Positive solutions for second-order superlinear repulsive singular Neumann boundary value problems

In this paper we establish the multiplicity of positive solutions to second-order superlinear repulsive singular Neumann boundary value problems. It is proved that such a problem has at least two positive solutions under reasonable conditions. Our nonlinearity may be repulsive singular in its dependent variable and superlinear at infinity. The proof relies on a nonlinear alternative of Leray-Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.      

Multiple solutions for three-point boundary value problem with nonlinear terms depending on the first order derivative

In this paper, we consider the following second-order three-point boundary value problem

The existence of three positive solutions for some nonlinear boundary value problems on the half-line

In this paper, by introducing a new operator, improving and generating a p-Laplace operator for some p > 1, we consider the existence of triple positive solutions for some nonlinear m-point boundary value problems on the half-line

BOUNDARY VALUE PROBLEM FOR A GENERALIZED LIENARD EQUATION

By coincidence degree, the existence of solution to the boundary value problem of a generalized Liénard equation

Boundary value problems for the one-dimensional Willmore equation

The one-dimensional Willmore equation is studied under Navier as well as under Dirichlet boundary conditions. We are interested in smooth graph solutions, since for suitable boundary data, we expect the stable solutions to be among these. In the first part, classical symmetric solutions for symmetric boundary data are studied and closed expressions are deduced. In the Navier case, one has existence of precisely two solutions for boundary data below a suitable threshold, precisely one solution on the threshold and no solution beyond the threshold. This effect reflects that we have a bending point in the corresponding bifurcation diagram and is not due to that we restrict ourselves to graphs. Under Dirichlet boundary conditions we always have existence of precisely one symmetric solution. In the second part, we consider boundary value problems with nonsymmetric data. Solutions are constructed by rotating and rescaling suitable parts of the graph of an explicit symmetric solution. One basic observation for the symmetric case can already be found in Euler’s work. It is one goal of the present paper to make Euler’s observation more accessible and to develop it under the point of view of boundary value problems. Moreover, general existence results are proved.     

A priori estimates for the vector p-Laplacian with potential boundary conditions

This paper deals with the solvability of the boundary value problem

Nonregular Boundary Problems for Elliptic Systems

We discuss explicit boundary value problems for solutions of the Fueter equation in which are normally solvable. The results extend to nonlinear first order elliptic systems. Received: October, 2007, Accepted: February, 2008.     

Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

We study the limit behaviour of solutions of with initial data k δ ₀ when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r ^β , β > N(p − 1) − 2, we prove that the limit function u _∞ is an explicit very singular solution, while such a solution does not exist if β ≤  N(p − 1) − 2. If lim inf _{r→ 0} r ² ln (1/h(r))  >  0, u _∞ has a persistent singularity at (0, t) (t ≥  0). If , u _∞ has a pointwise singularity localized at (0, 0).     

On two-point boundary value problems for systems of higher-order ordinary differential equations with singularities

The sufficient conditions of solvability and unique solvability of the two-point boundary value problems of Vallèe-Poussin and Cauchy-Niccoletti have been found for a system of ordinary differential equations of the form

Mixed Boundary Value Problems of Thermopiezoelectricity for Solids with Interior Cracks

We investigate asymptotic properties of solutions to mixed boundary value problems of thermopiezoelectricity (thermoelectroelasticity) for homogeneous anisotropic solids with interior cracks. Using the potential methods and theory of pseudodifferential equations on manifolds with boundary we prove the existence and uniqueness of solutions. The singularities and asymptotic behaviour of the mechanical, thermal and electric fields are analysed near the crack edges and near the curves, where the types of boundary conditions change. In particular, for some important classes of anisotropic media we derive explicit expressions for the corresponding stress singularity exponents and demonstrate their dependence on the material parameters. The questions related to the so called oscillating singularities are treated in detail as well. This research was supported by the Georgian National Science Foundation grant GNSF/ST07/3-170 and by the German Research Foundation grant DFG 436 GEO113/8/0-1.     

Boundary value problems for some nonlinear systems with singular \phi-laplacian

Systems of differential equations of the form

Existence of positive solutions for a class of second-order two-point boundary value problem

By using different convex functionals to compute fixed point index, the existence of positive solutions for a class of second-order two-point boundary value problem

Positive Solutions of Equations with Nonlinear Causal Mappings

We consider nonlinear scalar equations with causal mappings. These equations include differential, differential-delay, integral, integro-differential, difference and other traditional equations. Conditions that provide the existence of positive solutions are established. The suggested approach enables us to consider various classes of equations from the unified point of view. This research was supported by the Kamea Fund of the Israel     

Bimodules associated to vertex operator algebras

Let V be a vertex operator algebra and m, n ≥ 0. We construct an A _n (V)-A _m (V)-bimodule A _n,m (V) which determines the action of V from the level m subspace to level n subspace of an admissible V-module. We show how to use A _n,m (V) to construct naturally admissible V-modules from A _m (V)-modules. We also determine the structure of A _n,m (V) when V is rational. Chongying Dong was supported by NSF grants, China NSF grant 10328102 and a Faculty research grant from the University of California at Santa Cruz. Cuipo Jiang was supported in part by China NSF grant 10571119.     

On pointed Hopf algebras associated with the symmetric groups

Let G be the symmetric group . It is an important open problem whether the dimension of the Nichols algebra is finite when is the class of the transpositions and ρ is the sign representation, with m ≥ 6. In the present paper, we discard most of the other conjugacy classes showing that very few pairs might give rise to finite-dimensional Nichols algebras. This work was partially supported by CONICET, ANPCyT and Secyt (UNC).     

Vector Equilibrium Problems,Minimal Element Problems and Least Element Problems

In this paper we introduce some concepts of feasible sets for vector equilibrium problems and some classes of Z-maps for vectorial bifunctions. Under strict pseudomonotonicity assumptions, we investigate the relationship between minimal element problems of feasible sets and vector equilibrium problems. By using Z-maps, we further study the least element problems of feasible sets for vector equilibrium problems. Finally, we prove a generalized sublattice property of feasible sets for vector equilibrium problems associated with Z-maps. This work was supported by the National Natural Science Foundation of China and the Applied Research Project of Sichuan Province (05JY029-009-1). The authors thank Professor Charalambos D. Aliprantis and the referees for valuable comments and suggestions leading to improvements of this paper.     

Characterizations of Hankel multipliers

We give characterizations of radial Fourier multipliers as acting on radial L ^p functions, 1 < p < 2d/(d + 1), in terms of Lebesgue space norms for Fourier localized pieces of the convolution kernel. This is a special case of corresponding results for general Hankel multipliers. Besides L ^p  − L ^q bounds we also characterize weak type inequalities and intermediate inequalities involving Lorentz spaces. Applications include results on interpolation of multiplier spaces. G. Garrigós partially supported by grant “MTM2007-60952” and Programa Ramón y Cajal, MCyT (Spain). A. Seeger partially supported by NSF grant DMS 0652890.     

Elliptic Problems with Singular Potential and Double-Power Nonlinearity

We prove the existence of positive symmetric solutions to the semilinear elliptic problem

Convergence of LR algorithm for a one-point spectrum tridiagonal matrix

We prove convergence for the basic LR algorithm on a real unreduced tridiagonal matrix with a one-point spectrum—the Jordan form is one big Jordan block. First we develop properties of eigenvector matrices. We also show how to deal with the singular case.     

Hausdorff Dimension of Ruptures for Solutions of a Semilinear Elliptic Equation with Singular Nonlinearity

We consider the following semilinear elliptic equation with singular nonlinearity: