Entire solutions to a strongly competitive nonlinear Schr\"odinger system

TL;DR

This paper constructs infinitely many non-radial positive solutions for a strongly competitive nonlinear Schrödinger system, with solutions exhibiting complex peak arrangements as the competition parameter grows large.

Contribution

It introduces the first known examples of non-radial positive solutions for such systems in the entire space, with a novel peak configuration pattern.

Findings

Existence of infinitely many non-radial solutions.

Solutions' peaks arranged along polygons and rays.

Solutions' profiles are sums of shifted positive solutions.

Abstract

We build infinitely-many non-radial positive solutions to the Schr\"odinger system \begin{equation*} \left\{\begin{aligned} &-\Delta u_1+u_1=u_1^{{\mathfrak p} }-\Lambda u_1^{a_1} u_2^{a_2}\ \hbox{in}\ \mathbb R^N\\ &-\Delta u_2+u_2=u_2^{{\mathfrak p} }-\Lambda u_1^{b_1}u_2^{b_2} \ \hbox{in}\ \mathbb R^N\\ \end{aligned}\right. \end{equation*} with sub-critical $\mathfrak p$-growth as $\Lambda \to +\infty$. The profile of each component is the sum of several copies of the positive solution to $-\Delta U+U=U^{{\mathfrak p} }$ in $\mathbb R^N$, centered at suitable {\em peaks} whose mutual distances diverge as $\Lambda$ increases. More precisely, given two concentric regular polygons with $k$ sides and very large radii, the peaks of the first component are arranged along the edges of the {\em outer} polygon, alternated with those of the second component, and along the $k$ rays joining the vertices of the two polygons. To the best of our knowledge, this provides the first example of non-radial positive solutions for strongly competitive Schr\"odinger systems in the whole space.