MULTIPLE AND SIGN-CHANGING SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATION WITH CRITICAL POTENTIAL AND CRITICAL PARAMETER

Some embedding inequalities in Hardy-Sobolev space are proved.Furthermore,by the improved inequalities and the linking theorem,in a new k-order Sobolev-Hardy space,we obtain the existence of sign-changing solutions for the nonlinear elliptic equation {-△(k)u:=-△u-(((N-2)2)/4)U/︱X︱2-1/4 sum from i=1 to(k-1) u/(︱x︱2(In(i)R/︱x︱2))=f(x,u),x ∈Ω,u=0,x ∈Ω,where 0 ∈ΩBa(0)RN,N≥3,ln(i)=i éj=1 ln(j),and R=ae(k-1),where e(0)=1,e(j) = ee(j-1) for j≥1,ln(1)=ln,ln(j)=ln ln(j-1) for j≥2.Besides,positive andnegative solutions are obtained by a variant mountain pass theorem.     

A NOTE ON MORREY-NIKOL KII INEQUALITY

Let Q_0 be a Cube in R~n and u(x)∈L~p(Q_0).Suppose that∫_Q丨u(x t)-u(x)丨~pdx≤K~p丨t丨~(ap)丨Q丨~(1/β/n)for all parallel subcubes Q in Q_0 and for all t such that the integral makes sense with K≥0,0<α≤1, 0≤β≤n and p≥1.If αp=β,then u(x)is of bounded mean oscillation on Q_0(abbreviated to BMO(Q_0)),i.e.sup QQ_0 1/丨Q丨∫_Q丨u(x)-u_Q丨dx=‖u‖<∞,where u_Q is the mean value of u(x)over Q.     

REMARKS ON BIFURCATIONS OF u" μu-u~κ=0(4≤k∈Z~ )

1. IntroductionOne class of reaction-diffusion equations in one-spatial dimension reads as [1,2],where p is a parameter, 1 < k E Z .Li has ever considered the static (or steady-stste) bifurcations of (4) for the simplercases with k = 2, 3 in [3]. In this paper, we focus on studying the static bifurcations of (4)for the complicated cases with 4 5 k E Z . That is, we consider the bifurcations of thefollowing equationwhere p is a parameter and 4 5 k E Z .Let X = {u E C'[0,7] I u(0) ~ u(x) = …     

AVERAGE REGULARITY OF THE SOLUTION TO AN EQUATION WITH THE RELATIVISTIC-FREE TRANSPORT OPERATOR

Let u = u(t, x, p) satisfy the transport equation ?u/?t+p/p0 ?u/?x= f, where f =f(t, x, p) belongs to L~p((0, T) × R~3× R~3) for 1 p ∞ and ?/?t+p/p0 ?/?x is the relativisticfree transport operator from the relativistic Boltzmann equation. We show the regularity of ∫_(R~3) u(t, x, p)d p using the same method as given by Golse, Lions, Perthame and Sentis. This average regularity is considered in terms of fractional Sobolev spaces and it is very useful for the study of the existence of the solution to the Cauchy problem on the relativistic Boltzmann equation.     

REGULARITY AND SYMMETRY OF SOLUTIONS OF AN INTEGRAL SYSTEM

In this paper,we are concerned with the regularity and symmetry of positive solutions of the following nonlinear integral system u(x) = ∫R n G α(x-y)v(y) q/|y|β dy,v(x) = ∫R n G α(x-y)u(y) p/|y|β dy for x ∈ R n,where G α(x) is the kernel of Bessel potential of order α,0 ≤β < α < n,1 < p,q < n-β/β and 1/p + 1 + 1/q + 1 > n-α + β/n.We show that positive solution pairs(u,v) ∈ L p +1(R n) × L q +1(R n) are Ho¨lder continuous,radially symmetric and strictly decreasing about the origin.     

