Solutions with peaks near the critical points of Q(x) are constructed for the problem

We establish the existence of 2k −1 positive solutions when Q(x) has k non-degenerate critical points in ℝN

In this paper, we consider a planar system with two delays:

ẋ1(t) = −a0x1(t) + a1F1 (x1(t − τ1), x2(τ−t2)).

ẋ2(t) = −b0x2(t) + b1F2 (x1(t − τ1), x2(t − τxs2)).

Firstly, linearized stability and local Hopf bifurcations are studied. Then, existence conditions for non-constant periodic solutions are derived using degree theory methods. Finally, a simple neural network model with two delays is analysed as an example.

We consider the following special problem related to the optimal layout problems of materials: given two linear elastic materials, the elasticity tensors of which are C1 and C2, and a force f, find the strong closure of strains and stresses as the distribution of the materials varies, or, alternatively, find the sets of elasticity tensors which generate these strong closures. In this paper, it is shown that the local incompatibility conditions depending on C1, C2 and the local properties of strains or stresses completely characterize these sets. A connection to multiple-well problems is established.

We study the Jacobian determinants J = det(∂fi/∂xj) of mappings f: Ω ⊂ ℝn → ℝn in a Sobolev–Orlicz space W1,Φ (Ω,ℝn). Their natural generalizations are the wedge products of differential forms. These products turn out to be in the Hardy–Orlicz spaces ℌp (Ω). Other nonlinear quantities involving the Jacobian, such as J log |J|, are also studied. In general, the Jacobians may change sign and in this sense our results generalize the existing ones concerning positive Jacobians.

We obtain global existence and regularity of strong solutions to the incompressible Navier–Stokes equations for a variety of boundary conditions in such a way that the initial and forcing data can be large in the high-frequency eigenspaces of the Stokes operator. We do not require that the domain be thin as in previous analyses. But in the case of thin domains (and zero Dirichlet boundary conditions) our results represent a further improvement and refinement of previous results obtained.

A new model of random graphs – random intersection graphs – is introduced. In this model, vertices are assigned random subsets of a given set. Two vertices are adjacent provided their assigned sets intersect. We explore the evolution of random intersection graphs by studying thresholds for the appearance and disappearance of small induced subgraphs. An application to gate matrix circuit design is presented.

We shall prove multiplicity of non-trivial solutions for the following p-Laplacian problems

where f(x,u) is continuous, periodic in x and has superlinear growth in u.

We present a general algebraic technique and discuss some of its numerous applications in combinatorial number theory, in graph theory and in combinatorics. These applications include results in additive number theory and in the study of graph colouring problems. Many of these are known results, to which we present unified proofs, and some results are new.

In this note, we establish a variational setting for harmonic morphisms for target spaces of any dimension. We then extend this result to horizontally weakly conformal p-harmonic maps, such maps being p-harmonic morphisms.