MULTIPLE SOLUTIONS AND THEIR LIMITING BEHAVIOR OF COUPLED NONLINEAR SCHRDINGER SYSTEMS

We study the multiplicity of positive solutions and their limiting behavior as ε tends to zero for a class of coupled nonlinear Schrdinger system in RN . We relate the number of positive solutions to the topology of the set of minimum points of the least energy function for ε suffciently small. Also, we verify that these solutions concentrate at a global minimum point of the least energy function.     

THE RELATION BETWEEN SIGN-CHANGING SOLUTION AND POSITIVE-NEGATIVE SOLUTIONS FOR NONLINEAR OPERATOR EQUATIONS AND ITS APPLICATIONS

By fixed point index theory and a result obtained by Amann, existence of the solution for a class of nonlinear operator equations x=Ax is discussed. Under suitable conditions, a couple of positive and negative solutions are obtained. Finally, the abstract result is applied to nonlinear Sturm-Liouville boundary value problem, and at least four distinct solutions are obtained.     

POSITIVE PERIODIC SOLUTIONS OF NONLINEAR FUNCTIONAL DIFFERENCE EQUATION

Using the Krasnoselskii's fixed point theorem, the existence of positive periodic solutions to a class of nonlinear functional difference equations is studied in this paper. Some sufficient conditions for the existence of positive periodic solutions are presented.     

一类多项式型迭代函数方程在共振点附近的解析解

In this paper existence of local analytic solutions of a polynomial-like iterative functional equation is studied. As well as in previous work, we reduce this problem with the SchrSder transformation to finding analytic solutions of a functional equation without iteration of the unknown function f. For technical reasons, in previous work the constant α given in the Schroder transformation, i.e., the eigenvalue of the linearized f at its fixed point O, is required to fulfill that α is off the unit circle S^1 or lies on the circle with the Diophantine condition. In this paper, we obtain results of analytic solutions in the case of α at resonance, i.e., at a root of the unity and the case of α near resonance under the Brjuno condition.     

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS FOR A NONLINEAR ELLIPTIC EQUATION ARISING IN ASTROPHYSICS

The author first analyzes the existence of ground state solutions and cylindrically symmetric solutions and then the asymptotic behavior of the ground state solution of the equation -△u=φ(r)up-1,u>0 in RN, u ∈ D1,2(RN),where N≥ 3,x = (x',z)∈ RK×RN-K,2≤K≤N,r =|x'|.It is proved that for 2(N -s)/(N-2) < p < 2* = 2N/(N -2),0 < s < 2, the above equation has a ground state solution and a cylindrically symmetric solution. For p=2*, the above equation does not have a ground state solution but a cylindrically symmetric-solution, and when p close to 2*, the ground state solutions are not cylindrically symmetric. On the other hand, it is proved that as p close to 2*, the ground state solution up has a unique maximum point xp = (x'p,zp) and as p→2*, |x'p|→r0 which attains the maximum of φ on RN.The asymptotic behavior of ground state solution up is also given, which also deduces that the ground state solution is not cylindrically symmetric as p goes to 2*.     

A NEW APPROACH TO SOLVE PERTURBED NONLINEAR EVOLUTION EQUATIONS THROUGH LIE-BACKLUND SYMMETRY METHOD

A new method based on Lie-Backlund symmetry method to solve the perturbed nonlinear evolution equations is presented. New approximate solutions of perturbed nonlinear evolution equations stemming from the exact solutions of unperturbed equations are obtained. This method is a generalization of Burde＇s Lie point symmetry technique.     

Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity

In this paper, we consider the existence of homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity. The classical Ambrosetti–Rabinowitz superlinear condition is improved by a general superlinear one. The proof is based on the critical point theory in combination with periodic approximations of solutions.     

EXISTENCE OF SOLUTIONS OF STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS FOR NONLINEAR SECOND ORDER IMPULSIVE DIFFERENTIAL EQUATIONS IN BANACH SPACES

In this paper, the fixed point theorem is applied to investigate the existence of solutions of Sturm-Liouville boundary value problems for nonlinear second order impulsive differential equations in Banaeh spaces.     

POSITIVE SOLUTIONS TO A SECOND-ORDER m-POINT BOUNDARY VALUE PROBLEM ON TIME SCALES

By a fixed point theorem in a cone,the existence of at least three positive solutions to a class of second-order multi-point boundary value problem for dynamic equation on time scales with the nonlinear term depends on the first order derivative is studied.     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

Semilinear elliptic resonant problems at higher eigenvalue with unbounded nonlinear terms

In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems at higher eigenvalues with the nonlinear term which may be unbounded by making use of the Morse theory for aC ²-function at both isolated critical point and infinity.     

