Positive solutions for some non-autonomous Schrödinger-Poisson systems

In this paper we study the Schrödinger-Poisson system

(SP)     

Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity

In this paper we use a concentration and compactness argument to prove the existence of a nontrivial non-radial solution to the nonlinear Schrödinger-Poisson equations in R³, assuming on the nonlinearity the general hypotheses introduced by Berestycki and Lions.     

Schrödinger-Poisson equations without Ambrosetti-Rabinowitz condition

We prove the existence of ground state solutions for a stationary Schrödinger-Poisson equation in R³. The proof is based on the mountain pass theorem and it does not require the Ambrosetti-Rabinowitz condition.     

Asymptotically linear Schrödinger-Poisson systems with potentials vanishing at infinity

In this paper, we are concerned with the following nonlinear Schrödinger-Poisson equations

(P)     

Existence and multiplicity results for the nonlinear Schrödinger-Poisson systems

In this paper, we study the existence and multiplicity results for the nonlinear Schrödinger-Poisson systems

(∗)     

Multiple positive solutions for a Schrödinger-Poisson-Slater system

In this paper we investigate the existence of positive solutions to the following Schrödinger-Poisson-Slater system

     

The Schrödinger-Poisson equation under the effect of a nonlinear local term

In this paper we study the problem

     

Analytical solutions to the backward Kolmogorov PDE via an adiabatic approximation to the Schrödinger PDE

Analytical solutions to the backward Kolmogorov PDE

     

Solitary waves for non-homogeneous Schrödinger-Maxwell system

In this paper we study the Schrödinger-Maxwell system

     

拟线性Schrdinger-Poisson方程正解、负解、变号解的存在性

本文考虑拟线性Schrdinger-Poisson方程{-△u+V(x)u+Φu-1/2△(u~2)u=f(x,u),x∈R~3,-△Φ=u~2,x∈R~3,其中f是一个C~1超线性且次临界的非线性项,V是正的有界位势.利用扰动方法,我们证明了该方程非平凡解、正解、负解、变号解的存在性.     

Finite dimensional global attractor for a parametric nonlinear Schrödinger system with a trapping potential

We prove that a parametric nonlinear Schrödinger equation possesses a finite dimensional smooth global attractor in a suitable energy space.     

Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces

We study the well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations

     

Global existence of small classical solutions to nonlinear Schrödinger equations

We study the global Cauchy problem for nonlinear Schrödinger equations with cubic interactions of derivative type in space dimension n?3$n?3$. The global existence of small classical solutions is proved in the case where every real part of the first derivatives of the interaction with respect to first derivatives of wavefunction is derived by a potential function of quadratic interaction. The proof depends on the energy estimate involving the quadratic potential and on the endpoint Strichartz estimates.     

Semiclassical stationary states of Nonlinear Schrödinger equations with an external magnetic field

In this paper we obtain multiple solutions of the nonlinear Schrödinger equation with an external magnetic field

     

Inverse scattering for the magnetic Schrödinger operator

We show that fixed energy scattering measurements for the magnetic Schrödinger operator uniquely determine the magnetic field and electric potential in dimensions n?3. The magnetic potential, its first derivatives, and the electric potential are assumed to be exponentially decaying. This improves an earlier result of Eskin and Ralston (1995) [5] which considered potentials with many derivatives. The proof is close to arguments in inverse boundary problems, and is based on constructing complex geometrical optics solutions to the Schrödinger equation via a pseudodifferential conjugation argument.     

Rough solutions for the periodic Schrödinger-Korteweg-de Vries system

We prove two new mixed sharp bilinear estimates of Schrödinger-Airy type. In particular, we obtain the local well-posedness of the Cauchy problem of the Schrödinger-Kortweg-de Vries (NLS-KdV) system in the periodic setting. Our lowest regularity is H1/4×L², which is somewhat far from the naturally expected endpoint L²×H−1/2. This is a novel phenomena related to the periodicity condition. Indeed, in the continuous case, Corcho and Linares proved local well-posedness for the natural endpoint .Nevertheless, we conclude the global well-posedness of the NLS-KdV system in the energy space H¹×H¹ using our local well-posedness result and three conservation laws discovered by M. Tsutsumi.     

An endpoint space–time estimate for the Schrödinger equation

We obtain endpoint estimates for the Schrödinger operator f→eitΔf in with initial data f in the homogeneous Sobolev space . The exponents and regularity index satisfy and . For n=2 we prove the estimates in the range q>16/5, and for n?3 in the range q>2+4/(n+1).