω ⊂ℝ N is a bounded smooth domain, 1 < p 1 ≤ p 2 ≤ . . . ≤ p N and λ is a positive parameter. Using genus theory, we study the subcritical case g λ (u) = λ|u| q−2 u with q ∈ (1, p N ) and the critical case g λ (u) = λ|u| q−2 u +|u| p*−2 u with q ∈ (1, p 1 ) and p* = N| p̅/(N−p̅), with p̅ the harmonic mean of the p i ’s.

We study the existence and uniqueness of heteroclinic solutions to non-linear Allen-Cahn equation where G is a double-well potential. We investigate such a problem using variational methods after transforming the problem to an elliptic equation with a nonlinear Neumann boundary conditions.

We study Liouville-type theorem for polyharmonic Hénon-Lane-Emden system (−Δ) m u = |x| a v p , (−Δ) m v = |x| b u q when m, p, q ≥ 1, pq ≠ 1, and a, b ≥ 0. It is a natural conjecture that the nonexistence of positive solutions should be true if and only if (N + a)/(p + 1)+ (N + b)/(q + 1) > N − 2m. It is shown by Fazly[6] that the conjecture holds for radial solutions in all dimensions and for classical solutions in dimension N ≤ 2m + 1. We here give some partial results in dimension N ≥ 2m + 2.

We present new criteria for uniqueness and asymptotic stability of periodic solutions of a second order differential equation based on topological degree theory. As an application, we will study some well known equations and some illustrative examples.

In this work we construct a pseudo-gradient flow on a subspace C β of the dual Legendrian curves on a closed contact 3-manifold. These results improve and refine earlier results proved in this framework by A. Bahri [3]. The variational problem involved is not compact, thus asymptotes and critical points at infinity might occur. The flow in this work deforms curves in C β into a special set U k∈ℕ Γ 2k of curves at infinity and thus allows us a better understanding of this non-compact problem.