This paper deals with existence, uniqueness and multiplicity results of positive solutions for the quasilinear elliptic boundary-value problem , where Ω is a bounded open domain in RN with smooth boundary. Under suitable assumptions on the matrix A(x, s), and depending on the behaviour of the function f near u = 0 and near u = +∞, we can use bifurcation theory in order to give a quite complete analysis on the set of positive solutions. We will generalize in different directions some of the results in the papers by Ambrosetti et al., Ambrosetti and Hess, and Artola and Boccardo.

Stable homotopy decompositions of the classifying spaces of the gauge groups of principal SO(3) and U(2)-bundles over the sphere S2 are obtained using a fibrewise stable splitting theorem for the loop space of an unreduced suspension. The stable decomposition is related to a description of the integral cohomology ring.

In this paper we construct a compact set K of zero Hausdorff dimension that satisfies certain ‘arithmetic-type’ thickness properties. The concept of ‘arithmetic thickness’ has its origins in applications to harmonic analysis, introduced in a paper by Lebedev and Olevskiĭ. For example, there are no spectral sets whose ‘essential boundary’ can contain the above set K.

Let f(·, λ) : R→R be given so that f(0, λ) = 0 and f(x, λ) = (1 + λ)x + ax2 + bx3 + o(x3) as x → 0. We characterize those small values of ε > 0 and λ ∈ R for which there are periodic solutions of periods approximately 1/k with k ∈ N of the delay equations When a = 0, these periodic solutions approach square waves if b < 0 or pulses if b > 0 as ε → 0. These results are similar to those obtained by Chow et al. and Hale and Huang, where the case of f(x, λ) = −(1 + λ)x + ax2 + bx3 + o(x3) as x → 0 is considered. However, when a ≠ 0, all these periodic solutions approach pulses as ε → 0; an interesting phenomenon that cannot happen in the case considered by Chow et al. and Hale and Huang.

The functional F(u) = ∫Bf(x, Du) is considered, where B is the unit ball in R2, u varies in the set of the locally Lipschitz functions on R2 and f belongs to a family containing, as model case, the following integrand:

The computation of the relaxed functional F̄ is provided yielding an explicit representation formula.

This formula nevertheless is not integral, because F̄ is not a measure and does not coincide with the obvious extension of F over all W1,p(B).

This phenomenon is essentially due to the non-standard growth behaviour of f(x, z) in the variable z.

We examine the singularly perturbed variational problem in the plane. As ε → 0, this functional favours |∇ψ| = 1 and penalizes singularities where |∇∇ψ| concentrates. Our main result is a compactness theorem: if {Eε(ψε)}ε↓0 is uniformly bounded, then {∇ψε}ε↓0 is compact in L2. Thus, in the limit ε → 0, ψ solves the eikonal equation |∇ψ| = 1 almost everywhere. Our analysis uses ‘entropy relations’ and the ‘div-curl lemma,’ adopting Tartar's approach to the interaction of linear differential equations and nonlinear algebraic relations.

We present an L–A pair for the Hess–Apel'rot case of a heavy rigid three-dimensional body. Using it, we give an algebro-geometric integration procedure. Generalizing this L–A pair, we obtain a new completely integrable case of the Euler–Poisson equations in dimension four. Explicit formulae for integrals that are in involution are given. This system is a counterexample to one of Ratiu's theorems. A corrected version of this classification theorem is proved.

We define a new class of admissible solutions for the parabolic problems where the theory of viscosity solutions does not apply. A typical example is the porous medium equation complemented by lower-order terms, nonlinear with respect to the gradient. In the case, the nonlinearity in the equation fails to be proper in the sense of the theory of viscosity solutions. The admissible solutions satisfy a very general comparison principle and, consequently, the corresponding initial-boundary value problems are well-posed in this class. They coincide with the standard viscosity solutions provided the nonlinearity in the equation is proper.

The Kadomtsev–Petviashvili (KP) equation can be formally derived as an envelope equation for three-dimensional unidirectional water waves in the limit of long waves. As a first step towards a mathematical justification, we consider here a two-dimensional Boussinesq equation, which is a realistic model for three-dimensional water waves. Using rigorous estimates, we show that part of the dynamics of the KP equation can be found approximately in the two-dimensional Boussinesq equation. On the other hand, there exist initial data for the KP equation such that the corresponding solutions of the two-dimensional Boussinesq equation behave in no way according to the KP prediction. We expect that similar results hold for the three-dimensional water wave problem too.

We consider the eigenvalue problem for the one-dimensional (stationary) Dirac operator with some boundary conditions. We prove that if the spectrum is the same as the spectrum belonging to the zero potential, then the potential is actually zero. The analogous statement for the Schrödinger operator is due to Ambarzumian. The proof is based on the fact that the (generalized) moments of a function cannot have alternating signs unless the moments are zero (see §2).

We study the spreading of characteristics for a class of one-dimensional scalar conservation laws for which the flux function has one point of inflection. It is well known that in the convex case the characteristic speed satisfies a one-sided Lipschitz estimate. Using Dafermos' theory of generalized characteristics, we show that the characteristic speed in the non-convex case satisfies an Hölder estimate. In addition, we give a one-sided Lipschitz estimate with an error term given by the decrease of the total variation of the solution.

We study a problem from nonlinear optics that leads to an integro-differential equation which can be written as the abstract bifurcation problem Au = λLu + ∇φ(u). For such a type of equation, a general theorem on the bifurcation of solutions from the essential spectrum is proved. The hypothesis that φ is non-negative (as is usually assumed) could be omitted and replaced by other conditions that allow us to use the results for the particular example. We also indicate applications to semilinear elliptic equations.

A generalized Yosida approximation of monotone (and non-monotone) operators in Banach space is introduced. It uses a general potential that is not necessarily the square of the norm. It is therefore advisable to use it in cases where some other more convenient potentials are available, such as in Lp-spaces. As an illustration, the case of Nemyckii operators is considered.

In the plane R2, we classify all solutions for an elliptic problem of Liouville type involving a (radial) weight function. As a consequence, we clarify the origin of the non-radially symmetric solutions for the given problem, as established by Chanillo and Kiessling.

For a more general class of Liouville-type problems, we show that, rather than radial symmetry, the solutions always inherit the invariance of the problem under inversion with respect to suitable circles. This symmetry result is derived with the help of a 'shrinking-sphere' method.