Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

This paper is concerned with the existence and qualitative property of standing wave solutions  for the nonlinear Schrodinger equation  with E being a critical frequency in the sense that . We show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of V and whose amplitude goes to 0 as . Moreover, depending upon the local behaviour of the potential function V(x) near the minimum points, the limiting profile of the standing-wave solutions will be shown to exhibit quite different characteristic features. This is in striking contrast with the non-critical frequency case  which has been extensively studied in recent years.

Standing Waves with a Critical Frequency for Nonlinear Schrödinger Equations

We prove some results on the existence and compactness of solutions of a fractional Nirenberg problem.

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=16&iss=6&rank=1

On a fractional Nirenberg problem, part I: Blow up analysis and compactness of solutions

A classical mathematical proof is constructed using pencil and paper. However, there are many ways in which computers may be used in a mathematical proof. But ‘proof by computer’, or even the use of computers in the course of a proof, is not so readily accepted (the December 2008 issue of the Notices of the American Mathematical Society is devoted to formal proofs by computer).In the following we introduce verification methods and discuss how they can assist in achieving a mathematically rigorous result. In particular we emphasize how floating-point arithmetic is used.

Verification methods: Rigorous results using floating-point arithmetic

For elliptic equations e2Δu − V(x) u + f(u) = 0, x ∈ RN, N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of V, as e → 0, under conditions on f which we believe to be almost optimal.

/pdf/standing-waves-for-nonlinear-schrodinger-equations-with-a-508uaoykak.pdf

Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity

Infinitely many solutions for the prescribed scalar curvature problem on SN

We consider the following nonlinear problem in ${\mathbb {R}^N}$

$$- \Delta u +V(|y|)u = u^{p},\quad u > 0 \quad {\rm in}\, \mathbb {R}^N, \quad u \in H^1(\mathbb {R}^N), \quad \quad \quad (0.1)$$

where V(r) is a positive function, ${1< p < {\frac{N+2}{N-2}}}$. We show that if V(r) has the following expansion: 

$$V(r) = V_0+\frac a {r^m} +O \left(\frac1{r^{m+\theta}}\right),\quad {\rm as} \, r\to +\infty,$$

where a > 0, m > 1, θ > 0, and V0 > 0 are some constants, then (0.1) has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.

Shusen Yan

Papers

Infinitely many solutions for the prescribed scalar curvature problem on SN

Infinitely many positive solutions for the nonlinear Schrödinger equations in {\mathbb{R}^N}

On the prescribed scalar curvature problem in RN, local uniqueness and periodicity

Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth

Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth