Radial Symmetry of Entire Solutions of a Biharmonic Equation with Supercritical Exponent

2019 ◽  
Vol 19 (2) ◽  
pp. 291-316
Author(s):  
Zongming Guo ◽  
Long Wei
Keyword(s):  
Asymptotic Behavior ◽  
Biharmonic Equation ◽  
Sufficient Conditions ◽  
Entire Solution ◽  
Radial Symmetry ◽  
Entire Solutions ◽  
Moving Plane ◽  
Exact Asymptotic Behavior ◽  
Necessary And Sufficient ◽  
Positive Entire Solution

AbstractNecessary and sufficient conditions for a regular positive entire solution u of a biharmonic equation\Delta^{2}u=u^{p}\quad\text{in }\mathbb{R}^{N},\,N\geq 5,\,p>\frac{N+4}{N-4}to be a radially symmetric solution are obtained via the exact asymptotic behavior of u at {\infty} and the moving plane method (MPM). It is known that above equation admits a unique positive radial entire solution {u(x)=u(|x|)} for any given {u(0)>0}, and the asymptotic behavior of {u(|x|)} at {\infty} is also known. We will see that the behavior similar to that of a radial entire solution of above equation at {\infty}, in turn, determines the radial symmetry of a general positive entire solution {u(x)} of the equation. To make the procedure of the MPM work, the precise asymptotic behavior of u at {\infty} is obtained.

Radial symmetry of non-maximal entire solutions of a bi-harmonic equation with exponential nonlinearity

2019 ◽  
Vol 149 (6) ◽  
pp. 1603-1625
Author(s):  
Hongxia Guo ◽  
Zongming Guo ◽  
Fangshu Wan
Keyword(s):  
Sufficient Conditions ◽  
Entire Solution ◽  
Radial Symmetry ◽  
Necessary And Sufficient Conditions ◽  
Entire Solutions ◽  
Exponential Nonlinearity ◽  
Image Position ◽  
Harmonic Equation ◽  
Necessary And Sufficient

AbstractWe study radial symmetry of entire solutions of the equation0.1$$\Delta ^2u = 8(N-2)(N-4)e^u\quad {\rm in}\;R^N\;\;(N \ges 5).$$It is known that (0.1) admits infinitely many radially symmetric entire solutions. These solutions may have either a (negative) logarithmic behaviour or a (negative) quadratic behaviour at infinity. Up to translations, we know that there is only one radial entire solution with the former behaviour, which is called ‘maximal radial entire solution’, and infinitely many radial entire solutions with the latter behaviour, which are called ‘non-maximal radial entire solutions’. The necessary and sufficient conditions for an entire solutionuof (0.1) to be the maximal radial entire solution are presented in [7] recently. In this paper, we will give the necessary and sufficient conditions for an entire solutionuof (0.1) to be a non-maximal radial entire solution.

scholarly journals The exterior Dirichlet problems of Monge–Ampère equations in dimension two

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Unit Ball ◽  
Bounded Domain ◽  
Viscosity Solutions ◽  
Sufficient Conditions ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Necessary And Sufficient ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

AbstractIn this paper, we study the Monge–Ampère equations $\det D^{2}u=f$ det D 2 u = f in dimension two with f being a perturbation of $f_{0}$ f 0 at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.

Asymptotic behavior of the records of multivariate random sequences in a norm sense

2020 ◽  
Vol 70 (6) ◽  
pp. 1457-1468
Author(s):  
Haroon M. Barakat ◽  
M. H. Harpy
Keyword(s):  
Asymptotic Behavior ◽  
Weak Convergence ◽  
Sufficient Conditions ◽  
Record Values ◽  
Necessary And Sufficient Conditions ◽  
Random Sequences ◽  
Necessary And Sufficient

AbstractIn this paper, we investigate the asymptotic behavior of the multivariate record values by using the Reduced Ordering Principle (R-ordering). Necessary and sufficient conditions for weak convergence of the multivariate record values based on sup-norm are determined. Some illustrative examples are given.

scholarly journals On the oscillation of a nonlinear two-dimensional difference systems

2001 ◽  
Vol 32 (3) ◽  
pp. 201-209 ◽  
Author(s):  
E. Thandapani ◽  
B. Ponnammal
Keyword(s):  
Asymptotic Behavior ◽  
Sufficient Conditions ◽  
Real Sequence ◽  
Two Dimensional ◽  
Difference System ◽  
Nonoscillatory Solutions ◽  
Difference Systems ◽  
All Solutions ◽  
Necessary And Sufficient ◽  
Strongly Sublinear

The authors consider the two-dimensional difference system$$ \Delta x_n = b_n g (y_n) $$ $$ \Delta y_n = -f(n, x_{n+1}) $$where $ n \in N(n_0) = \{ n_0, n_0+1, \ldots \} $, $ n_0 $ a nonnegative integer; $ \{ b_n \} $ is a real sequence, $ f: N(n_0) \times {\rm R} \to {\rm R} $ is continuous with $ u f(n,u) > 0 $ for all $ u \ne 0 $. Necessary and sufficient conditions for the existence of nonoscillatory solutions with a specified asymptotic behavior are given. Also sufficient conditions for all solutions to be oscillatory are obtained if $ f $ is either strongly sublinear or strongly superlinear. Examples of their results are also inserted.

