OPTIMAL SUMMATION INTERVAL AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A DISCRETE SYSTEM

In this paper, we are concerned with properties of positive solutions of the following Euler-Lagrange system associated with the weighted Hardy-Littlewood-Sobolev inequality in discrete form{uj =∑ k ∈Zn u~q_k/(1 + |j|)α(1 + |k- j|)λ(1 + |k|)β,(0.1)vj =∑ k ∈Zn u~p_k/(1 + |j|)β(1 + |k- j|)λ(1 + |k|),where u, v 0, 1 p, q ∞, 0 λ n, 0 ≤α + β≤ n- λ,1p+1λ+αnand1p+1+1q+1≤λ+α+βn:=λˉn. We first show that positive solutions of(0.1) have the optimal summation interval under assumptions that u ∈ lp+1(Zn) and v ∈ lq+1(Zn). Then we show that problem(0.1) has no positive solution if 0 λˉ pq ≤ 1 or pq 1 and max{(n-)(q+1)pq-1,(n-λˉ)(p+1)pq-1} ≥λˉ.     

Symmetry and nonexistence of positive solutions to an integral system with weighted functions

Consider the system of integral equations with weighted functions in Rn,{u(x) =∫Rn|x-y|α-nQ(y)v(y)qdy1,v(x)=∫Rn|x-y|α-nK(y)u(y)pdy,where 0 < α < n,1/(p+1) + 1/(q+1)≥(n-α)/n1,α/(n-α) < p1q < ∞1,Q(x) and K(x) satisfy some suitable conditions.It is shown that every positive regular solution(u(x)1,v(x)) is symmetric about some plane by developing the moving plane method in an integral form.Moreover,regularity of the solution is studied.Finally,the nonexistence of positive solutions to the system in the case 0 <...     

Optimal integrability of some system of integral equations

We obtain the optimal integrability for positive solutions of the Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality in R^n ：{u（x）=1/｜x｜^α｜∫R^n v（y）^q｜y｜^β｜x-y｜^λdy,v（x）=1/｜x｜^β∫R^n u（y）^p｜y｜^α｜x-y｜^λdy.C. Jin and C. Li [Calc. Var. Partial Differential Equations, 2006, 26： 447-457] developed some very interesting method for regularity lifting and obtained the optimal integrability for p, q 〉 1. Here, based on some new observations, we overcome the difficulty there, and derive the optimal integrability for the case of p, q ≥1 and pq ≠1. This integrability plays a key role in estimating the asymptotic behavior of positive solutions when ｜x｜ →0 and when ｜x｜→∞.     

Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms

This work is concerned with a system of viscoelastic wave equations with nonlinear damping and source terms acting in both equations. We prove a global nonexistence theorem for certain solutions with positive initial energy.     

Global nonexistence of solutions with positive initial energy for a class of wave equations

In this study, we consider a class of wave equations with strong damping and source terms associated with initial and Dirichlet boundary conditions. We establish a blow up result for certain solutions with nonpositive initial energy as well as positive initial energy. This further improves the results by Yang (Math. Meth. Appl. Sci. 2002; 25 :825–833) and Messaudi and Houari (Math. Meth. Appl. Sci. 2004; 27 : 1687–1696). Copyright © 2008 John Wiley & Sons, Ltd.     

Global nonexistence of positive initial energy solution for coupled quasilinear system

In this paper, we consider global nonexistence of a solution for coupled quasilinear system with damping and source under Dirichlet boundary condition. We obtain a global nonexistence result of solution by using the perturbed energy method, where the initial energy is assumed to be positive. Copyright © 2017 John Wiley & Sons, Ltd.     

Asymptotic boundary and nonexistence of traveling waves in a discrete diffusive epidemic model

ABSTRACT

In this paper, we establish the asymptotic boundary of traveling waves (first-component) in a discrete diffusive epidemic model, whose existence is well-known. Meanwhile, we derive the nonexistence of traveling wave solutions with non-positive wave speed. We solve two open problems left in two papers [Fu et al., Nonlinear Convex A. 17 (2016), pp. 1739–1751; Wu, J. Differential Equations 262 (2017), pp. 272–282].     

一类拟线性椭圆方程整体解的不存在性

这篇文章主要利用常微分技术讨论了一个二阶拟线性椭圆方程Lpu≡div（｜Du｜^p-2Du）=f（x,u）,x∈R^N的整体解的不存在性.我们只考虑2≤p〈N的情況并且在假设f（x,u）关于第二个变量u没有单调性的情况下得到整体解的不存在性结果.     

Existence and nonexistence of global positive solutions for a weakly coupled P-Laplacian system

In this paper, we deal with a weakly coupled evolution P-Laplacian system with inhomogeneous terms. We obtain a critical criterion concerning existence and nonexistence of its global positive solutions. Such a criterion is different from that of the weakly coupled evolution P-Laplacian system with homogeneous terms. Further, we demonstrate existence and nonexistence of its global positive solutions.     

The nonexistence of positive solutions to some elliptic equations

The main result of the paper is the nonexistence of integer positive solutions in wide classes of quasilinear elliptic equations whose model examples are equations of the form {

Global existence and nonexistence of solutions for the nonlinear pseudo‐parabolic equation with a memory term

In this paper, we study the initial boundary value problem of nonlinear pseudo‐parabolic equation with a memory term with initial conditions and Dirichlet boundary conditions. By the combination of the Galerkin method and Potential well theory, the existence of global solutions is derived. Moreover, not only the finite time blow up of solutions with the negative initial energy (E(0) < 0) but also the finite time blow up results with the nonnegative initial energy (0≤E(0) < d_k) are obtained by using Concavity method and Potential well theory. Copyright © 2014 John Wiley & Sons, Ltd.     

Global existence and nonexistence results for a class of semilinear hyperbolic systems

In this paper, we investigate the Cauchy problem for a class of the system of semilinear hyperbolic equations with damping. With the use of the L_p → L_q type estimation for the corresponding linear problem and the method of comparison of functional, the existence and nonexistence criteria of global solutions are found. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence and nonexistence of positive solutions of a fractional thermostat model with a parameter

In this paper, we investigate positive solutions for boundary value problem of a fractional thermostat model with a parameter. Under different conditions of the function f, existence and nonexistence results for positive solutions are derived in terms of different values of λ. The results are illustrated with an example. Copyright © 2016 John Wiley & Sons, Ltd.