We prove a Schauder estimate for solutions of linear second order elliptic equations with linear Venttsel boundary conditions, and establish an existence result for classical solutions for such boundary value problems.

A certain plane sextic of genus 5 was encountered by Humbert and publicised by him [3] in 1894. Its striking geometrical properties clamour for elucidation; this was eventually supplied in 1951. For the canonical curve of genus 5 is the base curve C of a net N of quadrics in projective space [4], and C models a Humbert curve when all the quadrics of N have a common self-polar simplex [1]. The projection of C from one of its chords onto a plane is a 5-nodal sextic, the nodes all becoming cusps when the chord of C becomes a tangent. The properties to be elucidated become clear visually in the projection.

The sextic H described here is a specialisation of the cusped curve; it emerges as linearly dependent on a pair of reducible plane sextics concocted ad hoc.

In this paper we consider a simple model for coupled nonlinear oscillators in a continuous medium. There is shown to exist a maximal branch of phaselocked solutions connecting the trivial one to a limiting singular solution.

We consider nonlinear eigenvalue problems of the form Lu + F(u) = λu in a real Hilbert space, where L is a positive self-adjoint linear operator and F is a nonlinearity vanishing to higher order at u = 0. We suppose that there are gaps in the essential spectrum of L and use critical point theory for strongly indefinite functionals to derive conditions for the existence of non-zero solutions for λ belonging to such a gap, and for the bifurcation of such solutions from the line of trivial solutions at the boundary points of a gap. The abstract results are applied to the L2-theory of semilinear elliptic partial differential equations on ℝN. We obtain existence results for the general case and bifurcation results for nonlinear perturbations of the periodic Schrödinger equation.

The paper discusses properties of solutions of the differential equation

and in particular the existence and uniqueness of solutions to the boundary-value problem associated with the above equation and the boundary conditions

This equation, first introduced by Skyrme, is a spherically symmetric model for a soliton description of nucleons.

We introduce and begin an investigation of the family of hypergeometric groups. We determine which hypergeometric groups of Euclidean type are infinite.

We consider the non-linear differential equation

and state conditions for the function q such that (*) has infinitely many distinct pairs of (weak) solutions such that holds for all k ∈ ℕ. The main tools are results from critical point theory developed by A. Ambrosetti and P. H. Rabinowitz [1].

We analyse the set of nonnegative, global, and radial solutions (radial solutions, for short) of the equation

where 0 < p < 1, and is a radial and almost everywhere nonnegative function. We show that radial solutions of (E) exist if f(r) = o(r2p/1−1−p) or if f(r) ≈ cr2p/1−p as r → ∞, where

When f(r) = c*r2p/1−p + h(r) with h(r) = o(r2p/1−p) as r → ∞, radial solutions continue to exist if h(r) is sufficiently small at infinity. Existence, however, breaks down if h(r) > 0,

Whenever they exist, radial solutions are characterised in terms of their asymptotic behaviour as r → ∞.

We first study the Poisson equation Δu = f in Ώω, and , where Ωω = {(r cos θ, r sin θ): 0<r<1, θ ∈(0,ω)} is a sector in ℝ2, ω ∈ (0, 2π), Г0 = {(cos θ, sin θ): θ ∈ (0, ω)} and Г1 = ∂Ωω − Г0,b and λ are in ℝ1. We obtain Schauder-type estimates and Fredholm alternative theory for the problem. We then study the symmetry breaking problem for the Gel'fand equation Δu + λeu = 0 in Ωω and obtain a complete picture about the relationships among three parameters λ, b, and ω in the problem.

We prove the well-posedness for a one-dimensional free boundary problem arising from some reaction diffusion system. The interfacial point hits a boundary point in finite time or remains inside for all time. In the large diffusion limit, the system is reduced to ordinary differential equations of finite dimension.