REGULARITY OF THE SOLUTIONS FOR NONLINEAR BIHARMONIC EQUATIONS IN RN

The purpose of this article is to establish the regularity of the weak solutions for the nonlinear biharmonic equation
{△^2u ＋ a（x）u = g（x, u）
u∈ H^2（R^N）,
where the condition u∈ H^2（R^N） plays the role of a boundary value condition, and as well expresses explicitly that the differential equation is to be satisfied in the weak sense.     

EXISTENCE OF POSITIVE SOLUTIONS TO SEMILINEAR ELLIPTIC SYSTEMS IN IRN WITH ZERO MASS

In this paper, we prove the existence of at least one positive solution pairto the following semilinear elliptic systemby using a linking theorem, where K（x）is a positive function in L^s（R^N） for some s 〉 1and the nonnegative functions f, g ∈ C（R, R） are of quasicritical growth, superlinear atinfinity. We do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. Our main result can be viewed as a partial extension of a recent result of Alves, Souto and Montenegro in [1] concerning the existence of a positive solution to the following semilinear elliptic problemand a recent result of Li and Wang in [22] concerning the existence of nontrivial solutions to a semilinear elliptic system of Hamiltonian type in R^N.     

Existence of Renormalized Solutions for Nonlinear Parabolic Equations

We give an existence result of a renormalized solution for a class of nonlin- ear parabolic equations b（x,u）/ t-div （a（x,t,u, u））＋g（x,t,u,u ）＋H（x,t, u）=f,in QT, where the right side belongs to LP＇ （0,T;W-1,p＇（Ω）） and where b（x,u） is unbounded function of u and where - div （ a （ x, t, u, u） ） is a Leray-Lions type operator with growth ｜u ｜p- 1 in V u. The critical growth condition on g is with respect to u and no growth condition with re sp ect to u, while the function H （x, t, u） grows as｜ u ｜p - 1.     

A lower semicontinuity result for some integral functionals in the space SBD

In this paper, we obtain a lower semicontinuity result with respect to the strong L1-convergence of the integral functionals F（u,Ω）=Ωf（x,u（x）,εu（x））dx defined in the space SBD of special functions with bounded deformation. Here Su represents the absolutely continuous part of the symmetrized distributional derivative Eu. The integrand f satisfies the standard growth assumptions of order p 〉 1 and some other conditions. Finally, by using this result,we discuss the existence of an constrained variational problem.     

The existence of nontrivial solutions to a semilinear elliptic system on without the ambrosetti-rabinowitz condition

In this paper, we prove the existence of at least one positive solution pair （u, v）∈ H1（RN） × H1（RN） to the following semilinear elliptic system {-△u＋u=f（x,v）,x∈RN,-△u＋u=g（x,v）,x∈RN （0.1）,by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functions f, g ∈C0（RN× R1） are that, f（x, t） and g（x, t） are superlinear at t = 0 as well as at t =＋∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245（2008）, 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem {-△u＋u=f（x,u）,x∈Ω,u∈H0^1（Ω） where Ω ∩→RN is bounded and a result of Li and Yang [G. Li and J. Yang： Communications in P.D.E. Vol. 29（2004） Nos.5＆ 6.pp.925-954, 2004] concerning （0.1） when f and g are asymptotically linear.     

Profile of Blow-up Solution to Hyperbolic System with Nonlocal Term

This paper is concerned with a nonlocal hyperbolic system as follows utt = △u ＋ （∫Ωvdx ）^p for x∈R^N,t〉0 ,utt = △u ＋ （∫Ωvdx ）^q for x∈R^N,t〉0 ,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N, where 1≤ N ≤3, p ≥1, q ≥ 1 and pq 〉 1. Here the initial values are compactly supported and Ω belong to R^N is a bounded open region. The blow-up curve, blow-up rate and profile of the solution are discussed.     

Regularity of Weak Solutions of Nonlinear Equations with Discontinuous Coefficients

In this paper, we prove that the weak solutions u∈Wloc^1, p （Ω） （1 〈p〈∞） of the following equation with vanishing mean oscillation coefficients A（x）： -div[（A（x）△↓u·△↓u）p-2/2 A（x）△↓u＋│F（x）│^p-2 F（x）]=B（x, u, △↓u）, belong to Wloc^1, q （Ω）（A↓q∈（p, ∞）, provided F ∈ Lloc^q（Ω） and B（x, u, h） satisfies proper growth conditions where Ω ∪→R^N（N≥2） is a bounded open set, A（x）=（A^ij（x）） N×N is a symmetric matrix function.     

