We show the existence and exact asymptotic behaviour of the unique solution u ∈ C2(Ω)∩C(Ω̄) near the boundary to the singular nonlinear Dirichlet problem −Δu = k(x)g(u) + λ|∇u|q, u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded domain with smooth boundary in RN, λ ∈ R, q ∈ [0, 2], g(s) is non-increasing and positive in (0, ∞), lims→0+g(s) = +∞, k ∈ Cα(Ω) is non-negative non-trivial on Ω, which may be singular on the boundary.

In this paper we prove a sharp lower bound for the first non-trivial Neumann eigenvalue μ1(Ω) for the p-Laplace operator (p > 1) in a Lipschitz bounded domain Ω in ℝn. Our estimate does not require any convexity assumption on Ω and it involves the best isoperimetric constant relative to Ω. In a suitable class of convex planar domains, our bound turns out to be better than the one provided by the Payne—Weinberger inequality.

Solutions to the initial value problem for the mixed nonlinear Schrödinger equation

are considered. Conditions on the constants α,β, γ, function g(·) and initial data u(x, 0) are given so that, for this problem, the unique existence of smooth solutions is proved. In addition, the decay behaviours of the smooth solutions as |x|→+∞ are discussed.

In this work we are considering both the one-dimensional and the radially symmetric versions of the elliptic system Δu = vp, Δv = uq in Ω, where p, q > 0, under the boundary condition u|∂Ω = +∞, v|∂Ω = +∞. It is shown that no positive solutions exist when pq ≤ 1, while we provide a detailed account of the set of (infinitely many) positive solutions if pq > 1. The behaviour near the boundary of all solutions is also elucidated, and symmetric solutions (u, v) are completely characterized in terms of their minima (u(0), v(0)). Non-symmetric solutions are also deeply studied in the one-dimensional problem.

This paper deals with the oscillatory and asymptotic behaviour of all solutions of a class of nth order (n > 1) non-linear differential equations with deviating arguments involving the so called nth order r-derivative of the unknown function x defined by

where r1, (i = 0,1,…, n – 1) are positive continuous functions on [t0, ∞). The results obtained extend and improve previous ones in [7 and 15] even in the usual case where r0 = r1 = … = rn–1 = 1.

We introduce a new non-classical Riemann solver for scalar conservation laws with concave–convex flux-function. This solver is based on both a kinetic relation, which determines the propagation speed of (under-compressive) non-classical shock waves, and a nucleation criterion, which makes a choice between a classical Riemann solution and a non-classical one. We establish the existence of (non-classical entropy) solutions of the Cauchy problem and discuss several examples of wave interactions. We also show the existence of a class of solutions, called splitting–merging solutions, which are made of two large shocks and small bounded-variation perturbations. The nucleation solvers, as we call them, are applied to (and actually motivated by) the theory of thin-film flows; they help explain numerical results observed for such flows.

Let Ω be an open connected cone in ℝn with vertex at the origin. Assume that the Operator

is subcritical in Ω, where δΩ is the distance function to the boundary of Ω and μ ⩽ 1/4. We show that under some smoothness assumption on Ω the improved Hardy-type inequality

holds true, and the Hardy-weight λ(μ)|x|–2 is optimal in a certain definite sense. The constant λ(μ) > 0 is given explicitly.

We consider the nonlinear elliptic problem ± (Δu + λu) + f(x, u) = h(x) in Ω, u = 0 on ∂Ω, where Ω is a bounded smooth domain in ℝN, λ is near the first eigenvalue and h(x) is orthogonal to the first eigenfunction. We give some conditions of existence of positive solutions and of multiple solutions in terms of the primitive of f with respect to u.

This paper proves the existence of solutions to the spherical piston problem for isentropic gas dynamics with equation of state p(p) = Apγ, γ≧ 1. The method of analysis is to replace the usual viscosity ε with εt, thus permitting a search for self-similar viscous limits. The main result of the paper is that self-similar viscous limits are proved to exist and converge to a solution to the piston problem when

N = 1, 2, 3 space dimensions.

