We study the Cauchy problem for the spatially homogeneous Landau equation for Fermi–Dirac particles, in the case of hard and Maxwellian potentials. We establish existence and uniqueness of a weak solution for a large class of initial data.

We study a boundary-value problem for a particular semilinear elliptic equation on Rn (n ≥ 2), whose solutions represent generalized stream functions for steady axisymmetric ideal fluid flows. Solutions are shown to exist that generalize those already known in dimensions 2 and 3. An isoperimetric characterization is given for our solutions, which represent generalized spherical vortex-rings. As a corollary, a sharp Sobolev-type inequality is obtained.

Given an infinite-dimensional vector space V, we consider the semigroup GS (m, n) consisting of all injective linear α: V → V for which codim ran α = n, where dim V = m ≥ n ≥ ℵ0. This is a linear version of the well-known Baer–Levi semigroup BL (p, q) defined on an infinite set X, where |X| = p ≥ q ≥ ℵ0. We show that, although the basic properties of GS (m, n) are the same as those of BL (p, q), the two semigroups are never isomorphic. We also determine all left ideals of GS (m, n) and some of its maximal subsemigroups; in this, we follow previous work on BL (p, q) by Sutov and Sullivan as well as Levi and Wood.

The relationship of the Lp (Rn) estimate and two endpoint estimates for the multi-linear singular integral operators is considered. Some conditions are given under which one of these estimates implies the others.

In this paper, we study existence of solutions with boundary layer and peaks for an elliptic problem. We prove that the problem has a mountain-pass-type solution, which has a boundary layer and a single peak near the boundary. Moreover, we also study the existence of solutions with interior peak.

Convergent expansions are derived for three types of orthogonal polynomials: Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for large values of the degree. The expansions are given in terms of functions that are special cases of the given polynomials. The method is based on expanding integrals in one or two points of the complex plane, these points being saddle points of the phase functions of the integrands.

In the first part of this paper, a variational characterization of parts of the Fučík spectrum for the Laplacian in a bounded domain Ω is given. The proof uses a linking theorem on sets obtained through a suitable deformation of subspaces of H1 (Ω). In the second part, a nonlinear Sturm–Liouville equation with Neumann boundary conditions on an interval is considered, where the nonlinearity intersects all but a finite number of eigenvalues. It is proved that, under certain conditions, this equation is solvable for arbitrary forcing terms. The proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values related to the Fucík spectrum.

This paper considers the nonlinear stability oftravelling wavefronts of a time-delayed diffusive Nicholson blowflies equation. We prove that, under a weighted L2 norm, ifa solution is sufficiently close to a travelling wave front initially, it converges exponentially to the wavefront as t → ∞. The rate ofconvergence is also estimated.

We rigorously approach the Schrödinger equation of derivative type qt + iqxx + λ |q|2qx + μq2q̄x = 0, λ ∈ R, μ ∈ C, by the cubic nonlinear Schrödinger equation AT + i AXX + i k0 (λ − μ) |A|2A = 0. We also study the case of the KdV-like equation qt + i qxx + aqxxx + i |q|2q + λ̃ (|q|2q)x + μ̃ |q|2qx = 0, λ̃ μ̃ ∈ R, arising in optical physics.