Multiplicity results of nonlinear fractional magnetic Schrödinger equation with steep potential

In this article, we prove the existence and multiplicity of non-trivial solutions for an indefinite fractional elliptic equation with magnetic field and concave–convex nonlinearities. Our multiplicity results are based on studying the decomposition of the Nehari manifold.     

On fractional Schrödinger equation in α-dimensional fractional space

The Schrödinger equation is solved in $\alpha$-dimensional fractional space with a Coulomb potential proportional to $\frac{1}{r^{\beta ?2}}$, $2\le \beta \le 4$. The wave functions are studied in terms of spatial dimensionality $\alpha$ and $\beta$ and the results for $\beta =3$ are compared with those obtained in the literature.     

Existence and concentration results for some fractional Schrödinger equations in RN with magnetic fields

We consider some nonlinear fractional Schrödinger equations with magnetic field and involving continuous nonlinearities having subcritical, critical or supercritical growth. Under a local condition on the potential, we use minimax methods to investigate the existence and concentration of nontrivial weak solutions.     

Existence time for the semilinear Schrödinger equation

Based on the methods introduced by Klainerman and Ponce, and Cohn, a lower hounded estimate of the existence time for a kind of semilinear Schrödinger equation is ohtained in this paper. The implementation of this method depends on the L ^p ? L ^q estimate and the energy estimate.     

Fractional Schrödinger equation; solvability and connection with classical Schrödinger equation

We consider the Dirichlet boundary problem for semilinear fractional Schrödinger equation with subcritical nonlinear term. Local and global in time solvability and regularity properties of solutions are discussed. But our main task is to describe the connections of the fractional equation with the classical nonlinear Schrödinger equation, including convergence of the linear semigroups and continuity of the nonlinear semigroups when the fractional exponent α approaches 1.     

Hölder regularity for the time fractional Schrödinger equation

In this paper, we investigate the Hölder regularity of solutions to the time fractional Schrödinger equation of order 1<α<2, which interpolates between the Schrödinger and wave equations. This is inspired by Hirata and Miao's work which studied the fractional diffusion-wave equation. First, we give the asymptotic behavior for the oscillatory distributional kernels and their Bessel potentials by using Fourier analytic techniques. Then, the space regularity is derived by employing some results on singular Fourier multipliers. Using the asymptotic behavior for the above kernels, we prove the time regularity. Finally, we use mismatch estimates to prove the pointwise convergence to the initial data in Hölder spaces. In addition, we also prove Hölder regularity result for the Schrödinger equation.     

Global existence for the nonlinear fractional Schrödinger equation with fractional dissipation

We consider the initial value problem for the fractional nonlinear Schrödinger equation with a fractional dissipation. Global existence and scattering are proved depending on the order of the fractional dissipation.     

Existence Results for Nonlinear Schrödinger Equations With Electromagnetic Fields

 We consider the nonlinear Schr?dinger equation

Galerkin-Legendre spectral schemes for nonlinear space fractional Schrödinger equation

In the paper, we first propose a Crank-Nicolson Galerkin-Legendre (CN-GL) spectral scheme for the one-dimensional nonlinear space fractional Schrödinger equation. Convergence with spectral accuracy is proved for the spectral approximation. Further, a Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional nonlinear space fractional Schrödinger equation is developed. The proposed schemes are shown to be efficient with second-order accuracy in time and spectral accuracy in space which are higher than some recently studied methods. Moreover, some numerical results are demonstrated to justify the theoretical analysis.     

Existence of solutions for a nonperiodic semilinear Schrödinger equation

In this paper we consider a class of semilinear Schrödinger equation which terms are asymptotically periodic at infinity. Under a weaker superquadratic condition on the nonlinearity, the existence of a ground state solution is established. The main tools employed here to overcome the new difficulties are the concentration-compactness principle and the Local Mountain Pass Theorem.