Existence Of Entire Solutions Research Articles

Existence of entire radial solutions to Hessian type system

AbstractIn this paper, the Dirichlet problem of Hessian type system is studied. After converting the existence of an entire solution to the existence of a fixed point of a continuous mapping, the existence of entire radial solutions is established by Schaefer fixed point theorem.

Critical equations with Hardy terms in the Heisenberg group

In this paper, we are concerned with the study of a critical (p, q) equation with Hardy terms on the Heisenberg group. Existence of entire solutions is obtained via an application of some concentration–compactness type results and the mountain pass theorem. Our results are presented in the model case of the (p, q) horizontal Laplacian equations, but the method can be extended to deal with a more general class of problems with operators of (p, q) growth.

Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in C2

This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z), and the partial differential difference equations f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z12=eg(z),f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z2+∂f(z)∂z1+∂f(z)∂z22=eg(z). We establish some theorems about the forms of the finite order transcendental entire solutions of these functional equations. We also list a series of examples to explain the existence of the finite order transcendental entire solutions of such equations. Meantime, some examples show that there exists a very significant difference with the previous literature on the growth order of the finite order transcendental entire solutions. Our results show that some functional equations can admit the transcendental entire solutions with any positive integer order. These results make a few improvements of the previous theorems given by Xu and Cao, Liu and Yang.

Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph

<p style='text-indent:20px;'>We deal with the bistable reaction-diffusion equation in an infinite star graph, which consists of several half-lines with a common end point. The aim of our study is to show the existence of front-like entire solutions together with the asymptotic behaviors as <inline-formula><tex-math id="M1">\begin{document}$ t\to\pm\infty $\end{document}</tex-math></inline-formula> and classify the entire solutions according to their behaviors, where an entire solution is meant by a classical solution defined for all <inline-formula><tex-math id="M2">\begin{document}$ t\in(-\infty, \infty) $\end{document}</tex-math></inline-formula>. To this end, we give a condition under that the front propagation is blocked by the emergence of standing stationary solutions. The existence of an entire solution which propagates beyond the blocking is also shown.

Existence of entire solutions for critical Sobolev–Hardy problems involving magnetic fractional operator

In this paper, we deal with a class of Schrödinger–Kirchhoff problems in , driven by the fractional magnetic operator, involving critical Sobolev–Hardy nonlinearities and a nontrivial perturbation term. By variational approach, we obtain the existence of solutions which tend to zero under a suitable value of λ. The main feature and difficulty of our equations is the presence of the magnetic field and critical term as well as the possible degenerate nature of the Kirchhoff function M.

Entire solutions for a delayed lattice competitive system

In this paper, we investigate the existence of entire solutions for a delayed lattice competitive system. Here the entire solutions are the solutions that exist for all (n,t)in mathbb{Z}times mathbb{R}. In order to prove the existence, we firstly embed the delayed lattice system into the corresponding larger system, of which the traveling front solutions are identical to those of the delayed lattice system. Then based on the comparison theorem and the sup–sub solutions method, we construct entire solutions which behave as two opposite traveling front solutions moving towards each other from both sides of x-axis and then annihilating. Moreover, our conclusions extend the invading way, which the superior species invade the inferior ones from both sides of x-axis and then the inferior ones extinct, into the lattice and delay case.

Entire solutions for a reaction\u2013diffusion equation with doubly degenerate nonlinearity

This paper is concerned with the existence of entire solutions for a reaction–diffusion equation with doubly degenerate nonlinearity. Here the entire solutions are the classical solutions that exist for all (x,t)in mathbb{R}^{2}. With the aid of the comparison theorem and the sup-sub solutions method, we construct some entire solutions that behave as two opposite traveling front solutions with critical speeds moving towards each other from both sides of x-axis and then annihilating. In addition, we apply the existence theorem to a specially doubly degenerate case.