The Gradient Theory of the Phase Transitions with Dirichlet Boundary Condition

§ 1.Introduction　 In this paper,we study the asymptotic behavior of minimizers{ uε} ε>0 (ε→ 0 ) of thevariational probleminf∫Ω[εp- 1 | Du| p + 1εW(x,u) ] dx:u∈ W1 ,p(Ω ) ,u =g,x∈ Ω (Pε)where p>1 ,N≥ 2 ,andΩ RN is a bounded open domain with Ω∈C2 .　 This kind of problems comes from the theory of phase transitions and are widelystudied in recent years.In [1 ]— [1 4] ,the authors investigated these problems underthe integral constraint∫Ωudx =constant,In [1 5]— [1 6] ,…     

ON THE SINGULAR VARIATIONAL PROBLEMS

The authors deal with the singular variational problem S(a,b,λ0):=infu∈E,u(≡／)0 ∫RN(||X|-a(△)u|m ∫|x|-(a 1)m|u|m)dx/(∫RN||X|-bU|P dx)m/p as well as (S)=(S)(a,b,λ1,λ2):=u,ν,E∈,u(u,ν)(≡／)(1,1) ∫RN J(u,ν)dx/(∫RN|x|-bp|u|α|ν|βdx)m/p, whereJ(u, v) = ||x|- au|m λ1|x|- (a 1)m|u|m ||x|- av|m λ2|x|- (a 1)m|v|m,N ≥ m 1 ＞ 2, 0 ≤ a ＜ N-m/m, a ≤ b ＜ a 1 and p = p(a,b) = α β =Nm/N-m m(b-a), α, β≥ 1, E = D1,mα(RN). The aim of this paper is to show the existence of minimizer for S(a, b, A0) and S(a, b, λ1, λ2).     

REITERATED HOMOGENIZATION OF NONLINEAR MONOTONE OPERATORS

61. IntroductionLet us consider the clajss of partial differential equations of the form--div(a.(x, Du.)) ~ f on fi, us E Wt,p(fl), (1.1)where a6 is increasingly oscillating as E -- 0, fi is an open bounded subset of R", 1 < p < cot1/p 1/q = 1 and f E W--"q(fl). The homogenization problem for (l.1) consists of the studyof the asymptotic behavior of solutions net as e - 0. In many important cases nE convergesweakly in WI'p(n) to the solution no of the homogenized problem--div(b(Duo)) = f …     

THE REGULARITY OF QUASI-MINIMA AND ω-MINIMA OF INTEGRAL FUNCTIONALS

In this article, we have two parts. In the first part, we are concerned with the locally Hlder continuity of quasi-minima of the following integral functional ∫Ωf(x, u, Du)dx, (1) where Ω is an open subset of Euclidean N-space (N ≥ 3), u:Ω→ R,the Carath′eodory function f satisfies the critical Sobolev exponent growth condition |Du|p* |u|p*-a(x) ≤ f(x,u,Du) ≤ L(|Du|p+|u|p* + a(x)), (2) where L≥1, 1pN,p* = Np/N-p , and a(x) is a nonnegative function that lies in a suitable Lp space. In the second part, we study the locally Hlder continuity of ω-minima of (1). Our method is to compare the ω-minima of (1) with the minima of corresponding function determined by its critical Sobolev exponent growth condition. Finally, we obtain the regularity by Ekeland’s variational principal.     

On the adjacent vertex-distinguishing equitable edge coloring of graphs

Let G(V, E) be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if (1) C(u)≠C(v) for uv ∈ E(G), where C(u) = {f(uv)|uv ∈ E}, and (2) for any i, j = 1, 2,… k, we have ||Ei| |Ej|| ≤ 1, where Ei = {e|e ∈ E(G) and f(e) = i}. χáve (G) = min{k| there exists a k-AVEEC of G} is called the adjacent vertex-distinguishing equitable edge chromatic number of G. In this paper, we obtain the χáve (G) of some special graphs and present a conjecture.     