Entropy Unilateral Solution for Some Noncoercive Nonlinear Parabolic Problems Via a Sequence of Penalized Equations

We give an existence result of the obstacle parabolic equations(b(x,u))/(t)-div(a(x,t,u,▽u))+div(φ(x,t,u))=f in Q_T,where b(x,u) is bounded function of u,the term-div(a(x,t,u,▽u)) is a Leray-Lions type operator and the function φ is a nonlinear lower order and satisfy only the growth condition.The second term f belongs to L~1(Q_T).The proof of an existence solution is based on the penalization methods.     

Renormalized Solutions for Nonlinear Parabolic Systems with Three Unbounded Nonlinearities in Weighted Sobolev Spaces

We prove an existence result without assumptions on the growth of some nonlinear terms, and the existence of a renormalized solution. In this work, we study the existence of renormalized solutions for a class of nonlinear parabolic systems with three unbounded nonlinearities, in the form { b1（x,u1）/ t-div（a（x,t,u1,Du1））＋div（Ф1（u1））＋f1（x,u1,u2）=O in Q, b2（x,u2）/ t-div（a（x,t,u2,Du2））＋div（Ф2（u2））＋f2（x,u1,u2）=O in Q in the framework of weighted Sobolev spaces, where b（x,u） is unbounded function on u, the Carath6odory function ai satisfying the coercivity condition, the general growth condition and only the large monotonicity, the function Фi is assumed to be continuous on ]R and not belong to （Lloc1（Q））N.     

Entropy and renormalized solutions for nonlinear elliptic problem involving variable exponent and measure data

We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type:-div(a(x, u, u) + φ(u)) + g(x, u, u) = μ, where the right-hand side belongs to L1(Ω) + W-1,p(x)(Ω),-div(a(x, u, u)) is a Leray–Lions oper- ator defined from W-1,p(x)(Ω) into its dual and φ∈ C0(R, RN). The function g(x, u, u) is a non linear lower order term with natural growth with respect to |u| satisfying the sign condition, that is,g(x, u, u)u ≥ 0.     

A General Version of the Retract Method for Discrete Equations

We study a problem concerning the compulsory behavior of the solutions of systems of discrete equations u（k ＋ 1） = F（k, u（k））, k ∈ N（a） = {a, a ＋ 1, a ＋ 2 }, a ∈ N,N= {0, 1,... } and F ： N（a） × R^n→R^n. A general principle for the existence of at least one solution with graph staying for every k ∈ N（a） in a previously prescribed domain is formulated. Such solutions are defined by means of the corresponding initial data and their existence is proved by means of retract type approach. For the development of this approach a notion of egress type points lying on the defined boundary of a given domain and with respect to the system considered is utilized. Unlike previous investigations, the boundary can contain points which are not points of egress type, too. Examples are inserted to illustrate the obtained result.     

NONTRIVIAL SOLUTIONS FOR SEMILINEAR SCHR(O)DINGER EQUATIONS

The authors prove the existence of nontrivial solutions for the SchrSdinger equation -△u ＋ V（x）u =λf（x, u） in R^N, where f is superlinear, subcritical and critical at infinity, respectively, V is periodic.     

Intersections and polar functions of fractional brownian sheets

Let B^H={B^H（t）,t∈R^N＋}be a real-valued（N,d）fractional Brownian sheet with Hurst index H=（H1,…,HN）.The characteristics of the polar functions for B^H are discussed.The relationship between the class of continuous functions satisfying Lipschitz condition and the class of polar-functions of B^H is obtained.The Hausdorff dimension about the fixed points and the inequality about the Kolmogorov’s entropy index for B^H are presented.Furthermore,it is proved that any two independent fractional Brownian sheets are nonintersecting in some conditions.A problem proposed by LeGall about the existence of no-polar continuous functions satisfying the Holder condition is also solved.     

On an Anisotropic Equation with Critical Exponent and Non-Standard Growth Condition

Using variational methods, we prove the existence of a nontrivial weak solution for the problem
{-∑i=1^Nδxi（｜δxiu｜pi-2δxiu）=λα（x）｜u｜q（x）-2u＋｜u｜p＊-2u,in Ω,
u=0 inδΩ,
where Ω R^N（N≥3） is a bounded domain with smooth boundary δΩ,2≤pi〈N,i=1,N,q：Ω→（1,p＊）is a continuous function, p＊ =N/∑i=1^N 1/pi-1 is the critical exponent for this class of problem, and λ is a parameter.     

