Dirichlet problem with indefinite nonlinearities

We consider the following nonlinear elliptic equation in a bounded domain with the Dirichlet boundary condition, and , g₁(u)u and g₂(u)u are positive for |u| > > 1. Some existence results are given for superlinear g₁ and g₂ via the Morse theory.Received: 16 Januray 2003, Accepted: 26 August 2003, Published online: 24 November 2003Mathematics Subject Classification (2000): 35J20, 35J25, 58E05Parts of the work were completed while the authors were visiting the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. The authors thank the hospitality of ICTP. Both authors are supported by NSFC, RFDP, MCME, the second author is also supported by the Foundation for University Key Teacher of the Ministry of Education of China and the 973 project of the Ministry of Science and Technology of China.     

Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities

We consider a class of equations of the form By variational methods, we show the existence of families of positive solutions concentrating around local minima of the potential V(x), as . We do not require uniqueness of the ground state solutions of the associated autonomous problems nor the monotonicity of the function . We deal with asymptotically linear as well as superlinear nonlinearities.Received: 8 November 2003, Accepted: 18 November 2003, Published online: 2 April 2004Mathematics Subject Classification (2000): 35B25, 35J65, 58E05     

Existence and nonexistence of global solutions of some non-local degenerate parabolic systems

This paper establishes a new criterion for global existence and nonexistence of positive solutions of the non-local degenerate parabolic system
0, \end{align*}">
with homogeneous Dirichlet boundary conditions, where is a bounded domain with a smooth boundary and are positive constants. For all initial data, it is proved that there exists a global positive solution iff , where is the unique positive solution of the linear elliptic problem

     

Nonexistence of single blow-up solutions for a nonlinear Schrödinger equation involving critical Sobolev exponent

In this paper we shall consider the critical elliptic equation where and a(x) is a real continuous, non negative function, not identically zero. By using a local Pohozaev identity, we show that problem (0.1) does not admit a family of solutions which blows-up and concentrates as at some zero point x₀ of a(x) if the order of flatness of the function a(x) at x₀ is      

Infinitely many solutions of dirichlet problem for <Emphasis Type="Italic">p</Emphasis>-mean curvature operator

The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator: is considered, where Θ is a bounded domain in R ⁿ (n>p>1) with smooth boundary ∂Θ. Under some natural conditions together with some conditions weaker than (AR) condition, we prove that the above problem has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if . Supported by the National Natural Science Foundation of China (10171032) and the Guangdong Provincial Natural Science Foundation (011606).     

On Universal Representation of Random Graphs

It is shown that every probability measure on the interval [0, 1] gives rise to a unique infinite random graph g on vertices {v₁, v₂, . . .} and a sequence of random graphs gn on vertices {v₁, . . . , v_n} such that . In particular, for Bernoulli graphs with stable property Q, can be strengthened to: probability space (, F, P), set of infinite graphs G(Q) , F with property Q such that .AMS Subject Classification: 05C80, 05C62.     

Partial regularity of minimizers of polyconvex variational integrals

We prove partial regularity of vector-valued minimizers u of the polyconvex variational integral , where stands for the minors of the gradient Du. For the integrand, we assume f to be a continuous function of class C ², strictly convex and of polynomial growth in the minors, and g to be a bounded Carathéodory function. We do not employ a Caccioppoli inequality.Received: 19 March 2002, Accepted: 24 October 2002, Published online: 16 May 2003Mathematics Subject Classification (2000): 49N60, 35J50     

Heat kernels on metric measure spaces and an application to semilinear elliptic equations

We consider a metric measure space and a heat kernel on satisfying certain upper and lower estimates, which depend on two parameters and . We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space . Namely, is the Hausdorff dimension of this space, whereas , called the walk dimension, is determined via the properties of the family of Besov spaces on . Moreover, the parameters and are related by the inequalities .