Nonradial Entire Large Solutions of Semilinear Elliptic Equations

We consider the problem of whether the equation △u = p（x）f（u） on RN, N ≥ 3, has a positive solution for which lim ｜x｜→∞（x） = ∞ where f is locally Lipschitz continuous, positive, and nondecreasing on （0,oo） and satisfies ∫1∞[F （t）]^- 1/2dt = ∞ where F（t） = ∫0^tf（s）ds. The nonnegative function p is assumed to be asymptotically radial in a certain sense. We show that a sufficient condition to ensure such a solution u exists is that p satisfies ∫0∞ r min｜x｜=r P （x） dr = ∞. Conversely, we show that a necessary condition for the solution to exist is that p satisfies ∫0∞r1＋ε min ｜x｜=rp（x）dr =∞ for all ε〉0.     

NEW PROOFS OF THE DECAY ESTIMATE WITH SHARP RATE OF THE GLOBAL WEAK SOLUTION OF THE n-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Consider the Cauchy problem for the n-dimensional incompressible NavierStokes equations:??tu-α△u+(u·?)u+?p = f(x, t), with the initial condition u(x, 0) = u0(x) and with the incompressible conditions ? · u = 0, ? · f = 0 and ? · u0= 0. The spatial dimension n ≥ 2.Suppose that the initial function u0∈ L1(Rn) ∩ L2(Rn) and the external force f ∈ L1(Rn× R+) ∩ L1(R+, L2(Rn)). It is well known that there holds the decay estimate with sharp rate:(1 + t)1+n/2∫Rn|u(x, t)|2 dx ≤ C, for all time t 0, where the dimension n ≥ 2, C 0 is a positive constant, independent of u and(x, t).The main purpose of this paper is to provide two independent proofs of the decay estimate with sharp rate, both are complete, systematic, simplified proofs, under a weaker condition on the external force. The ideas and methods introduced in this paper may have strong influence on the decay estimates with sharp rates of the global weak solutions or the global smooth solutions of similar equations, such as the n-dimensional magnetohydrodynamics equations, where the dimension n ≥ 2.     

A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrödinger equations

The classical critical Trudinger-Moser inequality in R~2 under the constraint ∫_(R_2)(|▽u|~2+|u|~2)dx≤1 was established through the technique of blow-up analysis or the rearrangement-free argument:for any τ 0,it holds that ■ and 4π is sharp.However,if we consider the less restrictive constraint ∫_(R_2)(|▽u|~2+|u|~2)dx≤1,where V(x) is nonnegative and vanishes on an open set in R~2,it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π.The loss of a positive lower bound of the potential V(x) makes this problem become fairly nontrivial.The main purpose of this paper is twofold.We will first establish the Trudinger-Moser inequality ■ when V is nonnegative and vanishes on an open set in R~2.As an application,we also prove the existence of ground state solutions to the following Sciridinger equations with critical exponeitial growth:-Δu+V(x)u=f u) in R~2,(0.1)where V(x)≥0 and vanishes on an open set of R~2 and f has critical exponential growth.Having a positive constant lower bound for the potential V(x)(e.g.,the Rabinowitz type potential) has been the standard assumption when one deals with the existence of solutions to the above Schr?dinger equations when the nonlinear term has the exponential growth.Our existence result seems to be the first one without this standard assumption.     

Solutions to Some Open Problems in Fluid Dynamics

Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.     

Multiple solutions for a Schrdinger-Poisson-Slater equation with external Coulomb potential

Consider the following Schrdinger-Poisson-Slater system,(P)u+ω-β|x|u+λφ(x)u=|u|p-1u,x∈R3,-φ=u2,u∈H1(R3),whereω0,λ0 andβ0 are real numbers,p∈(1,2).Forβ=0,it is known that problem(P)has no nontrivial solution ifλ0 suitably large.Whenβ0,-β/|x|is an important potential in physics,which is called external Coulomb potential.In this paper,we find that(P)withβ0 has totally different properties from that ofβ=0.Forβ0,we prove that(P)has a ground state and multiple solutions ifλcp,ω,where cp,ω0 is a constant which can be expressed explicitly viaωand p.     

