Bifurcation problems for a class of degenerate quasilinear elliptic equations

In this paper we consider the bifurcation problem -div A（x, u）=λa（x）｜u｜^p-2u＋f（x,u,λ） in Ω with p 〉 1.Under some proper assumptions on A（x,ξ）,a（x） and f（x,u,λ）,we show that the existence of an unbounded branch of positive solutions bifurcating Irom the principal eigenvalue of the problem --div A（x, u）=λa（x）｜u｜^p-2u.     

ON NEWMAN-TYPE RATIONAL INTERPOLATION TO ｜x｜ AT THE CHEBYSHEV NODES OF THE SECOND KIND

Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary set of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods one could establish the exact order of approximation for some special nodes. In the present paper we consider the special case where the interpolation nodes are the zeros of the Chebyshev polynomial of the second kind and prove that in this case the exact order of approximation is O（1/n｜nn）     

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｜→∞.     

A Note to the Cauchy Problem for the Degenerate Parabolic Equations with Strongly Nonlinear Sources

In this note we study the nonexistence of nontrivial global solutions on S = R^N × （0,∞） for the following inequalities：｜u｜t≥△（｜u｜^m-1u）＋｜u｜^q and ｜u｜t≥div（｜△u｜^p-2△｜u｜）＋｜u｜^q.When m,p,q satisfy some given conditions, the nonexistence of nontrivial global solution is proved, without taking their traces on the hyperplans t = 0 into account.     

SIGN-CHANGING SOLUTIONS FOR SCHRODINGER EQUATIONS WITH VANISHING AND SIGN-CHANGING POTENTIALS

In this article, we study the existence of sign-changing solutions for the following SchrSdinger equation
-△u ＋ λV（x）u = K（x）｜u｜^p-2u x∈R^N, u→0 as ｜x｜→ ＋∞,
2N where N ≥ 3, λ〉 0 is a parameter, 2 〈 p 〈 2N/N-2, and the potentials V（x） and K（x） satisfy some suitable conditions. By using the method based on invariant sets of the descending flow, we obtain the existence of a positive ground state solution and a ground state sign-changing solution of the above equation for small λ, which is a complement of the results obtained by Wang and Zhou in [J. Math. Phys. 52, 113704, 2011].     

基于修正的第二类Chebyshev结点的Newman型有理插值函数对|x|的逼近

Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary sets of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods,one could establish the exact order of approximation for some special nodes.In the present note we consider the sets of interpolation nodes obtained by adjusting the Chebyshev roots of the second kind on the interval [0,1] and then extending this set to [-1,1] in a symmetric way.We show that in this case the exact order of approximation is O（ 1 n 2 ）.     

Exact Meromorphic Stationary Solutions of the Cubic-Quintic Swift-Hohenberg Equation

In this paper, we study an ODE of the form b0u（4）＋b1u′′＋b2u＋b3u3＋b4u5=0, ′= d/dz＇ , which includes, as a special case, the stationary case of the cubic-quintic Swift-Hohenberg equation. Based on Nevanlinna theory and Painleve analysis, we first show that all its meromorphic solutions are elliptic or degenerate elliptic. Then we obtain them all explicitly by the method introduced in [7].     

Periodic Solutions to a Kind of Second Order Neutral Functional Differential Equation in the Critical Case

In this paper, the authors consider the existence of periodic solutions for a kind of second neutral functional differential equation as follows：（x（t） - cx（t -τ）＂ = g（t, x（t - μ（t））） ＋ e（t）,in the critical case ｜c｜ = 1. By employing Mawhin＇s continuation theorem and some analysis techniques, some new results are obtained.     

Properties on solutions of some q-difference equations

We consider the properties on solutions of some q-difference equations of the form ∑ n j=0 aj（z）f（qj z）=an＋1（z）, where a0（z）,..., an＋1（z） are meromorphic functions, a0（z）an（z） ≠ 0 and q ∈ C such that 0 〈 ｜q｜ ≤ 1. We give estimates on the upper bound for the length of the gap in the power series of entire solutions of （＊） when the coefficients a0（z）,..., an＋1（z） are polynomials and 0 〈 ｜q｜ 〈 1. For some special cases, we give estimates of growth of f（z）. And we also show that the case 0 〈 ｜q｜ 〈 1 is different from the case ｜q｜=1.     

某个用Carlson-Shaffer算子定义的解析函数子类的性质(英文)

Let Hn（p）be the class of functions of the form f（z）=z p＋ ＋∞ Σ k=n akzk＋p,which are analytic in the open unit disk U={z：｜z｜1}.In the paper,we introduce a new subclass Bn（μ,a,c,α,p;φ）of Hn（p）and investigate its subordination relations,inclusion relations and distortion theorems.The results obtained include the related results of some authors as their special case.     

EXISTENCE AND NONEXISTENCE OF ENTIRE LARGE SOLUTIONS FOR SOME SEMILINEAR ELLIPTIC EQUATIONS

In this note, we consider positive entire large solutions for semilinear elliptic equations Au = p（x）f（u） in R^N with N ≥ 3. More precisely, we are interested in the link between the existence of entire large solution with the behavior of solution for --△u = p（x） in R^N. Especially for the radial case, we try to give a survey of all possible situations under Keller-Osserman type conditions.     

