Orbital stability of standing waves of two-component Bose–Einstein condensates with internal atomic Josephson junction

In this paper, we prove existence, symmetry and uniqueness of standing waves for a coupled Gross–Pitaevskii equations modeling component Bose–Einstein condensates BEC with an internal atomic Josephson junction. We will then address the orbital stability of these standing waves and characterize their orbit.     

Multiple Solutions for a Class of Nonhomogeneous Fractional Schrödinger Equations in $$\mathbb {R}^{N}$$

This paper is concerned with the following fractional Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s} u+u= k(x)f(u)+h(x) \text{ in } \mathbb {R}^{N}\\ u\in H^{s}(\mathbb {R}^{N}), \, u>0 \text{ in } \mathbb {R}^{N}, \end{array} \right. \end{aligned}$$

where \(s\in (0,1),N> 2s, (-\Delta )^{s}\) is the fractional Laplacian, k is a bounded positive function, \(h\in L^{2}(\mathbb {R}^{N}), h\not \equiv 0\) is nonnegative and f is either asymptotically linear or superlinear at infinity. By using the s-harmonic extension technique and suitable variational methods, we prove the existence of at least two positive solutions for the problem under consideration, provided that \(|h|_{2}\) is sufficiently small.

     

Symmetry of minimizers of some fractional problems

We establish the existence and symmetry of all minimizers of a constrained variational problem involving the fractional gradient. This problem is closely connected to some fractional kinetic equations.     

Characterization of Maximizers via Mass Transportation Techniques

In this paper, we address the question of existence and uniqueness of maximizers of a class of functionals under constraints, via mass transportation theory. We also determine suitable assumptions ensuring that balls are the unique maximizers. In both cases, we show that our hypotheses are optimal.     

On the Optimality of the Assumptions Used to Prove the Existence and Symmetry of Minimizers of Some Fractional Constrained Variational Problems

In this paper, we discuss the optimality of the assumptions used, in a previous paper, to prove existence and symmetry of minimizers of the fractional constrained variational problem: $$\inf \;\left\{\frac{1}{2} \int|\nabla_s u|^2 - \int F(|x|, u):\;u\in H^s (\mathbb{R}^N) \mbox{ and } \int u^2 = c^2\right\},$$ where c is a prescribed number. More precisely, we will show that if one of the conditions, used to prove that all minimizers of the above constrained variational problem, are radial and radially decreasing for all c, do not hold true, then there are several interesting situations:

There is no minimizer at all.

The infimum is achieved but no minimizer is radial.

For some values of c there is no minimizer. For large values, the minimizer is radial and radially decreasing.

In the fractional setting, such a study is more subtle than in the classical one. We take advantage of some brilliant results obtained recently in Cabre and Sire (Anal. PDEs, 2012), Dyda (Fractional calculus for power functions, 2012) and Hajaiej et al. (Necessary and sufficient conditions for the fractional Gagliardo–Nirenberg inequalities and applications to Navier–Stokes and generalized Boson equations, 2012).     

Variational Approach to the Orbital Stability of Standing Waves of the Gross-Pitaevskii Equation

This paper is concerned with the mathematical analysis of a masssubcritical nonlinear Schrödinger equation arising from fiber optic applications. We show the existence and symmetry of minimizers of the associated constrained variational problem. We also prove the orbital stability of such solutions referred to as standing waves and characterize the associated orbit. In the last section, we illustrate our results with few numerical simulations.     

Symmetrization Inequalities for Composition Operators of Caratheodory Type

Let F:(0, ) x [0, ) R be a function of Carathéodorytype. We establish the inequality where u^* denotes the Schwarz symmetrization of u, under hypotheseson F that seem quite natural when this inequality is used toobtain existence results in the context of elliptic partialdifferential equations. We also treat the case where R^N is replacedby a set of finite measure. The identity is also discussed under the assumption that G: [0,) R is a Borel function. 2000 Mathematics Subject Classification26D20, 42C20, 46E30.