Convexity of solutions and estimates for fully nonlinear elliptic equations

The starting point of this work is a paper by Alvarez, Lasry and Lions (1997) concerning the convexity and the partial convexity of solutions of fully nonlinear degenerate elliptic equations. We extend their results in two directions. First, we deal with possibly sublinear (but epi-pointed) solutions instead of 1-coercive ones; secondly, the partial convexity of C² solutions is extended to the class of continuous viscosity solutions. A third contribution of this paper concerns C^1,1 estimates for convex viscosity solutions of strictly elliptic nonlinear equations. To finish with, all the tools and techniques introduced here permit us to give a new proof of the Alexandroff estimate obtained by Trudinger (1988) and Caffarelli (1989).     

Inequality problems of quasi-hemivariational type involving set-valued operators and a nonlinear term

The aim of this paper is to establish the existence of at least one solution for a general inequality of quasi-hemivariational type, whose solution is sought in a subset K of a real Banach space E. First, we prove the existence of solutions in the case of compact convex subsets and the case of bounded closed and convex subsets. Finally, the case when K is the whole space is analyzed and necessary and sufficient conditions for the existence of solutions are stated. Our proofs rely essentially on the Schauder’s fixed point theorem and a version of the KKM principle due to Ky Fan (Math Ann 266:519–537, 1984).     

On Kneser solutions of higher order nonlinear ordinary differential equations

The equationx ⁽ⁿ⁾(t)=(−1) ⁿ │x(t)│ ^k withk>1 is considered. In the casen≦4 it is proved that solutions defined in a neighbourhood of infinity coincide withC(t−t₀)^−n/(k−1), whereC is a constant depending only onn andk. In the general case such solutions are Kneser solutions and can be estimated from above and below by a constant times (t−t ₀)^−n/(k−1). It is shown that they do not necessarily coincide withC(t−t₀)^−n/(k−1). This gives a negative answer to two conjectures posed by Kiguradze that Kneser solutions are determined by their value in a point and that blow-up solutions have prescribed asymptotics. Dedicated to Professor Vladimir Maz'ya on the occasion of his 60th birthday. The author was supported by the Swedish Natural Science Research Council (NFR) grant M-AA/MA 10879-304.     

Exact solutions for nonlinear diffusion with nonlinear loss

Exact similarity solutions are developed for nonlinear diffusion with nonlinear reactive or irreversible absorptive loss from an instantaneous source. The diffusivity is proportional to a powerm of concentration, with 0 < m 1; and loss rate is also proportional to a power of concentration,n, with 0 n < 1. The solutions are for an arbitrary number of dimensionss > 0 withs=1, 2, 3 in physical applications. All solutions give the slug radius finite, increasing to a maximum, and then decreasing to zero in finite time. Withn < 1, the loss rate at small concentrations is large enough to ensure slug extinction in finite time. The corresponding exact solutions for gain, not loss, are given also. They become independent of initial slug quantity in the limit of infinite time.     

Generalized nonlinear ordinary differential equations linear in measures

Summary In this paper one treats the initial value problem for nonlinear measure differential equations. Under various hypotheses one gets existence of global solutions and sometimes uniqueness too. There is also an example giving nonexistence or bifurcation depending on the initial datum and the measure in the equation. The results are generalizations to the nonlinear case of earlier linear results by the author. One main feature is the procedure which shows how the solutions act at points where the involved measures have point masses of arbitrary magnitude.     

Bubbles and droplets in nonlinear physics models: Analysis and numerical simulation of singular nonlinear boundary value problem

For a second-order nonlinear ordinary differential equation (ODE), a singular Boundary value problem (BVP) is investigated which arises in hydromechanics and nonlinear field theory when static centrally symmetric bubble-type (droplet-type) solutions are sought. The equation, defined on a semi-infinite interval 0 < r < ∞, possesses a regular singular point as r→ 0 and an irregular one as r→ ∞. We give the restrictions to the parameters for a correct mathematical statement of the limit boundary conditions in singular points and their accurate transfer into the neighborhoods of these points using certain results for singular Cauchy problems and stable initial manifolds. The necessary and sufficient conditions for the existence of bubble-type (droplet-type) solutions are discussed (in the form of additional restrictions to the parameters) and some estimates are obtained. A priori detailed analysis of a singular nonlinear BVP leads to efficient shooting methods for solving it approximately. Some results of the numerical experiments are displayed and their physical interpretation is discussed. This article was submitted by the author in English.     

On a class of nonlinear Neumann problems

Summary Existence theorems for nonlinear Neumann problems with inhomogeneous boundary conditions are established. It is then investigated under which conditions the solutions are uniformly bounded. Uniqueness results for positive solutions are given and the asymptotic behavior of the solutions of the corresponding parabolic equation is discussed. The main tools are fixed point theorems and the method of upper and lower solutions.     

On the concentration properties for the nonlinear Schrödinger equation with a Stark potential

In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdinger equation without any potential, we obtain some concentration properties of blow-up solutions, including that the origin is the blow-up point of the radial blow-up solutions, the phenomenon of L2-concentration and rate of L2-concentration of blow-up solutions.