CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS

In this paper, we are concerned with the following Hardy-Sobolev type system{(-?)~(α/2) u(x) =v~q(x)/|y|~(t_2) (-?)α/2 v(x) =u~p(x)/|y|~(t_1),x =(y, z) ∈(R ~k\{0}) × R~(n-k),(0.1)where 0 α n, 0 t_1, t_2 min{α, k}, and 1 p ≤τ_1 :=(n+α-2t_1)/( n-α), 1 q ≤τ_2 :=(n+α-2 t_2)/( n-α).We first establish the equivalence of classical and weak solutions between PDE system(0.1)and the following integral equations(IE) system{u(x) =∫_( R~n) G_α(x, ξ)v~q(ξ)/|η|t~2 dξ v(x) =∫_(R~n) G_α(x, ξ)(u~p(ξ))/|η|~(t_1) dξ,(0.2)where Gα(x, ξ) =(c n,α)/(|x-ξ|~(n-α))is the Green's function of(-?)~(α/2) in R~n. Then, by the method of moving planes in the integral forms, in the critical case p = τ_1 and q = τ_2, we prove that each pair of nonnegative solutions(u, v) of(0.1) is radially symmetric and monotone decreasing about the origin in R~k and some point z0 in R~(n-k). In the subcritical case (n-t_1)/(p+1)+(n-t_2)/(q+1) n-α,1 p ≤τ_1 and 1 q ≤τ_2, we derive the nonexistence of nontrivial nonnegative solutions for(0.1).     

Ground states of a nonlinear Schrdinger system with nonconstant potentials

In the case where either the potentials Vj,μj andβare periodic or Vj are well-shaped andμj andβare anti-well-shaped,existence of a positive ground state of the Schrdinger system -Δu1+V1(x)u1=μ1(x)u31+β(x)u1u22in RN,-Δu2+V2(x)u2=β(x)u21u2+μ2(x)u32in RN,uj∈H1(RN),j=1,2,where N=1,2,3,is proved provided thatβis either small or large in terms of Vj andμj.The system with constant coefficients has been studied extensively in the last ten years,and the nonconstant coefficients case has seldom been studied.It turns out that new technical machineries in the setting of variational methods are needed in dealing with the nonconstant coefficients case.     

Liouville type theorems for a system of integral equations on upper half space

In this paper,we consider the following system of integral equations on upper half space {u(x) = ∫Rn + (1/|x-y|n-α-1/|-y|n-α) λ1up1(y) + μ1vp2(y) + β1up3(y)vp4(y) dy;v(x) = ∫Rn + (1/|x-y|n-α-1/|-y|n-α)(λ2uq1(y) + μ2vq2(y) + β2uq3(y)vq4(y) dy,where Rn + = {x =(x1,x2,...,xn) ∈ Rn|xn 0}, =(x1,x2,...,xn-1,-xn) is the reflection of the point x about the hyperplane xn= 0,0 α n,λi,μi,βi≥ 0(i = 1,2) are constants,pi≥ 0 and qi≥ 0(i = 1,2,3,4).We prove the nonexistence of positive solutions to the above system with critical and subcritical exponents via moving sphere method.     

“具有稀疏效应的捕食-食饵模型”的注记

§ 1　IntroductionA predator-prey system with undercrowding effectis studied in [1 ] ,dxdτ=x(x -l) (1 -x)m + x -xy,dydτ=ny(x -g0 ) ,(1 )where m,n,g0 and l are non-negative constants.There are singular points O(0 ,0 ) ,A(l,0 )and B(1 ,0 ) in system (1 ) .As g0 locates between 1 and l,system (1 ) has a unique positivesingular point C(g0 ,(g0 -l) (1 -g0 )m+ g0) .Without loss of generality,assume l相似文献   

Mapping properties of certain oscillatory integrals on modulation spaces

Suppose β1 α1 ≥0,β2 α2 ≥ 0 and(k,j) ∈R2. In this paper, we mainly investigate the mapping properties of the operator T_αβf(x,y,z)=∫_Q~2f(x-t,y-s,z-t~ks~j)e~(-2πit-β1_s-β2)t~(-1-α1)s~(-1-α2)dtds on modulation spaces, where Q~2 = [0,1] x [0,1] is the unit square in two dimensions.     

A positive solution of a nonlinear Schrdinger system with nonconstant potentials

In this paper, we study the existence of positive solutions to the following Schr¨odinger system:{-?u + V_1(x)u = μ_1(x)u~3+ β(x)v~2u, x ∈R~N,-?v + V_2(x)v = μ_2(x)v~3+ β(x)u~2v, x ∈R~N,u, v ∈H~1(R~N),where N = 1, 2, 3; V_1(x) and V_2(x) are positive and continuous, but may not be well-shaped; and μ_1(x), μ_2(x)and β(x) are continuous, but may not be positive or anti-well-shaped. We prove that the system has a positive solution when the coefficients Vi(x), μ_i(x)(i = 1, 2) and β(x) satisfy some additional conditions.     

