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least energy solutions for semilinear Schrdinger equation with electromagnetic fields and critical growth

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Science China Mathematics 2015年第11期58卷 2317-2328页

作者： TANG ZhongWei WANG YanLi School of Mathematical Sciences Beijing Normal University

We study a class of semilinear SchrSdinger equation with electromagnetic fields and the nonlinearity term involving critical growth. We assume that the potential of the equation includes a parameter A and can be negative in some domain. Moreover, the potential behaves like potential well when the parameter A is large. Using variational methods combining Nehari methods, we prove that the equation has a least energy solution which, as the parameter A becomes large, localized near the bottom of the potential well. Our result is an extension of the corresponding result for the SchrSdinger equation which involves critical growth but does not involve electromagnetic fields.

关键词： semilinear Schr6dinger equation least energy solution critical growth electromagnetic fields

least energy solutions of nonlinear Schr odinger equations involving the fractional Laplacian and potential wells

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Science China Mathematics 2017年第2期60卷 261-276页

作者： NIU MiaoMiao TANG ZhongWei School of Mathematical Sciences Beijing Normal University Beijing 100875 China

We are concerned with the existence of least energy solutions of nonlinear Schrodinger equations involving the fractional Laplacian（-△）%s u（x）＋λV（x）u（x）=u（x）^（p-1）,u（x）〉=0,x∈R^N,for sufficiently large λ,2〈p〈N-2s^-2N for N≥2. V（x） is a real continuous function on RN. Using variational methods we prove the existence of least energy solution uλ（x） which localizes near the potential well int V-1 （0） for A large. Moreover, if the zero sets int V-1 （0） of V（x） include more than one isolated component, then ux（x） will be trapped around all the isolated components. However, in Laplacian case s = 1, when the parameter A is large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrarily small in other components of int V^-1（0）. This is the essential difference with the Laplacian problems since the operator （-△）s is nonlocal.

关键词： nonlinear SchrSdinger equation least energy solution fractional Laplacian variational methods

On a Class of Infinite-Dimensional Hamiltonian Systems with Asymptotically Periodic Nonlinearities

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Chinese Annals of Mathematics,Series B 2011年第1期32卷 45-58页

作者： Minbo YANG Zifei SHEN Yanheng DING Department of Mathematics Zhejiang Normal University Jinhua 321004 Zhejiang China Institute of Mathematics AMSS Chinese Academy of Sciences Beijing 100}90 China

The authors study the existence of homoclinic type solutions for the following system of diffusion equations on R × RN：{atu-xu ＋ b ·▽xu ＋ au ＋ V（t,x）v = Hv（t,x,u,v）,-atv-xv-b·▽xv ＋ av ＋ V（t,x）u = Hu（t,x,u,v）,where z =（u,v）：R × RN → Rm × Rm,a 〉 0,b =（b1,···,bN） is a constant vector and V ∈ C（R × RN,R）,H ∈ C1（R × RN × R2m,R）.Under suitable conditions on V（t,x） and the nonlinearity for H（t,x,z）,at least one non-stationary homoclinic solution with least energy is obtained.

关键词： Variational methods least energy solution Hamiltonian system