We prove the existence of multiple bound states of the nonlinear Schrödinger equation −Δu + V(x)u = f(u). Here the linear potential V is continuous and bounded from below, and the nonlinearity f is of asymptotically linear type. We show that, under certain assumptions on the spectrum of the Schrödinger operator −Δ + V and the asymptotic behaviour of f(u)/u, the above equation has at least four non-trivial solutions, two of them sign changing.

We consider a class of convex non-linear boundary value problems of the form

where L is a linear, uniformly elliptic, self-adjoint differential expression, f is a given non-linear function, B is a boundary differential expression of either Dirichlet or Neumann type and D is a bounded open domain with boundary ∂D. Particular problems of this class arise in the process of thermal combustion [8].

In this paper we show that stable solutions of this class can be bounded from below (above) by a monotonically increasing (decreasing) sequence of Newton (Picard) iterates. The possibility of using these schemes to construct unstable solutions is also considered.

Global weak solutions to the continuous Smoluchowski coagulation equation (SCE) are constructed for coagulation kernels featuring an algebraic singularity for small volumes and growing linearly for large volumes, thereby extending previous results obtained in Norris (1999) and Cueto Camejo & Warnecke (2015). In particular, linear growth at infinity of the coagulation kernel is included and the initial condition may have an infinite second moment. Furthermore, all weak solutions (in a suitable sense) including the ones constructed herein are shown to be mass-conserving, a property which was proved in Norris (1999) under stronger assumptions. The existence proof relies on a weak compactness method in L1 and a by-product of the analysis is that both conservative and non-conservative approximations to the SCE lead to weak solutions which are then mass-conserving.

We prove some new results which justify the use of interval truncation as a means of regularising a singular fourth-order Sturm–Liouville problem near a singular endpoint. Of particular interest are the results in the so-called lim-3 case, which has no analogue in second-order singular problems.

In this paper we analyse a singular perturbation problem for linear wave equations with interior and boundary damping. We show how the solutions converge to the formal parabolic limit problem with dynamic boundary conditions. Conditions are given for uniform convergence in the energy space.

We examine the regularity of the solution of a quasilinear system involving the curl of vector fields. This system arises in the mathematical theory of superconductivity. The C2+α regularity was obtained by Bates and Pan under the condition that Ω is simply connected and has no holes, and that the normal component of the curl of the boundary data vanishes. The aim of this paper is to remove these technical restrictions on the topology of the domain and on the boundary data. By carefully studying the related quasilinear Neumann problem, we obtain the C2+α regularity without assuming these technical conditions.

We consider an inverse spectral problem for Sturm–Liouville boundary-value problems on a graph with formally self-adjoint boundary conditions at the nodes, where the given information is the M-matrix. Based on the authors' previous results, using Green's function, we prove that the poles of the M-matrix are at the eigenvalues of the associated boundary-value problem and are simple, located on the real axis, and that the residue at a pole is a negative semi-definite matrix with rank equal to the multiplicity of the eigenvalue. We define the so-called norming constants and relate them to the spectral measure and the M-matrix. This enables us to recover, from the M-matrix, the boundary conditions and the potential, up to a unitary equivalence for co-normal boundary conditions.

We look for standing waves for nonlinear Schrödinger equations

with cylindrically symmetric potentials g vanishing at infinity and non-increasing, and a C1 nonlinear term satisfying weak assumptions. In particular, we show the existence of standing waves with non-vanishing angular momentum with prescribed L2 norm. The solutions are obtained via a minimization argument, and the proof is given for an abstract functional which presents a lack of compactness. As a specific case, we prove the existence of standing waves with non-vanishing angular momentum for the nonlinear hydrogen atom equation.

We study certain perturbations of the differential equation Δu − u + up = 0 on all of n-dimensional Euclidean space. Conditions are obtained which ensure the existence of a solution to the perturbed equation near a given solution to the unperturbed equation. We have to overcome degeneracy of the unperturbed solution and lack of smooth dependence on the perturbation parameter. An abstract version of the argument is sketched in a functional-analytic setting related toequivariant bifurcation theory. We consider also a smooth perturbation with several parameters and study the singularities of the mapping which maps each solution to its associated parameters.

In the course of a search for criteria which would assist in deciding whether a particular series of numbers was randomly arranged, it was found necessary to investigate some of the more obvious characters of a random series of numbers. If the series is an infinite one and all the numbers are unequal (the series of decimals between o and 1 selected in some random fashion may be considered) certain of the numbers will be maximal, that is greater than their immediate neighbours, whilst others will be minimal, that is less than their immediate neighbours. Every maximal number is succeeded by a run down, that is by a series of descending numbers ending in the succeeding minimal number. The length of a run down is the number of numbers it contains, including the maximal number which begins it and the minimal number which ends it. The length of a run up is similarly defined. Clearly the minimum length of a run is 2.

