Infinitely many positive solutions for a Schrödinger–Poisson system

We are interested in the existence of infinitely many positive non-radial solutions of a Schrödinger–Poisson system with a positive radial bounded external potential decaying at infinity.     

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.     

Ground states for Schrödinger–Poisson type systems

In this paper we consider the following elliptic system in \mathbbR³{\mathbb{R}^3}

On a class of nonlinear Schrödinger equations

This paper concerns the existence of standing wave solutions of nonlinear Schrödinger equations. Making a standing wave ansatz reduces the problem to that of studying the semilinear elliptic equation:

Homogenized Diffusion Limit of a Vlasov–Poisson–Fokker–Planck Model

The approximation by diffusion and homogenization of the initial-boundary value problem of the Vlasov–Poisson–Fokker–Planck model is studied for a given velocity field with spatial macroscopic and microscopic variations. The L¹-contraction property of the Fokker–Planck operator and a two-scale Hybrid-Hilbert expansion are used to prove the convergence towards a homogenized Drift–Diffusion equation and to exhibit a rate of convergence.     

On a nonlinear Schrödinger equation with a localizing effect

We consider the nonlinear Schrödinger equation associated to a singular potential of the form $a{|u|}^{?(1?m)}u+bu$, for some $m\in (0,1)$, on a possible unbounded domain. We use some suitable energy methods to prove that if $\mathrm{Re}(a)+\mathrm{Im}(a)>0$ and if the initial and right hand side data have compact support then any possible solution must also have a compact support for any $t>0$. This property contrasts with the behavior of solutions associated to regular potentials $(m?1)$. Related results are proved also for the associated stationary problem and for self-similar solution on the whole space and potential $a{|u|}^{?(1?m)}u$. The existence of solutions is obtained by some compactness methods under additional conditions. To cite this article: P. Bégout, J.I. Díaz, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

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.     

On a class of nonlinear inhomogeneous Schrödinger equation

In this paper, we study the inhomogeneous Schrödinger equation $$i\varphi_{t}=-\triangle\varphi -|x|^{b}|\varphi|^{p-1}\varphi,\quad x\in \mathbb{R}^{N}.$$ By using variational methods and a refined interpolation inequality, we establish some simple but sharp conditions on the solutions which exist globally or blow up in a finite time. An interesting result is that we obtain the existence of global solution for arbitrarily large data.     

On Characterization of Poisson Integrals of Schrödinger Operators with Morrey Traces

Let L be a Schr?dinger operator of the form L =-Δ + V acting on L~2(R~n) where the nonnegative potential V belongs to the reverse H?lder class B_q for some q ≥ n. In this article we will show that a function f ∈ L~(2,λ)(R~n), 0 λ n, is the trace of the solution of L_u =-u_(tt) + L_u =0, u(x, 0) = f(x), where u satisfies a Carleson type condition sup x_B,r_Br_B~(-λ)∫_0~(rB)∫_(B(x_B,r_B))t|u(x,t)|~2dxdt≤C∞.Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces L_L~(2,λ)(R~n) associated to the operator L, i.e.L_L~(2,λ)(R~n)=L~(2,λ)(R~n).Conversely, this Carleson type condition characterizes all the L-harmonic functions whose traces belong to the space L~(2,λ)(R~n) for all 0 λ n.     

On Combinatorics of Schrödinger Perturbations

We give a tight upper bound for Schrödinger-type perturbations of integral kernels.     

On a periodic Schrödinger equation with nonlocal superlinear part

We consider the Choquard-Pekar equation and focus on the case of periodic potential V. For a large class of even functions W we show existence and multiplicity of solutions. Essentially the conditions are that 0 is not in the spectrum of the linear part –+V and that W does not change sign. Our results carry over to more general nonlinear terms in arbitrary space dimension N2.Mathematics Subject Classification (2000):35Q55, 35Q40, 35J10, 35J20, 35J60, 46N50, 49J35, 81V70in final form: 14 November 2003