We consider weakly-coupled elliptic systems of the type

with each fi being either an increasing or a decreasing function of each uj. Assuming the existence of coupled super- and subsolutions, we prove the existence of solutions, and provide a constructive iteration scheme to approximate the solutions. We then apply our results to study the steady-states of two-species interaction in the Volterra–Lotka model with diffusion.

In this paper, it is proved that the problem (∂2u/∂t2) + ∆xu = v, u(x, 0) = u0(x), ut(x, 0) = u1(x), with homogeneous Dirichlet conditions on the boundary, is well posed provided v, u0, u1 belong to a suitable space of functions.

Let d ≧ 1 be an integer and ω ⊂ℝd a smooth bounded domain and consider the elliptic equation − Δu = g(u) on Ω = ℝ2 × ω. We prove that under (almost) necessary and sufficient conditions on the continuous function g: ℝm→ ℝm the above equation has a minimum-action solution.

We consider the characterisation of a class of free boundary problems arising in the flow of a viscous liquid in a porous medium (or, in two dimensions, a Hele–Shaw cell). Injected air forms a bubble which grows as time increases; it is shown that three kinds of behaviour can occur. Firstly, the solution may cease to exist in finite time; secondly, the solution may exist for all time and the free boundary may have one or more limit points as t tends to infinity; and thirdly, the bubble may exist for all time and fill the whole space as t tends to infinity. Two-dimensional explicit examples arc given of all three types of behaviour, and it is proved that the only solutions of the third kind are those in which the bubble is always elliptical; the proof uses the theory of null quadrature domains. It is shown that solutions for ellipsoidal bubbles exist in three dimensions and it is conjectured that the only three-dimensional null quadrature domains with finite complement are those whose complement is an ellipsoid.

Averaging techniques on Hamiltonian dynamical systems can often be used to establish the existence of hyperbolic periodic orbits. In equilibrium situations, it is then often difficult to show that there are homoclinic/heteroclinic connections between these hyperbolic orbits in the original unaveraged system. This existence problem is solved in this paper for a class of Hamiltonian systems admitting a sufficient number of symmetries (including reversing symmetries). Under isoenergetic reduction, the problem is reduced to one involving reversible vector fields under time-dependent perturbations admitting the same reversing symmetries. Applications are made to the one-parameter Hénon-Heiles family. The paper concludes with remarks on the problem of showing transversality of these homoclinic/heteroclinic orbits.

In this paper we introduce the concept of fractional powers of a pair of operators between two Banach spaces. The operators need not be closed, but form a closed pair. The properties of the fractional powers are studied. An application of the theory is briefly discussed.

This paper proves the existence of a solution of a non-linear Goursat problem for a partial differential equation of order 2p (p ≧ 2) with the boundary conditions given on 2p curves emanating from a common point. The problem is reduced to a system of integro-differential-functional equations and then Schauder's fixed point theorem is applied.

A sequence of transformations of the type Y = (I + o(1))Z is developed for the system Y′(x) = {Λ(x) + R(x)}Y(x), where Λ is diagonal. The transformations bring in the derivatives of R in succession until the Levinson form is obtained when a given derivative is reached. This theory covers rapidly varying coefficients and it extends results which are known for constant Λ.