EXISTENCE OF GENERALIZED SOLUTION AND ORBITAL STABILITY OF STANDING WAVES FOR NONLINEAR SINGULAR SCHRDINGER EQUATION

The existence of generalized solution to the initial value problem iu_t △u k/(x_N)u_X_N q(x)u |u|~(p-1)u=0 on R~N is studied, By Galerkin method, we prove that the solution always exists for every initial value in H~1(R~N; k) if 1相似文献   

Singularly perturbed Neumann problem for fractional Schrdinger equations

This paper is concerned with a Neumann type problem for singularly perturbed fractional nonlinear Schrdinger equations with subcritical exponent. For some smooth bounded domain ?  R~n, our boundary condition is given by∫_?u(x)-u(y)/|x-y|~(n+2s)dy = 0 for x ∈ R~n\?.We establish existence of non-negative small energy solutions, and also investigate the integrability of the solutions on Rn.     

On a nonlinear Sturm-Liouville problem arising in the study of soliton

In this article,we consider the following nonlinear Sturm-Liouville problem(?)and prove the existence of the eigenvalue and the eigenfunction by using Schauder's fixed opinttheorem.This problem arises from finding the solutions of solitons and stationary states of thenonlinear Schr(?)dinger equation (NLS Eq.) with external fields.Using the result obtained,we provethe existence of solitons and stationary states of the NLS equation for the oscillater.     

ORBITAL INSTABILITY OF STANDING WAVES FOR THE GENERALIZED 3D NONLOCAL NONLINEAR SCHR?DINGER EQUATIONS

The existence and orbital instability of standing waves for the generalized threedimensional nonlocal nonlinear Schr¨odinger equations is studied. By defining some suitable functionals and a constrained variational problem, we first establish the existence of standing waves, which relys on the inner structure of the equations under consideration to overcome the drawback that nonlocal terms violate the space-scale invariance. We then show the orbital instability of standing waves. The arguments depend upon the conservation laws of the mass and of the energy.     

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.     

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.     

ON THE CONCENTRATION PROPERTIES FOR THE NONLINEAR SCHRDINGER 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.     

ORBITAL INSTABILITY OF STANDING WAVES FOR THE COUPLED NONLINEAR KLEIN-GORDON EQUATIONS

This paper deals with a type of standing waves for the coupled nonlinear Klein-Gordon equations in three space dimensions. First we construct a suitable constrained variational problem and obtain the existence of the standing waves with ground state by using variational argument. Then we prove the orbital instability of the standing waves by defining invariant sets and applying some priori estimates.     

EXISTENCE OF SOLUTIONS TO A THIRD-ORDER THREE-POINT BOUNDARY VALUE PROBLEM

In this paper,we study the existence of solutions to a third-order three-point boundary value problem.By imposing certain restrictions on the nonlinear term,we prove the existence of at least one solution to the boundary value problem by the method of lower and upper solutions.We are interested in the construction of lower and upper solutions.     

Existence and Concentration of Bound States of a Class of Nonlinear Schrdinger Equations in R~2 with Potential Tending to Zero at Infinity

In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schrdinger equation -ε2 Δuε + V(x)uε = K(x)|uε|p-2 uεeα0 |uε|γ,uε0, uε∈H 1(R2),where 2 p ∞, α0 0, 0 γ 2. When the potential function V (x) decays at infinity like (1 + |x|)-α with 0 α≤ 2 and K(x) 0 are permitted to be unbounded under some necessary restrictions, we will show that a positive H1 (R2 )-solution uε exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schrdinger equation-Δu + V (ξ)u = K(ξ)|u| p-2 ue α0 |u|γ has local minimum points. Furthermore, the concentration property of uε is also established as ε tends to zero.     

Stability of solutions for nonlinear Schrdinger equations in critical spaces

We consider the Cauchy problem for nonlinear Schrdinger equation iut + Δu = ±|u|pu,4/d< p <4 /d-2 in high dimensions d 6. We prove the stability of solutions in the critical space H˙xsp , where sp = d/2-p/2 .     

EXISTENCE OF SOLUTIONS FOR NON-PERIODIC SUPERLINEAR SCHRODINGER EQUATIONS WITHOUT （AR） CONDITION

The existence of solutions is obtained for a class of the non-periodic Schrdinger equation -Δu + V (x)u = f (x, u), x ∈ R N , by the generalized mountain pass theorem, where V is large at infinity and f is superlinear as |u| →∞.     

