For a singularly perturbed nonlinear elliptic equation ε2Δu − V(x)u + up = 0, x ∈ ℝN, there are solutions concentrating around critical points of V with different energy scales with respect to small ε > 0. We combine the solutions when .

For every 0 < k < min{m,n} and any linear subspace E of real m × n matrices whose non-zero elements have rank greater than k, we show that there is a maximal extension Emax satisfying the same rank condition, and that the dimension of Emax is not less than (m – k)(n – k). We apply this result to the study of quasiconvex functions defined on the complement E⊥ of E in the form F(X) = f(PE⊥(X)), where PE⊥ is the orthgonal projection to E⊥.

Let D be a domain in C and let f be a transcendental meromorphic function on C such that C* \f(C) = ∅, {∞} or {α, β}, where α and β are two distinct values in C* = C ∪ {∞}. Then the family of functions g that are analytic on D and such that f ∘ g has no fixpoints on D is normal.

The distribution of stress in the neighbourhood of an infinite row of collinear cracks in an elastic body is calculated for general distributions of internal pressure. The case in which each of the cracks is opened out by the same constant pressure is considered in detail; the variation with the ratio of the width of the crack to the distance between two cracks of quantities of physical interest is shown in a series of diagrams.

We study the effects of a small symmetry breaking perturbation on a system of differential equations at a coupled Hopf bifurcation with O(2) symmetry, where the perturbation breaks the continuous rotation symmetry, but retains a reflection (Z2) symmetry. It is shown that for a large range of parameter values, the invariant manifolds of the unperturbed bifurcation persist and that for some values of normal form coefficients there are secondary bifurcations of nonsymmetric periodic standing wave solutions.

This paper treats the global qualitative behaviour of all bifurcating configurations of whirling nonlinearly elastic strings with ends fixed on the axis of rotation.

In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem

\[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \]

$(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$

$N\geq 3$

$(N+\alpha )/p + (N+\beta )/q \geq 2N$

$N=2$

$I_{\alpha }$

$I_{\beta }$

\[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \]

$\alpha ,\,\beta \in (0,\,2)$

$f,\,g$

We generalize the ‘sup+C inf’ inequality obtained by Shafrir to the solutions of

with Ω ⊂ ℝ2 open and bounded, α ∈ (0,1) and V any measurable function which satisfies 0 < a ≤ V ≤ b < + ∞.

Some of the formulae obtained in this paper are likely to find application in problems concerning a rectangular lattice of “atoms”, each of which is under the influence of its near neighbours. Some of the determinants considered apply to cases in which both the nearest and the next nearest neighbours are operative. The inverses of certain types of matrices are found, and these may prove to be of value either in solving systems of linear equations such as arise in relaxation problems, or in determining the latent roots of matrices which may occur in problems in applied mathematics.

We consider boundary value problems for second order differential systems of the form (1)x” = A(t)x′ + f(t, x) and (2) x” = A(t)x′ + f(t, x) + q(t, x). By assuming the existence of a solution to (1) with a given region in (t, x) space, we derive conditions under which there exists a solution to (2) which stays in a certain neighbourhood of and satisfies given boundary conditions.

We show the existence of a timelike periodic trajectory for a time-dependent Lorentzian metric on ℝn × ℝ.

We consider first-order hyperbolic systems on an interval with dynamic boundary conditions. These systems occur when the ordinary differential equation dynamics on the boundary interact with the waves in the interior. The well-posedness for linear systems is established using an abstract Friedrichs theorem. Due to the limited regularity of the coefficients, we need to introduce the appropriate space of test functions for the weak formulation. It is shown that the weak solutions exhibit a hidden regularity at the boundary as well as at interior points. As a consequence, the dynamics of the boundary components satisfy an additional regularity. Neither result can be achieved from standard semigroup methods. Nevertheless, we show that our weak solutions and the semigroup solutions coincide. For illustration, we give three particular physical examples that fit into our framework.

Suppose that f and g are transcendental entire functions such that the composition F = f(g) has finite order, and suppose that Q is a nonconstant rational function. We show that N(r, 1/(F – Q)) ≠ o(T(r, F)). The theorem is related to results of Bergweiler, Goldstein and others.

Let denote a subring of the complex field that contains 1 and is closed under complex conjugation. It is shown that, with respect to the involution induced by word-reversal, the algebra over of a free monoid admits a trace and a separating family of star matrix representations. From the existence of a trace it is deduced that the aforementioned involution is special, in the sense of Easdown and Munn. Similar results hold for the algebra over of a free monoid with involution.

In this work we consider the Cauchy problem associated with dissipative perturbations of infinite-dimensional Hamiltonian systems. we describe abstract conditions under which the problem is locally and globally well posed. Moreover, we establish the existence of global attractor. Finally, we present several applications of the theory.

A recently developed asymptotic theory of higher-order differential equations is applied to problems of right-definite type to determine the numbers M+, M− of linearly independent solutions with a convergent Dirichlet integral, M+ and M− referring to the usual upper and lower λ.-half-planes. Particular attention is given to the phenomenon noted by Karlsson in which one of M+ and M− is maximal but not the other. Conditions are given under which M+ (say) is maximal and M− is the same, one less, and two less.

We study the Cauchy problem for the degenerate and uniformly parabolic equations with gradient term. The local existence, global existence and non-existence of solutions are obtained. In the case of global solvability, we get the exact estimates of a solution. In particular, we obtain the global existence of solutions in the limiting case.

We discuss a new class of Banach spaces which are wider than the strongly convex spaces introduced by Congxin Wu and Yongjin Li. We prove that the new class of Banach spaces lies strictly between either the class of uniformly convex spaces and strongly convex spaces or the class of fully k-convex spaces and strongly convex spaces. The new class of Banach spaces has inclusive relations with neither the class of locally uniformly convex spaces nor the class of nearly uniformly convex spaces. We obtain in addition some characterizations of this new class of Banach spaces.

In this paper we deal with a singular elliptic problem involving an asymptotically linear nonlinearity and depending on two positive parameters. We investigate the existence, uniqueness and non-existence of the minima of the functional associated with the problem and, by employing a natural and very general definition of a weak solution, we also obtain a bifurcation-type result.

In the former parts of this investigation we studied the physiological action of the salts of a considerable number of ammonium bases—that is, of the salts formed by the union of an ether with the nitride of one or more alcohol radicals. Thus—

The examination of the physiological action of such salts proved that, while differing from one another in many respects, there are two points in which they agree—they all paralyse the end-organs of the motor nerves, and none of them possess that stimulating action of the spinal cord which we observe in such a substance as strychnia.