We prove also the embedding theorems for the space , and use them to obtain the existence results for weak solutions to semilinear elliptic equations on of the form

where is the generator of the semigroup associated with .

The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpinski carpet in .

     

Strongly indefinite functionals and multiple solutions of elliptic systems

We study existence and multiplicity of solutions of the elliptic system

where , is a smooth bounded domain and . We assume that the nonlinear term

where , , and . So some supercritical systems are included. Nontrivial solutions are obtained. When is even in , we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if 2$"> (resp. ). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.

     

Some extremal results on circles containing points

We define (n) to be the largest number such that for every setP ofn points in the plane, there exist two pointsx, y P, where every circle containingx andy contains (n) points ofP. We establish lower and upper bounds for (n) and show that [n/27]+2(n)[n/4]+1. We define for the special case where then points are restricted to be the vertices of a convex polygon. We show that .     

Estimates on the Green’s Function and Existence of Positive Solutions of Nonlinear Singular Elliptic Equations in the Half Space

We establish a new 3G-Theorem for the Green’s function for the half space We exploit this result to introduce a new class of potentials that we characterize by means of the Gauss semigroup on . Next, we define a subclass of and we study it. In particular, we prove that properly contains the classical Kato class . Finally, we study the existence of positive continuous solutions in of the following nonlinear elliptic problem

A sharp attainment result for nonconvex variational problems

We consider the problem of minimizing autonomous, multiple integrals like

Infinitely many bound states for some nonlinear scalar field equations

In this paper we consider the problem in , where p > 2 and if N > 2. Assuming that the potential a(x) is a regular function such that 0$" align="middle" border="0"> and that verifies suitable decay assumptions, but not requiring any symmetry property on it, we prove that the problem has infinitely many solutions.Received: 3 December 2003, Accepted: 10 May 2004, Published online: 22 December 2004     

Berger-Shaw's theorem forp-hyponormal operators

For an-multicyclicp-hyponormal operatorT, we shall show that |T|^2p –|T ^*|^2p belongs to the Schatten and that tr Area ((T)).     

Blowup for

In this note we consider the global regularity of smooth solutions to the vector-valued Cauchy problem

We show that if , the gradient-blowup phenomenon occurs in finite time for suitably chosen vanishing at infinity. We also present a simple example of the -blowup solutions for for any 0$">, if .

     

Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity

This paper is concerned with the inhomogeneous nonlinear Shrödinger equation (INLS-equation)

In the critical and supercritical cases with it is shown here that standing-wave solutions of (INLS-equation) on perturbation are nonlinearly unstable or unstable by blow-up under certain conditions on the potential term V with a small 0.$">

     

A Littlewood-Paley type inequality

In this note we prove the following theorem: Let u be a harmonic function in the unit ball and . Then there is a constant C = C(p, n) such that

一类四阶边值问题的正解

§ 1　IntroductionThe deformations of an elastic beam are described by a fourth-order two-pointbound-ary value problem[1 ] .The boundary conditions are given according to the controls at theends of the beam. For example,the nonlinear fourth order problemu(4) (x) =λa(x) f(u(x) ) ,u(0 ) =u′(0 ) =u′(1 ) =u (1 ) =0 (1 .1 ) λdescribes the deformations of an elastic beam whose one end fixed and the other slidingclamped.The existence of solutions of (1 .1 ) λhas been studied by Gupta[1 ] . But …     

On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity

We prove the existence of continuously differentiable solutions with required asymptotic properties as t +0 and determine the number of solutions of the following Cauchy problem for a functional differential equation:

Partial regularity for minima of higher order functionals with p(x)-growth

For higher order functionals $\int_\Omega f(x, \delta u(x), {D^m}u(x))\,dxFor higher order functionals with p(x)-growth with respect to the variable containing D ^m u, we prove that D ^m u is H?lder continuous on an open subset of full Lebesgue-measure, provided that the exponent function itself is H?lder continuous.