A NOTE ON THE UNIQUENESS AND THE NON-DEGENERACY OF POSITIVE RADIAL SOLUTIONS FOR SEMILINEAR ELLIPTIC PROBLEMS AND ITS APPLICATION

In this paper, we are concerned with the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Using detailed ODE anal- ysis, we extend previous results to cases where nonlinear terms may have sublinear growth. As an application, we obtain the uniqueness and the non-degeneracy of ground states for modified SchrSdinger equations.

REVISITING A NON-DEGENERACY PROPERTY FOR EXTREMAL MAPPINGS

We extend an earlier result obtained by the author in [7].

Non-degeneracy of extremal points in multi-dimensional space

For a family of smooth functions defined in multi-dimensional space,we show that,under certain generic conditions,all minimal and maximal points are non-degenerate.

Invariant Hyperbolic Tori for Gevrey-smooth Hamiltonian Systems Under Rfissmann's Non-degeneracy Condition

In this paper, we prove the persistence of hyperbolic lower dimensional invariant tori for Gevrey-smooth perturbations of partially integrable Hamiltonian systems under Riissmann＇s nondegeneracy condition by an improved KAM iteration, and the persisting invariant tori are Gevrey smooth, with the same Gevrey index as the Hamiltonian.

Non-degeneracy of Extremal Points

For a family of smooth functions, the author shows that, under certain generic conditions, all extremal(minimal and maximal) points are non-degenerate.

On reducibility of a class of nonlinear quasi-periodic systems with small perturbational parameters near equilibrium

Consider the reducibility of a class of nonlinear quasi-periodic systems with multiple eigenvalues under perturbational hypothesis in the neighborhood of equilibrium. That is, consider the following system x = （A ＋ εQ（ t） ）x ＋ eg（t） ＋ h（x, t）, where A is a constant matrix with multiple eigenvalues; h = O（x2） （x-4）） ; and h（x, t）, Q（t）, and g（t） are analytic quasi-periodic with respect to t with the same frequencies. Under suitable hypotheses of non-resonance conditions and non-degeneracy conditions, for most sufficiently small ε, the system can be reducible to a nonlinear quasi-periodic system with an equilibrium point by means of a quasi-periodic transformation.

Infinitely many dichotomous solutions for the Schrödinger-Poisson system

In this paper,we consider the following Schrodinger-Poisson system{-ε^(2)Δu+V(x)u+K(x)Φ(x)u=|u|^(p-1)u in R^(N),-ΔΦ(x)=K(x)u^(2)in RN,,where e is a small parameter,1<p<N+2/N-2,N∈[3,6],and V(x)and K(x)are potential functions with different decay at infinity.We first prove the non-degeneracy of a radial low-energy solution.Moreover,by using the non-degenerate solution,we construct a new type of infinitely many solutions for the above system,which are called“dichotomous solutions”,i.e.,these solutions concentrate both in a bounded domain and near infinity.

New existence of multi-spike solutions for the fractional Schrodinger equations

We consider the following fractional Schr¨odinger equation:(-Δ)^(s)u+V(y)u=u^(p);u>0 in R^(N);(0.1)where s ∈(0,1),1<p<N+2s/N-2s,and V(y)is a positive potential function and satisfies some expansion condition at infinity.Under the Lyapunov-Schmidt reduction framework,we construct two kinds of multi-spike solutions for(0.1).The first k-spike solution uk is concentrated at the vertices of the regular k-polygon in the(y1;y2)-plane with k and the radius large enough.Then we show that uk is non-degenerate in our special symmetric workspace,and glue it with an n-spike solution,whose centers lie in another circle in the(y3;y4)-plane,to construct infinitely many multi-spike solutions of new type.The nonlocal property of(-Δ)^(s)is in sharp contrast to the classical Schr¨odinger equations.A striking difference is that although the nonlinear exponent in(0.1)is Sobolev-subcritical,the algebraic(not exponential)decay at infinity of the ground states makes the estimates more subtle and difficult to control.Moreover,due to the non-locality of the fractional operator,we cannot establish the local Pohozaev identities for the solution u directly,but we address its corresponding harmonic extension at the same time.Finally,to construct new solutions we need pointwise estimates of new approximate solutions.To this end,we introduce a special weighted norm,and give the proof in quite a different way.

Closed characteristics on non-degenerate star-shaped hypersurfaces in R^(2n)

In this paper, firstly we establish the relation theorem between the Maslov-type index and the index defined by C. Viterbo for star-shaped Hamiltonian systems. Then we extend the iteration formula of C. Viterbo for non-degenerate star-shaped Hamiltonian systems to the general case. Finally we prove that there exist at least two geometrically distinct closed characteristics on any non-degenerate star-shaped compact smooth hypersurface on R2n with n ＞ 1. Here we call a hypersurface non-degenerate, if all the closed characteristics on the given hypersurface together with all of their iterations are non-degenerate as periodic solutions of the corresponding Hamiltonian system. We also study the ellipticity of closed characteristics when n=2.

Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems

In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus,numerical solutions are quasi-periodic with a diophantine frequency vector of time step size dependence. These results generalize Shang's previous ones(1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov.

An Improved Result for Positive Measure Reducibility of Quasi-periodic Linear Systems

In this paper, by the KAM method, under weaker small denominator conditions and nondegeneracy conditions, we prove a positive measure reducibility for quasi-periodic linear systems close to constant： X = （A（λ） ＋ F（ψ, λ））X, ψ=ωwhere the parameter λ∈ （a, b）, w is a fixed Diophantine vector, which is a generalization of jorba ＆ Simo＇s positive measure reducibility result.

Existence of Invariant Tori in Reversible Mappings

In this paper, we study reversible diffeomorphisms with n angular variables and only one action variable. Under some reasonable non-degeneracy conditions, we prove that the reversible diffeomorphisms sufficiently close to the integrable ones preserve a large set of n-dimensional invariant tori.