The object of the paper is to investigate solutions of equations of the form

with

and in particular to look at the asymptotic behaviour of these solutions as γ ↑∞. It is found that, if tγ is the first zero of ϑ, then

while tγ is bounded below if p < 2. This answers questions raised by Brezis concerning solutions of −Δu = eup in R2

Suppose that a space form is immersed into another Riemannian manifold as a totally umbilical hypersurface with constant mean curvature. Then, in the ambient manifold, the lengthof the curvature tensor, that of the Ricci tensor and the scalar curvature must satisfy an inequality. In this paper the authors proved the inequality. As applications of the inequality, some immersibility problems are investigated. For example, it is proved that if a space form is immersed in an Einstein manifold as a totally umbilical hypersurface, then the Einstein manifold has constant sectional curvature along the hypersurface. Moreover, it is proved that a space form cannot be immersed into some Kaehlerian manifolds as a totally umbilical hypersurface with constant mean curvature.

Iffis a continuous even function which is decreasing on (0,∞) and such that±α are its only zeros and are simple, then in three-dimensional phase spacethe unstable manifold of the equilibrium u = −α and the stable manifold of u = α are both two dimensional. If λ<0 it is shown that there is a unique bounded orbit of the equation λu‴ + u′ = f(u), and that this is a heteroclinic orbit joining these two equilibria. Other results on the existence and uniqueness of heteroclinic orbits are also established when f is not even and when f is not monotone on (0, ∞).

The lattice of subalgebras of a Malcev algebra determines to a great extent the structureof the algebra. It is shown that conditions such as nilpotency, solvability or semisimplicity are almost characterised by means of conditions on this lattice. This enables us to study the relationship between Malcev algebras with isomorphic lattices of subalgebras.

In this paper, we study problems of the form:

We obtain some existence and regularity results when Ω is either a ball or an annulus, without convexity hypothesis on g. We then apply these results to some shear problems in nonlinear elasticity.

This paper gives a sufficient condition for almost-everywhere injectivity for nonlinear three dimensional elasticity similar to that of Claret-Necas [8], namely.

We prove that this relation is maintained under the weak convergence of minimising sequences for nonlinear elasticity problems. The existence and partial regularity of an “inverse” function are proved.

The stability of several natural subsets of the bounded non-semi-Fredholm operators undercompact perturbations were studied by R. Bouldin [2] in separable Hilbert spaces and by M. Gonzales and V. M. Onieva [6] in Banach spaces. The aim of this paper is to study this problem for closed operators in operator ranges. The main results are a characterisation of the non-semi-Fredholm operators with respect to α-closed and α-compact operators as well as a generalisation of a result of M. Goldman [5]. We also give some applications of the theory developed to ordinary differential operators.

In this paper we apply the Painlevé tests to the damped, driven nonlinear Schrödinger equation

where a(x, t) and b(x, t) are analytic functions of x and t, todetermine under what conditions the equation might be completely integrable. It is shown that (0.1) can pass the Painlevé tests only if

where α0(t),α1(t) and β(t) are arbitrary, real analytic functions of time. Furthermore, it is shown that in this special case, (0.1) may be transformed into the original nonlinear Schrödinger equation, which is known to be completely integrable.

Sufficient oscillation criteria of Nehari-type are established for the differential equation −uʺ(t) + q(t)u(t) = 0, 0<t<∞, with and without sign restrictions on q(t), respectively. These results are extended to Sturm-Liouville equations and elliptic differential equations of second order.In Section 7 we present conclusions for the lower spectrum of elliptic differential operators and also for the discreteness of the spectrum of certain ordinary differential operators of second order.

We consider time-dependent perturbations of generators of strongly continuous semigroups on a Banach space. The perturbations map the Banach space into a bigger space, which is the second dual of the original space in a specific semigroup sense. Using the theory of dual semigroups we show that the solutions of a generalised variation-of-constants formuladefine an evolutionary system. We investigate continuity and differentiability propertiesof this evolutionary system and its dual system and examine in what sense the perturbed generator and its adjoint generate these evolutionary systems. It is shown that the results apply naturally to retarded functional differential equations and age structured population dynamics.

A Levinson theorem is proved for a Dirac system with one singular endpoint. The number ofbound state is expressed in terms of the change in asymptotic phase of an appropriate solution and in terms of factors whose values depend on the presence of half-bound states. The behaviour of the asymptotic phase is used to determine the asymptotic behaviour of the Titchmarsh-Weyl m-function.

We study the class of polynomial vector fields of the form = αx — y + Pn(x, y), = x + αy + Qn(x, y), where Pn and Qn are homogeneous polynomials of degree n. If we define the functions f(x, y) = xPn(x, y) + yQn(x, y) and g(x, y) = xQn(x, y)−yPn(x, y), we characterise the number of limit cycles for this class when the function g(αg − f) does not change sign.