MULTIPLICITY AND CONCENTRATION BEHAVIOUR OF POSITIVE SOLUTIONS FOR SCHRDINGER-KIRCHHOFF TYPE EQUATIONS INVOLVING THE p-LAPLACIAN IN R~N

In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of Schrdinger-Kirchhoff type -εpMεp_N∫RN|▽u|p△pu+V(x)|u|p-2u=f(u) in R~N, where △_p is the p-Laplacian operator, 1 p N, M :R~+→R~+ and V :R~N→R~+are continuous functions,ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and LyusternikSchnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation.     

SIGN-CHANGING SOLUTIONS FOR THE STATIONARY KIRCHHOFF PROBLEMS INVOLVING THE FRACTIONAL LAPLACIAN IN R~N

In this paper, we study the existence of least energy sign-changing solutions for a Kirchhoff-type problem involving the fractional Laplacian operator. By using the constraint variation method and quantitative deformation lemma, we obtain a least energy nodal solution ubfor the given problem. Moreover, we show that the energy of ubis strictly larger than twice the ground state energy. We also give a convergence property of ubas b↘0, where b is regarded as a positive parameter.     

The defocusing energy-supercritical Hartree equation

In this paper,we study the global well-posedness and scattering problem for the energysupercritical Hartree equation iut+Δu.(|x|.γ.|u|2)u=0 with γ4 in dimension d γ.We prove that if the solution u is apriorily bounded in the critical Sobolev space,that is,u ∈Lt∞(I;Hxsc(Rd)) with sc:= γ/2.11,then u is global and scatters.The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation(NLW) and nonlinear Schrdinger equation(NLS).We utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios:finite time blowup;soliton-like solution and low to high frequency cascade.Making use of the No-waste Duhamel formula,we deduce that the energy of the finite time blow-up solution is zero and so get a contradiction.Finally,we adopt the double Duhamel trick,the interaction Morawetz estimate and interpolation to kill the last two scenarios.     

MEAN-FIELD LIMIT OF BOSE-EINSTEIN CONDENSATES WITH ATTRACTIVE INTERACTIONS IN R~2

Starting with the many-body Schrdinger Hamiltonian in R~2, we prove that the ground state energy of a two-dimensional interacting Bose gas with the pairwise attractive interaction approaches to the minimum of the Gross-Pitaevskii energy functional in the meanfield regime, as the particle number N→∞ and however the scattering length κ→0. By fixing N|κ|, this leads to the mean-field approximation of Bose-Einstein condensates with attractive interactions in R~2.     

A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS

In this article, we consider the dynamics of N two-dimensional boson systems interacting through a pair potential N-1Va(xi-xj) where Va(x) = a-2V (x/a). It is well known that the Gross-Pitaevskii (GP) equation is a nonlinear Schrdinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k-particle density matrices {k ut, k ≥ 1} solves the GP hierarchy. Denote by ψN,t the solution to the N-particle Schrdinger equation. Under the assumption that a = N-ε for 0 ε 3/4, we prove that as N →∞ the limit points of the k-particle density matrices of ψN,t are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by ∫V (x) dx.     

Standing Waves of the Coupled Nonlinear Schrdinger Equations

In this paper, we study the existence of standing waves of the coupled nonlinear Schr dinger equationsThe proofs of which rely on the Lyapunov-Schmidt methods and contraction mapping principle are due to FWeinstein in [1].     

The Nehari Manifold and Application to a Quasilinear Elliptic Equation with Multiple Hardy-Type Terms

In this paper, by using the Nehari manifold and variational methods, we study the existence and multiplicity of positive solutions for a multi-singular quasilinear elliptic problem with critical growth terms in bounded domains. We prove that the equation has at least two positive solutions when the parameters A belongs to a certain subset of JR.     

ON THE CAUCHY PROBLEM OF A COHERENTLY COUPLED SCHR?DINGER SYSTEM

In this article, we consider the well-posedness of a coherently coupled Schrdinger system with four waves mixing in space dimension n ≤ 4. The Cauchy problem for the cubic system is studied in L~2 for n ≤ 2 and in H~1 for n ≤ 4. We obtain two sharp conditions between global existence and blow up.