Attractor and dimension for strongly damped nonlinear wave equation

1. IntroductionConsider the strongly damped nonlinear wave equationwith the Dirichlet boundary conditionand the initial value conditionswhere u = u(x, t) is a real--valued function on fi x [0, co), fi is an open bounded set of R"with smooth boundary off, or > 0, g e L'(fl), D(--Q) ~ Ha(~~) n H'(fl).We assume for the function f(u) as follows'f(u) E CI (R, R) satisfiesfor any ig al, uZ E R, where k, hi > 0, i ~ 0, 1, 2, 61 > 0 and 0 5 6o < 1'The type of equation (1) can be regarded as the…     

Mordell-Weil groups and Selmer groups of twin- prime elliptic curves

Let E = Eσ : y2 = x(x + σp)(x + σq) be elliptic curves, where σ = ±1, p and q are primenumbers with p+2 = q. (i) Selmer groups S(2)(E/Q), S(φ)(E/Q), and S(φ)(E/Q) are explicitly determined,e.g. S(2)(E+1/Q)= (Z/2Z)2, (Z/2Z)3, and (Z/2Z)4 when p ≡ 5, 1 (or 3), and 7(mod 8), respectively. (ii)When p ≡ 5 (3, 5 for σ = -1) (mod 8), it is proved that the Mordell-Weil group E(Q) ≌ Z/2Z Z/2Z,symbol, the torsion subgroup E(K)tors for any number field K, etc. are also obtained.     

EXISTENCE OF INFINITELY MANY SOLUTIONS FOR ELLIPTIC PROBLEMS WITH CRITICAL EXPONENT

This paper is concerned with the following nonlinear Dirichlet problem:{-Δpu=|u|^p*-2 u λf(x,u) x∈Ω；u=0 x∈эΩ} whereΔp^u = div(|∧u|^p-2∧u) is the p-Laplacian of u,Ω is a bounded in R^n(n≥3），1＜p＜n, p=pn/n-p is the critical exponent for the Sobolev imbedding,λ＞0 and f(x,u)satisfies some conditions. It reaches the conclusion that this problem has infinitely many solutions. Some results as p=2 or f(x,u) = |u|^q-2 u, where 1＜q＜p, are generalized.     

Liouville theorem for Choquard equation with finite Morse indices

In this article, we study the nonexistence of solution with finite Morse index for the following Choquard type equation-△u=∫RN|u(y)|p|x-y|αdy|u(x)|p-2u(x) in RN where N ≥ 3, 0 α min{4, N}. Suppose that 2 p (2 N-α)/(N-2),we will show that this problem does not possess nontrivial solution with finite Morse index. While for p=(2 N-α)/(N-2),if i(u) ∞, then we have ∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~α dxdy ∞ and ∫_RN|▽u|~2 dx=∫_RN∫_RN|u(x)p(u)(y)~p/|x-y|~αdxdy.     

UNIFORM BLOWUP PROFILES FOR DIFFUSION EQUATIONS WITH NONLOCAL SOURCE AND NONLOCAL BOUNDARY

Long time behavior of solutions to semilinear parabolic equations with nonlocal nonlinear source ut - △u = ∫Ω g(u)dx inΩ× (0, T) and with nonlocal boundary condition u(x, t) = ∫Ω f(x, y)u(y, t)dy on(e) Ω× (0, T) is studied. The authors establish local existence, global existence and nonexistence of solutions and discuss the blowup properties of solutions. Moveover, they derive the uniform blowup estimates for g(s) = sp(p ＞ 1) and g(s) = es under the assumption fΩ f(x, y)dy ＜ 1 for x ∈(e)Ω.