Global Existence of Classical Solutions of the Nordström–Vlasov System in Two Space Dimensions

ABSTRACT

The dynamics of a self-gravitating ensemble of collisionless particles is modeled by the Nordström–Vlasov system in the framework of the Nordström scalar theory of gravitation. For this system in two space dimensions, integral representations of the first-order derivatives of the field are derived. Using these representations we show global existence of smooth solutions for large data.     

On the Existence of Ground State Solutions for Fractional Schrödinger–Poisson Systems with General Potentials and Super-quadratic Nonlinearity

In this article, we are concerned with the following fractional Schrödinger–Poisson system:

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u+V(x)u+\phi u=f(u)&{} \quad \hbox {in}~\mathbb {R}^{3},\\ (-\Delta )^{t}\phi =u^2&{} \quad \hbox {in}~\mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$

where \(0<s\le t<1\), \(2s+2t>3\), and \(f\in C(\mathbb {R},\mathbb {R})\). Under more relaxed assumptions on potential V(x) and f(x), we obtain the existence of ground state solutions for the above problem by adopting some new tricks. Our results here extend the existing study.

     

Schrödinger–Poisson systems in the 3-sphere

We investigate nonlinear Schrödinger–Poisson systems in the 3-sphere. We prove existence results for these systems and discuss the question of the stability of the systems with respect to their phases. While, in the subcritical case, we prove that all phases are stable, we prove in the critical case that there exists a sharp explicit threshold below which all phases are stable and above which resonant frequencies and multi-spikes blowing-up solutions can be constructed. Solutions of the Schrödinger–Poisson systems are standing waves solutions of the electrostatic Maxwell–Schrödinger system. Stable phases imply the existence of a priori bounds on the amplitudes of standing waves solutions. Unstable phases give rise to resonant states.     

Multiple Existence of Nonradial Positive Solutions for a Coupled Nonlinear Schrödinger System

In this paper, we consider the multiple existence of nonradial positive solutions of coupled nonlinear Schr?dinger system

Dispersive Estimates of Solutions to the Schrödinger Equation

We prove time decay L¹ → L^∞ estimates for the Schr?dinger group e^{it(−Δ + V)} for real-valued potentials satisfying V (x) = O (|x|^−δ), |x| ≫ 1, with δ > 5/2. Communicated by Bernard Helffer submitted 27/11/04, accepted 29/04/05     

The Existence of Infinitely Many Solutions for the Nonlinear Schrödinger–Maxwell Equations

In this paper, by using variational methods and critical point theory, we shall mainly be concerned with the study of the existence of infinitely many solutions for the following nonlinear Schrödinger–Maxwell equations $$\left\{\begin{array}{l@{\quad}l}-\triangle u + V(x)u + \phi u = f(x, u), \quad \; \, {\rm in} \, \mathbb{R}^{3},\\ -\triangle \phi = u^{2}, \quad \quad \qquad \quad \quad \quad \quad {\rm in} \, \mathbb{R}^{3},\end{array}\right.$$ where the potential V is allowed to be sign-changing, under some more assumptions on f, we get infinitely many solutions for the system.     

Quantitative properties on the steady states to a Schrödinger–Poisson–Slater System

In this paper, we obtain the existence, uniqueness and asymptotic behavior of steady states to a class of Schrödinger-Poisson-Slater System.     

Existence and multiplicity of positive solutions to Schrödinger–Poisson type systems with critical nonlocal term

The existence, nonexistence and multiplicity of positive radially symmetric solutions to a class of Schrödinger–Poisson type systems with critical nonlocal term are studied with variational methods. The existence of both the ground state solution and mountain pass type solutions are proved. It is shown that the parameter ranges of existence and nonexistence of positive solutions for the critical nonlocal case are completely different from the ones for the subcritical nonlocal system.     

Approximation and Optimal Control of Dissipative Solutions to the Ericksen–Leslie System

Abstract

We analyze the Ericksen–Leslie system equipped with the Oseen–Frank energy in three space dimensions. Recently, the author introduced the concept of dissipative solutions. These solutions show several advantages in comparison to the earlier introduced measure-valued solutions. In this article, we argue that dissipative solutions can be numerically approximated by a relatively simple scheme, which fulfills the norm-restriction on the director in every step. We introduce a semi-discrete scheme and derive an approximated version of the relative-energy inequality for solutions of this scheme. Passing to the limit in the semi-discretization, we attain dissipative solutions. Additionally, we introduce an optimal control scheme, showing the existence of an optimal control and a possible approximation strategy. We prove that the cost functional is lower semi-continuous with respect to the convergence of this approximation and argue that an optimal control is attained in the case that there exists a solution admitting additional regularity.     

Convergence of Nonlinear Schrödinger–Poisson Systems to the Compressible Euler Equations

Abstract

The combined semi-classical and quasineutral limit in the bipolar defocusing nonlinear Schrödinger–Poisson system in the whole space is proven. The electron and current densities, defined by the solution of the Schrödinger–Poisson system, converge to the solution of the compressible Euler equation with nonlinear pressure. The corresponding Wigner function of the Schrödinger–Poisson system converges to a solution of a nonlinear Vlasov equation. The proof of these results is based on estimates of a modulated energy functional and on the Wigner measure method.     

Existence and Stability of Solitary Waves of an M-Coupled Nonlinear Schrödinger System

In this paper, the existence and stability results for ground state solutions of an m-coupled nonlinear Schrödinger system $$i\frac{∂}{∂ t}u_j+\frac{∂²}{∂x²}u_j+\sum\limits^m_{i=1}b_{ij}|u_i|^p|u_j|^{p-2}u_j=0,$$ are established, where $2 ≤ m, 2≤p<3$ and $u_j$ are complex-valued functions of $(x,t) ∈ \mathbb{R}^2, j=1,...,m$ and $b_{ij}$ are positive constants satisfying $b_{ij}=b_{ji}$. In contrast with other methods used before to establish existence and stability of solitary wave solutions where the constraints of the variational minimization problem are related to one another, our approach here characterizes ground state solutions as minimizers of an energy functional subject to independent constraints. The set of minimizers is shown to be orbitally stable and further information about the structure of the set is given in certain cases.