Well-posedness in the energy space for non-linear system of wave equations with critical growth

The authors consider the well-posedness in energy space of the critical non-linear system of wave equations with Hamiltonian structure{utt-△u=-F1（｜u｜^2,｜v｜^2）u,utt-△u=-F2（｜u｜^2,｜v｜^2）u where there exists a function F（λ,μ） such that δF（λ,μ）/δλ=F1（λ,μ）.δF（λ,μ）/δμ=F2（λ,μ） By showing that the energy and dilation identities hold for weak solution under some assumptions on the non-linearities, we prove the global well-posedness in energy space by a similar argument to that for global regularity as shown in ＂Shatah and Struwe＇s paper, Ann. of Math. 138, 503-518 （1993）＂.     

EXISTENCE OF A NONTRIVIAL SOLUTION FOR CHOQUARD＇S EQUATION

In this article, the authors consider the existence of a nontrivial solution for the following equation： -△u＋u=q（x）（｜u｜^2＊1/｜x｜）u, x∈R^3, where q（x） satisfies some conditions. Using a Min-Max method, the authors prove that there is at least one nontrivial solution for the above equation.     

Quasilinear elliptic equations with Neumann boundary and singularity

Let Ω be a bounded domain with a smooth C2 boundary in RN（N ≥ 3）, 0 ∈Ω, and n denote the unit outward normal to ЭΩ.We are concerned with the Neumann boundary problems： -div（｜x｜α｜△u｜p-2△u）=｜x｜βup（α,β）-1-λ｜x｜γup-1,u（x）〉0,x∈Ω,Эu/Эn=0 on ЭΩ,where 1〈p〈N and α〈0,β〈0 such that p（α,β）△=p（N＋β）/N-p＋α〉p,y〉α-p.For various parameters α,βorγ,we establish certain existence results of the solutions in the case 0∈Ω or 0∈ЭΩ.     

Existence of Solutions for Schrodinger-Poisson Systems with Sign-Changing Weight

We study the existence of solutions for the SchrOdinger-Poisson system
{-△u＋u＋k（x）φu=a（x）｜u｜p-1u,in R3,
-△φ=k（x）u2, in R3,
where 3 G p 〈 5, a （x） is a sign-changing function such that both the supports of a＋ and a- may have infinite measure. We show that the problem has at least one nontrivial solution under some assumptions.     

The abc-conjecture for Algebraic Numbers

The abe-conjecture for the ring of integers states that, for every ε 〉 0 and every triple of relatively prime nonzero integers （a, b, c） satisfying a ＋ b = c, we have max（｜a｜, ｜b｜, ｜c｜） 〈 rad（abc）^1＋ε with a finite number of exceptions. Here the radical rad（m） is the product of all distinct prime factors of m. In the present paper we propose an abe-conjecture for the field of all algebraic numbers. It is based on the definition of the radical （in Section 1） and of the height （in Section 2） of an algebraic number. From this abc-conjecture we deduce some versions of Fermat＇s last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat＇s equation in fields of small degrees over Q.     

EXISTENCE OF GLOBAL ATTRACTORS FOR A NONLINEAR EVOLUTION EQUATION IN SOBOLEV SPACE Hk

In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation ut = △u ＋ λu - u^3 possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded domain of H^k（Ω） in the H^k-norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k ∈ [0, 1] to the case k∈ [0, ∞）.     

Bahadur representation of nonparametric M-estimators for spatial processes

Under some mild conditions, we establish a strong Bahadur representation of a general class of nonparametric local linear M-estimators for mixing processes on a random field. If the socalled optimal bandwidth hn = O（｜n｜^-1/5）, n ∈ Z^d, is chosen, then the remainder rates in the Bahadur representation for the local M-estimators of the regression function and its derivative are of order O（｜n｜^-4/5 log ｜n｜）. Moreover, we derive some asymptotic properties for the nonparametric local linear M-estimators as applications of our result.     

The L（3,2, 1）-labeling on Bipartite Graphs

An L（3, 2, 1）-labeling of a graph G is a function from the vertex set V（G） to the set of all nonnegative integers such that ｜f（u）-f（v）｜≥3 if dG（u,v） = 1, ｜f（u）-f（v）｜≥2 if dG（u,v） = 2, and ｜f（u）-f（v）｜≥1 if dG（u,v） = 3. The L（3, 2,1）-labeling problem is to find the smallest number λ3（G） such that there exists an L（3, 2,1）-labeling function with no label greater than it. This paper studies the problem for bipartite graphs. We obtain some bounds of λ3 for bipartite graphs and its subclasses. Moreover, we provide a best possible condition for a tree T such that λ3（T） attains the minimum value.     

Existence and Nonexistence of Weak Positive Solution for a Class of p-Laplacian Systems

In this work, we are interested to obtain some result of existence and nonex- istence of positive weak solution for the following p-Laplacian system
{-△piui=λifi（u1,^…,um）,inΩ, i=1,...,m, ui=0,
on δΩ,Vi=1,…,m,
where △piz = div（｜△z｜^pi-2△Z）, Pi ≥ 1,λi,1 ≤ i ≤ m are a positive parameter, and Ω is a bounded domain in IR^N with smooth boundary δΩ. The proof of the main results is based to the method of sub-supersolutions.