Existence of entire solutions of some non-linear differential-difference equations

In this paper, we investigate the admissible entire solutions of finite order of the differential-difference equations (f'(z))^{2}+P^{2}(z)f^{2}(z+c)=Q(z)e^{alpha(z)} and (f'(z))^{2}+[f(z+c)-f(z)]^{2}=Q(z)e^{alpha(z)}, where P(z), Q(z) are two non-zero polynomials, alpha(z) is a polynomial and cinmathbb{C}backslash{0}. In addition, we investigate the non-existence of entire solutions of finite order of the differential-difference equation (f'(z))^{n}+P(z)f^{m}(z+c)=Q(z), where P(z), Q(z) are two non-constant polynomials, cinmathbb{C}backslash{0}, m, n are positive integers and satisfy frac{1}{m}+frac{1}{n}<2 except for m=1, n=2.

Entire solutions for some reaction–diffusion systems

This paper is concerned with the existence of entire solutions of some reaction–diffusion systems. We first consider Belousov–Zhabotinskii reaction model. Then we study a general model. Using the comparing argument and sub-super-solutions method, we obtain the existence of entire solutions which behave as two wavefronts coming from the both sides of x-axis, where an entire solution is meant by a classical solution defined for all space and time variables. At last, we give some examples to explain our results for the general models.

On Properties of Entire Solutions of Difference Equations and Difference Polynomials

Abstract In this paper, we consider the existence of entire solutions of certain type of non-linear difference equation of the form: f(z)n + p(z)f(z + c)m = q(z). We also consider value distribution problems of difference polynomials of entire functions such as f(z)n(f(z)−1)f(z +c) for n = 1, which is a supplement of previous results.

Entire solutions of delayed reaction‐diffusion equations

AbstractThis paper is concerned with entire solutions of delayed reaction‐diffusion equations. Using the comparing argument and sub‐super solutions method, we obtain the existence of entire solutions which behave as two wave fronts coming from the both sides of x‐axis, where an entire solution is meant by a classical solution defined for all space and time variables.

Asymptotic behavior of traveling fronts and entire solutions for a nonlocal monostable equation

This paper is concerned with the asymptotic behavior of traveling fronts and entire solutions for a nonlocal monostable equation. The asymptotic behavior of traveling fronts is obtained using Ikehara’s Theorem. By the sub–super-solution method, the existence of entire solutions is obtained; they behave as two opposite wave fronts approaching each other from either side of the x -axis and then annihilating in a finite time. By an entire solution we mean one which is defined over the whole space for all time t ∈ R .

Solutions of quasilinear elliptic equations in R N decaying at infinity to a non-negative number

We deal with existence of entire solutions for the quasilinear elliptic problem where 1 < p < N, N ≥ 3, ℓ ≥ 0, Δ p is the p-Laplacian operator, λ > 0 is a parameter, a : R N → (0, ∞) and f : (0, ∞) → (0, ∞) are suitable functions. When ℓ = 0, f is allowed to behave at 0 like . The potential a(x) will be required to decay to zero at infinity fast enough. Our technique explores variational principles, symmetry arguments as well as lower and upper solutions.

On the existence of entire solutions to a class of semilinear elliptic equations on noncompact Riemann manifolds

On the existence of entire solutions to a class of semilinear elliptic equations on noncompact Riemann manifolds

Existence of Entire Solutions of a Singular Semilinear Elliptic Problem

In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang’s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems.

Entire Solutions of Nonhomogeneous Linear Differential Equations

In this paper we give conditions on the existence of entire solutions of second order linear nonhomogeneous ordinary differential equations.

Analytic solutions of nonlinear neutral and advanced differential equatios

A study is made of local existence and uniqueness theorems for analytic solutions of nonlinear differential equations of neutral and advanced types. These results are of special interest for advanced eauations whose solutions, in general, lose their margin of smoothness. Furthermore, existence of entire solutions is established for linear advanced differential systems with polynomial coefficients.