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 ...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.展开更多
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 impro...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.展开更多
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 con...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.展开更多
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 no...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.展开更多
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) ...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.展开更多
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 clos...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.展开更多
基金supported by JSPS Grant-in-Aid for Scientific Research(C)(15K04970)
文摘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.
文摘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.
基金supported by National Natural Science Foundation of China(Grant No.11771469)Yuxia Guo was supported by National Natural Science Foundation of China(Grant No.11771235)Shuangjie Peng was supported by National Natural Science Foundation of China(Grant No.11831009).
文摘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.
基金This work was partially supported by the 973 Program of the Ministryof Science and Technology, the Mathematical Center of the Ministry of Education, the Research Fund for the Doctorial Program of High Education, the Research Fund for the Doctorial Prog
文摘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.
基金The work is supported by the National Natural Science Foundation of China (19925107) and the Special Funds for Major State Basic Research Projects (973 Projects)
文摘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.
文摘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.