Calculus of Variations and Geometric Measure Theory

N. Soave

On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition.

created by soave on 29 Jul 2015


Published Paper

Inserted: 29 jul 2015

Journal: Calc. Var. Partial Differential Equations
Volume: 53
Pages: 689--718
Year: 2015
Doi: 10.1007/s00526-014-0764-3
Links: link preliminary version


We study existence and phase separation, and the relation between these two aspects, of positive bound states for the nonlinear elliptic system \[ \begin{cases} - \Delta u_i + \lambda_i u_i = \sum_{j=1}^d \beta_{ij} u_j^2 u_i & \text{in $\Omega$} \\ u_1 =\cdots = u_d=0 & \text{on $\pa \Omega$}. \end{cases} \] This system arises when searching for solitary waves for the Gross-Pitaevskii equations. We focus on the case of simultaneous cooperation and competition , that is, we assume that there exist two pairs $(i_1,j_1)$ and $(i_2,j_2)$ such that $\beta_{i_1 j_1}>0$ and $\beta_{i_2 j_2}<0$. Our first main results establishes the existence of solutions with at least $m$ positive components for every $m \leq d$; any such solution is a minimizer of the energy functional $J$ restricted on a Nehari-type manifold $\mathcal{N}$. At a later stage, by means of level estimates on the constrained second differential of $J$ on $\mathcal{N}$, we show that, under some additional assumptions, any minimizer of $J$ on $\mathcal{N}$ has all nontrivial components. In order to prove this second result, we analyse the phase separation phenomena which involve solutions of the system in a not completely competitive framework.