On a degree associated with the Gross-Pitaevskii system with a large parameter

E Norman Dancer

doi:10.3934/dcdss.2019120

Abstract

In a number of cases we calculate the sum of the degrees of the small positive solutions of the Gross-Pitaevskii system when the interaction is strong.

Highlights

We want to study the following system for large positive k−∆u = f (u) − kv2u −∆v = g(v) − ku2v in Ω, (1)where u = v = 0 on ∂Ω and u, v ≥ 0 in Ω
−∆v = dv − ku2v for large k on the ball with centre 0 and radius 1 in K ⊕ K, where K is the cone of non-negative functions in C0(Ω)
We prove these results by deforming the problem (1) to the population model and using degree results for mappings with a rotational symmetry

Summary

Introduction

Ω is a smooth bounded domain in Rn. We assume f (0) = g(0) = 0, f and g are C1, and f (y) and g(y) are negative for large positive y. It was proved in [2] and [6] that positive solutions of (1) for large k are of two types: (u, v) is close to (z+, z−) where z is a non-trivial sign changing solution of

Objectives

Results

Conclusion