LIMIT CYCLE F0R CUBIC SYSTEM OF KOLMOGOROV TYPE WITH A CONIC ALGEBRAIC TRAJECTORY

In this paper we study the existence of limit cycle for cubic system (E)_3, of Kolmogorov typewith a conic algebraic trajectoryF_2(x,y)=ax~2 2bxy cy~2 dx ey f=0 It has been proved in my former papers that (E)_3 doesn't have any limit cycle on the whole planeIf b~2-ac≠0, Now we are investigating the case where b~2-ac=0. We prove the sufficient andnecessary formula (2) or (13) witb which (E)_3 must have a parabolic trajectory F_2(x,y)=0. Thenthere will not be any limit cycle on the full plane. On the basis of this, we conclude: The cubic system of Kolmogorov type with a non-degenerated quadratic algebraic trajectory onthe whole plane has no limit cycle.     

INFINITELY MANY SOLUTIONS WITH PEAKS FOR A FRACTIONAL SYSTEM IN R~N

In this article, we consider the following coupled fractional nonlinear Schrodinger system in R~N■,where N ≥ 2, 0 s 1, 1 p N/(N-2 s), μ1 0, μ2 0 and β∈ R is a coupling constant.We prove that it has infinitely many non-radial positive solutions under some additional conditions on P(x), Q(x), p and β. More precisely, we will show that for the attractive case,it has infinitely many non-radial positive synchronized vector solutions, and for the repulsive case, infinitely many non-radial positive segregated vector solutions can be found, where we assume that P(x) and Q(x) satisfy some algebraic decay at infinity.     

GLOBAL C~1 SOLUTION OF CAUCHY PROBLEM FOR TWO-DIMENSIONAL GAS DYNAMICS SYSTEM

Using the method of characteristic lines this paper considers the global C~1 solution of the Cauchy problem for two-dimensional gas dynamics system. When the initial data degenerate to the special case φ_0(x, y)=const, the global C~1 solution is obtained. For the case of isentropic exponent γ=1, a transformation about variables is introduced, which changes the system to a first order linear hyperbolic system with constant coefficients and the global C~1 solution is also obtained in this case when the initial data of the forms (φ_0(x, y), u_0(x, y), u_0(x, y))=(exp(w_(01) (c_1x d_1y) w_(02)(c_2x d_2y)), u_(01)(c_1x d_1y) u_(02)(c_2x d_2y), u_(01)(c_1x d_1y) u_(02)(c_2x d_2y)), where c_i and d_i(i=1, 2) are constants.     

Certain oscillatory integrals on unit square and their applications

Let Q2 = [0, 1]2 be the unit square in two dimension Euclidean space R2. We study the Lp boundedness properties of the oscillatory integral operators Tα,β defined on the set S(R3) of Schwartz test functions f by Tα,βf(x,y,z) = Q2 f(x - t,y - s,z - tksj)e-it-β1s-β2t-1-α1s-1-α2dtds, where β1 > α1 0, β2 > α2 0 and (k, j) ∈ R2. As applications, we obtain some Lp boundedness results of rough singular integral operators on the product spaces.     

On the period function of a class of reversible quadratic centers

This paper is devoted to the study of the period function for a class of reversible quadratic system

Generalized sine equations,II

In this paper we solve the equations

Finite difference schemes of the nonlinear pseudo-parabolic system

ＦＩＮＩＴＥＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＳＯＦＴＨＥＮＯＮＬＩＮＥＡＲＰＳＥＵＤＯ－ＰＡＲＡＢＯＬＩＣＳＹＳＴＥＭＤＵＭＩＮＧＳＨＥＮＧ（杜明笙）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＰｈｙｓｉｃｓａｎｄＣｏｍｐｕｔａｔｉｏｎａｌＭａｔｈｅｍａｔ...     

Interlineation of functions of two variables on M(M>-2) lines preserving the class Cr(R2)

An algorithm for constructing the operator O_Mn({_kx}; x, y) with the properties

Combinatorial identity obtained by the differentiation of convolution

The combinatorial identity

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

The critical exponent for a weakly coupled system of the generalized Fujita type reaction-diffusion equations

We study nonnegative solutions of the initial value problem for a weakly coupled system

Extensions of some inequalities

Abstract. By using a simple analytic method the following inequalities are proved:     

Uniform meyer solution to the three dimensional cauchy problem for laplace equation

We consider the three dimensional Cauchy problem for the Laplace equation uxx(x,y,z)+ uyy(x,y,z)+ uzz(x,y,z) = 0, x ∈ R,y ∈ R,0 z ≤ 1, u(x,y,0) = g(x,y), x ∈ R,y ∈ R, uz(x,y,0) = 0, x ∈ R,y ∈ R, where the data is given at z = 0 and a solution is sought in the region x,y ∈ R,0 z 1. The problem is ill-posed, the solution (if it exists) doesn't depend continuously on the initial data. Using Galerkin method and Meyer wavelets, we get the uniform stable wavelet approximate solution. Furthermore, we shall give a recipe for choosing the coarse level resolution.     

Stabilization of discrete systems by dynamic regulator

We consider the system