In this contribution an account is given of the dominant plants of aquatic and semi-aquatic habit existing in and about the lakes of the following districts of Scotland:—

1. The Loch Ness area.

2. The lakes in the Island of Lismore, Argyll.

3. Lakes situated between Nairn and the Culbin Sand Hills.

In the first decade of the present century Bloomer reported in a series of concise papers (for which see the list of references) the results of his investigations into the structure of the majority of the lamellibranchs which were then classified in the family Solenidæ. In addition to a general description of the external anatomy of these bivalves he paid especial attention to the details of the musculature, to the course of the alimentary canal and the structure of the stomach, while he gave rather generalised accounts of the nervous system and of the arrangement of the circulation, in which he almost entirely followed the previous work of Ménégaux (1890). Bloomer made little attempt to view as a whole the information which had been gathered together as a result of these years of work; the only papers which he published that can be regarded as dealing with his results from a general standpoint being a short one (1903b) on the classification of the British species of the genus Solen, and a second (1903c), equally brief, on the origin and function of the small fourth pallial aperture which is a marked feature of the anatomy of certain members of the family. This he proved to be a separated portion of the pedal gape, not homologous with the similar structure lying in a more posterior position in some of the Anatinacea and in Lutraria.

The existence and uniqueness of travelling-wave solutions is investigated for a system of two reaction–diffusion equations where one diffusion constant vanishes. The system arises in population dynamics and epidemiology. Travelling-wave solutions satisfy a three-dimensional system about (u, u′, ν), whose equilibria lie on the u-axis. Our main result shows that, given any wave speed c > 0, the unstable manifold at any point (a, 0, 0) on the u-axis, where a ∈ (0, γ) and γ is a positive number, provides a travelling-wave solution connecting another point (b, 0, 0) on the u-axis, where b:= b(a) ∈ (γ, ∞), and furthermore, b(·): (0, γ) → (γ, ∞) is continuous and bijective

In this paper we introduce the two possible formulations of the -functional calculus that are based on the Fueter–Sce mapping theorem in integral form and we introduce the pseudo--resolvent equation. In the case of dimension 3 we prove the -resolvent equation and we study the analogue of the Riesz projectors associated with this calculus. The case of dimension 3 is also useful to study the quaternionic version of the -functional calculus.

Considered is a nonlinear model which describes the dynamic behaviour of the not necessarily small longitudinal and transverse displacements of a thin, cantilevered beam. The motion of the beam is driven by a bending moment, an axial force and a vertical shear force that act at the free end of the beam. The main goal is to describe the reachable set of the system, that is, the set of all states that can be reached by varying the end forces within a certain set of admissible controls.

First-order scalar linear delay differential equations with periodic coefficients and constant delays are considered, where the coefficients have a common period and the delays are multiples of this period. A basic asymptotic criterion is given. Moreover, some results on the nonoscillation and on the stability of the trivial solution are obtained. An equation, which is in a sense the characteristic equation, plays an important role in establishing the results of the paper.

This note gives a simple proof of uniqueness for positive solutions of certain non-linear boundary value problems on ℝ+ which are typified by the equation

with boundary conditions u′(0) = u(+∞) = 0. In the autonomous case (r ≡ 1), this is easy to see, by quadrature. The proof here supposes r to be non-increasing on ℝ+.

The exact number of positive solutions of Δu + f(u) = 0 on finite balls in ℝN is determined. The assumptions about f(u) are similar to those imposed by Serrin and the second author in a previous study of uniqueness of the positive solution when the spatial domain is all of ℝN (see [7, 8]). For finite balls of sufficiently large radius it is shown here that there are exactly two positive and, hence, radial solutions. To this end, we first prove the linear nondegeneracy of the positive solution of ℝN. This is obtained by applying the technique of monotone separation of graphs [7] to the linearised equations. Somewhat sharper estimates are required here (see Part I, Section 2).

In this paper we consider quasilinear hemivariational inequalities at resonance. We obtain existence theorems using Landesman-Lazer-type conditions and multiplicity theorems for problems with strong resonance at infinity. Our method of proof is based on the non-smooth critical point theory for locally Lipschitz functions and on a generalized version of the Ekeland variational principle.

We show that if the leading coefficient p in a Sturm-Liouville expression is negative on a set E with positive Lebesgue measure, then the minimal operator (and hence any self-adjoint realization of the Sturm-Liouville expression) is not bounded below. The case when E contains an interval follows from standard methods but these methods fail when E contains no interval, e.g. when E is a Cantor-like set. This is the case considered here.

Sufficient conditions are given which ensure nonexistence of spherically symmetric entire solutions of Δpu = f(u), p ≧ 2. Sufficient conditions for existence of spherically symmetric entire solutions of Δpu = f(r, u) are also given.