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We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form \begin{equation*} (\Delta - \lambda^2) u = N[u], \end{equation*} where $\Delta = -\sum_j \partial^2_j$ is the…
It has been proven by Rosu and Cornejo-Perez in 2005 that for some nonlinear second-order ODEs it is a very simple task to find one particular solution once the nonlinear equation is factorized with the use of two first-order differential…
We derive a priori second order estimates for fully nonlinear elliptic equations which depend on the gradients of solutions in critical ways on Hermitian manifolds. The global estimates we obtained apply to an equation arising from a…
We investigate the growth of the Nevanlinna Characteristic of f(z+\eta) for a fixed \eta in this paper. In particular, we obtain a precise asymptotic relation between T(r,f(z+\eta) and T(r,f), which is only true for finite order meromorphic…
Using a new definition of generalized divisors we prove that the lattice of such divisors for a given linear partial differential operator is modular and obtain analogues of the well-known theorems of the Loewy-Ore theory of factorization…
The existence of meromorphic solutions to various difference equations has been extensively studied in recent years, the precise functional forms of such solutions -- particularly when the function and its difference operators share values…
We derive a priori second order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under some very general structure conditions. We treat both equations on closed manifolds, and the Dirichlet…
This article complements recent results of the papers [J. Math. Phys. 41 (2000), 480; 45 (2004), 336] on the symmetry classification of second-order ordinary difference equations and meshes, as well as the Lagrangian formalism and…
In this paper second order elliptic boundary value problems on bounded domains $\Omega\subset\dR^n$ with boundary conditions on $\partial\Omega$ depending nonlinearly on the spectral parameter are investigated in an operator theoretic…
We consider second-order elliptic equations in non-divergence form with oblique derivative boundary conditions. We show that any strong solutions to such problems are twice continuously differentiable up to the boundary provided that the…
It is proven that second-order vectorial nonlinear differential systems y''=f(y) , possess a continuum of symmetric solutions. They are shown to possess a continuum of even solutions. If f(y) is an odd function of y , then y''=f(y) is shown…
A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic…
The method of double extension, introduced by A.~Medina and Ph.~Revoy, is a procedure which decomposes a Lie algebra with an invariant symmetric form into elementary pieces. Such decompositions were developed for other algebras, for…
Identifying integrable coupled nonlinear ordinary differential equations (ODEs) of dissipative type and deducing their general solutions are some of the challenging tasks in nonlinear dynamics. In this paper we undertake these problems and…
General solutions of nonlinear ordinary differential equations (ODEs) are in general difficult to find although powerful integrability techniques exist in the literature for this purpose. It has been shown that in some scalar cases…
Point singularities of solutions to the classical Lane-Emden-Serrin equation have a polyhomogeneous asymptotic expansion whose logarithmic corrections are determined by a first order ODE. Surprisingly, we are able to discover such an ODE…
We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully…
This paper is part of a series of papers in which the asymptotic theory and appropriate symbolic computer code are developed to compute the asymptotic expansion of the solution of an n-th order ordinary differential equation. The paper…
In this paper, we will give suitable conditions on differential polynomials $Q(f)$ such that they take every finite non-zero value infinitely often, where $f$ is a meromorphic function in complex plane. These results are related to Problem…
In this article, we study the growth of meromorphic solutions of linear delay-differential equation of the form \begin{equation*} \sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}(z)f^{(j)}(z+c_{i})=F(z), \end{equation*}% where $A_{ij}(z)$ $(i=0,1,\ldots…