Least energy solutions for semilinear Schrdinger equation with electromagnetic fields and critical growth

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.

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

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.

Uniqueness and Radial Symmetry of Least Energy Solution for a Semilinear Neumann Problem

Consider the following Neumann problem d△u- u ＋ k（x）u^p = 0 and u 〉 0 in B1, δu/δv =0 on OB1,where d 〉 0, B1 is the unit ball in R^N, k（x） = k（｜x｜） ≠ 0 is nonnegative and in C（-↑B1）, 1 〈 p 〈 N＋2/N-2 with N≥ 3. It was shown in [2] that, for any d 〉 0, problem （＊） has no nonconstant radially symmetric least energy solution if k（x） ≡ 1. By an implicit function theorem we prove that there is d0 〉 0 such that （＊） has a unique radially symmetric least energy solution if d 〉 d0, this solution is constant if k（x） ≡ 1 and nonconstant if k（x） ≠ 1. In particular, for k（x） ≡ 1, do can be expressed explicitly.

EXISTENCE AND UNIQUENESS OF THE WEAK SOLUTION TO THE INCOMPRESSIBLE NAVIER-STOKES-LANDAU-LIFSHITZ MODEL IN 2-DIMENSION

In this paper, we prove the existence and uniqueness of the weak solution to the incompressible Navier-Stokes-Landau-Lifshitz equations in two-dimension with finite energy.The main techniques is the Faedo-Galerkin approximation and weak compactness theory.

Solution for Energies and Mixing of Two 0^+ States in ^(10)He

Using results from various reactions that populate 10He, I conclude that the ground state has E2n = 1.07（7） MeV and the excited 0＋ state is in the region of 2.1-3.1 MeV. The amount of the （sd）2 component in the ground state is less than about 0.075.

Towards a (More) Electronic Transmission and Distribution (eT&D)

The challenges and the path towards a(more)electronic transmission and distribution(eT&D)is presented in this paper.The challenges are first identified together with key stakeholders in the drive for grid modernization.A fundamental question is then asked about the investment priority.Six basic characteristics are reviewed,leading to the composition of structured microgrids as the basic functional cell of a modern grid.One example of fractal radial structure and one fractal meshed structure are presented.The likely evolution path is then proposed together with basic technology sets.Specific foundation technologies are discussed in detail,including adiabatic power conversion,3MC technology,medium voltage conversion,distribution-level electronic power transformer and FACTs hardware integration,and back-to-back converters as a universal interconnect element.The rapidly emerging on-wire sensing technology is also discussed.It is pointed out that the distribution-level large electronic power transformer will provide a key component to enable hybrid ac/dc grid flow control and ancillary support for a flexible electronic transmission and distribution(eT&D)systems.

Local L^(2) theory of the fractional Navier-Stokes equations and the self-similar solution

This paper is concerned with two aspects of the fractional Navier-Stokes equation. First, we establish the local L^(2)theory of the hypo-dissipative Navier-Stokes system. More precisely, the existence of local-in-time as well as global-in-time local energy weak solutions to the hypo-dissipative Navier-Stokes system is proved.In particular, in order to construct a pressure with an explicit representation, some technical innovations are required due to the lack of known results on the local regularity of the non-local Stokes operator. Secondly, as an important application to the local L^(2)theory, we give a second construction of large self-similar solutions of the hypo-dissipative Navier-Stokes system along with the Leray-Schauder degree theory.

Periodic Solutions of Prescribed Energy for a Class of Symmetric Singular Dynamical Systems

We consider the following Hamiltonian system: q″(t)+V′(q(t))=0 q ∈C^2(R,R^n\{0}) (HS) where V ∈C^2(R^n\{0},R)is an even function.By looking for closed geodesics,we prove that (HS)has a nonconstant periodic solution of prescribed energy under suitable assumptions.Our main assumption is related with the strong force condition of Gordon.

GLOBAL FINITE ENERGY WEAK SOLUTION TO THE VISCOUS QUANTUM NAVIERSTOKES-LANDAU-LIFSHITZ-MAXWELL MODEL IN 2-DIMENSION

In this paper,we prove the global existence of the weak solution to the viscous quantum Navier-Stokes-Landau-Lifshitz-Maxwell equations in two-dimension for large data.The main techniques are the Faedo-Galerkin approximation and weak compactness theory.

Well-posedness in the Energy Space for Non-linear System of Wave Equations with Critical Growth

The authors consider the well-posedness in energy space of the critical non-linear system of wave equations with Hamiltonian structure{utt-△u=-F1（｜u｜^2,｜v｜^2）u,utt-△u=-F2（｜u｜^2,｜v｜^2）u where there exists a function F（λ,μ） such that δF（λ,μ）/δλ=F1（λ,μ）.δF（λ,μ）/δμ=F2（λ,μ） By showing that the energy and dilation identities hold for weak solution under some assumptions on the non-linearities, we prove the global well-posedness in energy space by a similar argument to that for global regularity as shown in ＂Shatah and Struwe＇s paper, Ann. of Math. 138, 503-518 （1993）＂.

Global Existence and Energy Decay for a Coupled System of Kirchhoff Type Equations with Damping and Source Terms

This paper deals with the global existence and energy decay of solutions to some coupled system of Kirchhoff type equations with nonlinear dissipative and source terms in a bounded domain. We obtain the global existence by defining the stable set in H0^1 （Ω） × H1 （Ω）, and the energy decay of global solutions is given by applying a lemma of V. Komornik.

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

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.

EXPONENTIAL STABILITY CRITERIA FOR STOCHASTIC DELAY PARTIAL DIFFERENTIAL EQUATIONS

In this paper,by constructing proper Lyapunov functions,exponential stability criteria for stochastic delay partial differential equations are obtained. An example is shown to illustrate the results.