Static Vacuum Solutions from Convergent Null Data Expansions at Space-Like Infinity

We study formal expansions of asymptotically flat solutions to the static vacuum field equations which are determined by minimal sets of freely specifyable data referred to as ‘null data’. These are given by sequences of symmetric trace free tensors at space-like infinity of increasing order. They are 1 : 1 related to the sequences of Geroch multipoles. Necessary and sufficient growth estimates on the null data are obtained for the formal expansions to be absolutely convergent. This provides a complete characterization of all asymptotically flat solutions to the static vacuum field equations. Submitted: October 26, 2006. Accepted: October 29, 2006.     

On Convergent Series Expansions of Solutions of the Riccati Equation

Mathematical Notes - The Riccati equation with coefficients expandable in convergent power series in a neighborhood of infinity are considered. Extendable solutions of such equations are studied....     

On Convergent Series Expansions for Solutions of Nonlinear Ordinary Differential Equations

Doklady Mathematics - We consider a large class of nonlinear ordinary differential equations of arbitrary order with coefficients in the form of power series that converge in a neighborhood of the...     

Solutions of mKdV in Classes of Functions Unbounded at Infinity

Using P. Lax’s concept of a Lax pair we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and in particular, may tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of the Schrödinger operator under the KdV flow and the invariance of the spectrum and the unitary type of the impedance operator under the mKdV flow for potentials in these classes.     

Comparison Principle for Viscosity Solutions with High Growth at Infinity

This paper is concerned with the comparison principle for viscosity solutions of the nonlinear elliptic equation F(Du, D²u} + |u|^{s-1}u =f in R^N, where f is uniformly continuous and F satisfies some conditions about p (p > 2}. We got the comparison principle for the viscosity solutions with some high growth at infinity, which relies on the relation between p and s.     

Asymptotic Behavior at Infinity of Solutions to the Nonhomogeneous Sobolev Equation

Under study is the asymptotic behavior at infinity of solutions to the Cauchy problem to the nonhomogeneous Sobolev equation. We obtain the form of the limit function and the convergence rate.     

Solutions Almost Periodic at Infinity to Differential Equations With Unbounded Operator Coefficients

The new class of functions almost periodic at infinity is defined using the subspace of functions with integrals decreasing at infinity. We obtain spectral criteria for almost periodicity at infinity of bounded solutions to differential equations with unbounded operator coefficients. For the new class of asymptotically finite operator semigroups we prove the almost periodicity at infinity of their orbits.     

Nontrivial Solutions for p-Laplace Equations with Right-Hand Side Having p-Linear Growth at Infinity

ABSTRACT

The existence of a nontrivial solution for quasi-linear elliptic equations involving the p-Laplace operator and a nonlinearity with p-linear growth at infinity is proved. Techniques of Morse theory are employed.     

Multiple Solutions to p-Laplacian Problems with Asymptotic Nonlinearity as up-1 at Infinity

The paper studies the existence of multiple solutions to thefollowing p-Laplacian type elliptic problem (p > 1): where is a bounded domain in R^N(N 1) with smooth boundary, and f(x, u) goes asymptotically in u to |u|^p–2u at infinity.It is well known that this kind of nonlinear term creates somedifficulties in the application of the mountain pass theorembecause of the lack of an Ambrosetti–Rabinowitz type superlinearcondition on f(x, u). An improved mountain pass theorem is usedto prove that the above problem possesses multiple solutionsunder some natural conditions on f(x, u), and some known resultsare generalized.     

On Complicated Expansions of Solutions to ODES

Polynomial ordinary differential equations are studied by asymptotic methods. The truncated equation associated with a vertex or a nonhorizontal edge of their polygon of the initial equation is assumed to have a solution containing the logarithm of the independent variable. It is shown that, under very weak constraints, this nonpower asymptotic form of solutions to the original equation can be extended to an asymptotic expansion of these solutions. This is an expansion in powers of the independent variable with coefficients being Laurent series in decreasing powers of the logarithm. Such expansions are sometimes called psi-series. Algorithms for such computations are described. Six examples are given. Four of them are concern with Painlevé equations. An unexpected property of these expansions is revealed.     

Convergent Expansions for Solutions of Linear Ordinary Differential Equations Having a Simple Turning Point, with an Application to Bessel Functions

Second-order linear ordinary differential equations with a large parameter u are examined. Classic asymptotic expansions involving Airy functions are applicable for the case where the argument z lies in complex domain containing a simple turning point. In this article, such asymptotic expansions are converted into convergent series, where u appears in an inverse factorial, rather than an inverse power. The domain of convergence of the new expansions is rigorously established and is found to be an unbounded domain containing the turning point. The theory is then applied to obtain convergent expansions for Bessel functions of complex argument and large positive order.     

Convergent Expansions for Solutions of Linear Ordinary Differential Equations Having a Simple Pole, with an Application to Associated Legendre Functions

Second-order linear ordinary differential equations with a large parameter u are examined. Asymptotic expansions involving modified Bessel functions are applicable for the case where the coefficient function of the large parameter has a simple pole. In this paper, we examine such equations in the complex plane, and convert the asymptotic expansions into uniformly convergent series, where u appears in an inverse factorial, rather than an inverse power. Under certain mild conditions, the region of convergence containing the simple pole is unbounded. The theory is applied to obtain exact connection formulas for general solutions of the equation, and also, in a special case, to obtain convergent expansions for associated Legendre functions of complex argument and large degree.     

Limits at Infinity of Superharmonic Functions and Solutions of Semilinear Elliptic Equations of Matukuma Type

In an unbounded domain Ω in ℝ ⁿ (n ≥ 2) with a compact boundary or Ω = ℝ ⁿ , we investigate the existence of limits at infinity of positive superharmonic functions u on Ω satisfying a nonlinear inequality like as

带有共振的二阶哈密顿系统非平凡解的存在性

本文研究二阶哈密顿系统的非平凡解问题.假设系统中的非线性项V′是渐近线性的.利用变分法,通过系统对应泛函的小扰动的临界点来建立系统的Palais-Smale序列,进而说明该序列的有界性.与一般做法不同的是,本文对V′不限定Landesman-Lazer条件.     

Asymptotic Expansions of Solutions to the Riccati Equation

Doklady Mathematics - Scalar real Riccati equations with coefficients expandable in convergent power series in a neighborhood of infinity are considered. Extendable solutions to equations of this...     

Existence of Solutions to Nonlinear Schrödinger Equations Involving N-Laplacian and Potentials Vanishing at Infinity

We study the existence of solutions for the following class of nonlinear Schrödinger equations -Δ_Nu + V (x)u=K(x)f(u) in R^N where V and K are bounded and decaying potentials and the nonlinearity f(s) has exponential critical growth. The approaches used here are based on a version of the Trudinger-Moser inequality and a minimax theorem.     

Asymptotic Expansions for Stationary Distributions of Nonlinearly Perturbed Semi-Markov Processes. 2

Methodology and Computing in Applied Probability - Asymptotic expansions with explicit upper bounds for remainders are given for stationary distributions of nonlinearly perturbed semi-Markov...