From the Proceedings of the Royal Society of Edinburgh, No. 106, p. 501, it appears the colouring of maps is receiving attention. This note bears chiefly upon the history of the matter.

Some thirty years ago, when I was attending Professor De Morgan's class, my brother, Francis Guthrie, who had recently ceased to attend them (and who is now professor of mathematics at the South African University, Cape Town), showed me the fact that the greatest necessary number of colours to be used in colouring a map so as to avoid identity of colour in lineally contiguous districts is four. I should not be justified, after this lapse of time, in trying to give his proof, but the critical diagram was as in the margin. With my brother's permission I submitted the theorem to Professor De Morgan, who expressed himself very pleased with it; accepted it as new; and, as I am informed by those who subsequently attended his classes, was in the habit of acknowledging whence he had got his information.

In the course of an investigation on the effects of intravitam staining with trypan blue on the uterus and ovaries of the guinea-pig at different periods of the œstrous cycle in virgin animals and after parturition, it was found necessary to have definite information with regard to possible variations in the lengths of the cycles. Stockard and Papanicolaou (1917 and 1919) reviewed the earlier work on the œstrous cycle in this rodent, and introduced the vaginal smear technique by which successive cycles could be studied in the living animal. The results of their researches seemed to indicate that the non-pregnant female guinea-pig comes into “heat” regularly about every sixteen days, and that there are only slight time-variations in the periodic rhythm of a given female, and that among different individuals there is a limited variation in the lengths of the cycles. Practically similar results were obtained by Selle (1922), though Voss (1927) indicated that his cases did not confirm the very regular periodicity.

1. It is pointed out that for very small adsorptions the adsorption law appears to be

for both gases and solutions.

2. The general form of the adsorption curve for solutions is deduced from the above conclusion and found to agree with the results of different observers.

The author wishes to thank Professor James Walker, F.R.S., for his assistance in the presentation of this paper.

Let p ∈ (1, ∞) and let Ω ⊆ ℝN be a bounded domain with Lipschitz continuous boundary. We characterize on L2(Ω) all order-preserving semigroups that are generated by convex, lower semicontinuous, local functionals and are sandwiched between the semigroups generated by the p-Laplace operator with Dirichlet and Neumann boundary conditions. We show that every such semigroup is generated by the p-Laplace operator with Robin-type boundary conditions.

A periodic connection is constructed for a double well potential defined in the plane. This solution violates Modica's estimate as well as the corresponding Liouville theorem for general phase transition potentials. Gradient estimates are also established for several kinds of elliptic systems. They allow us to prove the Liouville theorem in some particular cases. Finally, we give an alternative form of the stress–energy tensor for solutions defined in planar domains. As an application, we deduce a (strong) monotonicity formula.

A general continuation theorem for isolated sets in infinite-dimensional dynamical systems is proved for a class of semiflows. This result is then used to prove the existence of continua of full bounded solutions bifurcating from infinity for systems of reaction—diffusion equations.

Upon formalising an analogy between two-dimensional Stokes flow and two-dimensional isotropic conductivity, we exhibit a class of fourth order equations which behave “isomorphically” like isotropic conductivity from the standpoint of homogenisation and from that of corresponding bounding methods on possible effective behaviours. In particular, Lipton's result on the G-closure problem for mixtures of two incompressible elastic materials is recovered in the two-dimensional case.

Most of the convergence results appearing so far for delayed Lotka–Volterra-type systems require that undelayed negative feedback dominate both delayed feedback and interspecific interactions. Such a requirement is rarely met in real systems. In this paper we present convergence criteria for systems without instantaneous feedback. Roughly, our results suggest that in a Lotka–Volterra-type system if some of the delays are small, and initial functions are small and smooth, then the convergence of its positive steady state follows that of the undelayed system or the corresponding system whose instantaneous negative feedback dominates. In particular, we establish explicit expressions for allowable delay lengths for such convergence to sustain.

The solutions of the differential equation Lny + p(x)y = 0, where Lny = ρn(ρn−1 … (ρ1(ρ0y)′)′ …)′ and p(x) is of one sign, are classified according to their behaviour as x → ∞. The solution space is decomposed into disjoint, non-empty sets Sk, 0≦K≦n, such that (−1)n−kp(x)≦0. We study the growth properties and the density of the zeros of the solutions which belong to the different sets Sk, the structure of the sets and its connection with (k, n − k)-disfocality.

The nonlinear evolution problem [Bu(t)]′ = A(t, Bu)u + f(t, Bu) with B a constant linear operator and A = A(t, Bu) a time-dependent nonlinear operator from one Banach space to another, is studied. Existence and uniqueness results are obtained by making use of the theory of B-evolutions and the fractional powers of A and B. Two examples are presented in which the theory is applied to nonlinear equations with dynamic boundary conditions.