Existence Results for a Class of Semilinear Elliptic Systems

In this paper, we study the existence of nontrivial solutions for the problem
{-△u=f（x,u,v）＋h1（x）in Ω
-△v=g（x,u,v）＋h2（x）inΩ
u=v=0 onδΩ
where Ω is bounded domain in R^N and h1,h2 ∈ L^2 （Ω）. The existence result is obtained by using the Leray-Schauder degree under the following condition on the nonlinearities f and g：
{lim s,｜t｜→＋∞f（x,s,t）/s=lim ｜s｜,t→＋∞g（x,s,t）/t=λ＋1 uniformly on Ω,
lim -s,｜t｜→＋∞f（x,s,t）/s=lim ｜s｜,-t→＋∞g（x,s,t）/t=λ-,uniformly on Ω,
where λ＋,λ-∈（0）∪σ（-△）,σ（-△）denote the spectrum of -△. The cases （i） where λ＋ = λ_ and （ii） where λ＋≠λ_ such that the closed interval with endpoints λ＋,λ_ contains at most one simple eigenvatue of -△ are considered.     

脉冲发展方程初值问题的正解

This paper deals with the existence of e-positive mild solutions（see Definition 1）for the initial value problem of impulsive evolution equation with noncompact semigroup u（t）＋ Au（t）= f（t,u（t））,t ∈ [0,＋∞）,t = tk,u-t=tk = Ik（u（tk））,k = 1,2,...,u（0）= x0 in an ordered Banach space E.By using operator semigroup theory and monotonic iterative technique,without any hypothesis on the impulsive functions,an existence result of e-positive mild solutions is obtained under weaker measure of noncompactness condition on nonlinearity of f.Particularly,an existence result without using measure of noncompaceness condition is presented in ordered and weakly sequentially complete Banach spaces,which is very convenient for application.An example is given to illustrate that our results are more valuable.     

A remark on partly dissipative reaction diffusion systems on IRn

The long time behavior of the solutions of some partly dissipative reaction diffusion systems is studied. We prove the existence of a compact （L^2 × L^2 - H^1 × L^2） attractor for a partly dissipative reaction diffusion system in Rn. This improves a previous result obtained by A. Rodrigues-Bernal and B. Wang concerning the existence of a compact （L^2 × L^2 - L^2 × L^2） attractor for the same system.     

Problem with Critical Sobolev Exponent and with Weight

The authors consider the problem: -div(p▽u) = uq-1 λu, u > 0 inΩ, u = 0 on (?)Ω, whereΩis a bounded domain in Rn, n≥3, p :Ω→R is a given positive weight such that p∈H1 (Ω)∩C(Ω),λis a real constant and q = 2n/n-2, and study the effect of the behavior of p near its minima and the impact of the geometry of domain on the existence of solutions for the above problem.     

THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON RN WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION

In this paper, we prove the existence of at least one positive solution pair (u, v) ∈ H 1 (R N ) × H 1 (R N ) to the following semilinear elliptic system{-u + u = f(x, v), x ∈RN ,-v + v = g(x,u), x ∈ R N ,(0.1) by using a linking theorem and the concentration-compactness principle. The main con-ditions we imposed on the nonnegative functions f, g ∈ C 0 (R N × R 1 ) are that, f (x, t) and g(x, t) are superlinear at t = 0 as well as at t = +∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245(2008), 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem{-u + u = f(x, u), x ∈Ω,u ∈H10(Ω)where ΩRN is bounded and a result of Li and Yang [G. Li and J. Yang: Communications in P.D.E. Vol. 29(2004) Nos.5 6.pp.925–954, 2004] concerning (0.1) when f and g are asymptotically linear.     

A note on solutions for asymptotically linear elliptic systems

In this paper, we are concerned with the elliptic system of
{ -△u＋V（x）u=g（x,v）, x∈R^N,
-△v＋V（x）v=f（x,u）, x∈R^N,
where V（x） is a continuous potential well, f, g are continuous and asymptotically linear as t→∞. The existence of a positive solution and ground state solution are established via variational methods.