Asymptotic behavior of Markov systems

1982 ◽  
Vol 19 (04) ◽  
pp. 851-857 ◽  
Author(s):  
P.-C. G. Vassiliou
Keyword(s):  
Asymptotic Behavior ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weak Ergodicity ◽  
Markov Systems ◽  
Homogeneous Systems ◽  
Necessary And Sufficient

In this paper we study the asymptotic behavior of Markov systems and especially non-homogeneous Markov systems. It is found that the limiting structure and the relative limiting structure exist under certain conditions. The problem of weak ergodicity in the above non-homogeneous systems is studied. Necessary and sufficient conditions are provided for weak ergodicity. Finally, we discuss the application of the present results in manpower systems.

scholarly journals Entire Solutions of an Integral Equation in R5

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Xin Feng ◽  
Xingwang Xu
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Integral Form ◽  
Entire Solution ◽  
Entire Solutions ◽  
Positive Entire Solution ◽  
Moving Sphere Method

We will study the entire positive C0 solution of the geometrically and analytically interesting integral equation: u(x)=1/C5∫R5‍|x-y|u-q(y)dy with 0<q in R5. We will show that only when q=11, there are positive entire solutions which are given by the closed form u(x)=c(1+|x|2)1/2 up to dilation and translation. The paper consists of two parts. The first part is devoted to showing that q must be equal to 11 if there exists a positive entire solution to the integral equation. The tool to reach this conclusion is the well-known Pohozev identity. The amazing cancelation occurred in Pohozev’s identity helps us to conclude the claim. It is this exponent which makes the moving sphere method work. In the second part, as normal, we adopt the moving sphere method based on the integral form to solve the integral equation.

Best Polynomial Approximation with Linear Constraints

1992 ◽  
Vol 44 (6) ◽  
pp. 1289-1302
Author(s):  
K. Pan ◽  
E. B. Saff
Keyword(s):  
Asymptotic Behavior ◽  
Polynomial Approximation ◽  
Asymptotic Relation ◽  
Sufficient Conditions ◽  
Linear Constraints ◽  
Necessary And Sufficient Conditions ◽  
Compact Set ◽  
Approximation By Polynomials ◽  
The Matrix ◽  
Necessary And Sufficient

AbstractLet A be a (k + 1) × (k + 1) nonzero matrix. For polynomials p ∈ Pn, set and . Let E ⊂ C be a compact set that does not separate the plane and f be a function continuous on E and analytic in the interior of E. Set and . Our goal is to study approximation to f on E by polynomials from Bn(A). We obtain necessary and sufficient conditions on the matrix A for the convergence En(A,f) → 0 to take place. These results depend on whether zero lies inside, on the boundary or outside E and yield generalizations of theorems of Clunie, Hasson and Saff for approximation by polynomials that omit a power of z. Let be such that . We also study the asymptotic behavior of the zeros of and the asymptotic relation between En(f) and En(A,f).

scholarly journals Existence and Asymptotic Behavior of Traveling Wave Fronts for a Time-Delayed Degenerate Diffusion Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-20
Author(s):  
Weifang Yan ◽  
Rui Liu
Keyword(s):  
Asymptotic Behavior ◽  
Time Delay ◽  
Diffusion Equation ◽  
Traveling Wave ◽  
Sufficient Conditions ◽  
Degenerate Diffusion ◽  
Wave Fronts ◽  
Traveling Wave Fronts ◽  
Necessary And Sufficient ◽  
Degenerate Diffusion Equation

This paper is concerned with traveling wave fronts for a degenerate diffusion equation with time delay. We first establish the necessary and sufficient conditions to the existence of monotone increasing and decreasing traveling wave fronts, respectively. Moreover, special attention is paid to the asymptotic behavior of traveling wave fronts connecting two uniform steady states. Some previous results are extended.

scholarly journals Asymptotic Behavior of Solutions to Half-Linearq-Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Pavel Řehák
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Positive Solutions ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Special Limit ◽  
Bounded Functions ◽  
Necessary And Sufficient

We derive necessary and sufficient conditions for (some or all) positive solutions of the half-linearq-difference equationDq(Φ(Dqy(t)))+p(t)Φ(y(qt))=0,t∈{qk:k∈N0}withq>1,Φ(u)=|u|α−1sgn⁡uwithα>1, to behave likeq-regularly varying orq-rapidly varying orq-regularly bounded functions (that is, the functionsy, for which a special limit behavior ofy(qt)/y(t)ast→∞is prescribed). A thorough discussion on such an asymptotic behavior of solutions is provided. Related Kneser type criteria are presented.

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