The Toda system and multiple-end solutions of autonomous planar elliptic problems

TL;DR

This paper constructs a new class of positive solutions for planar elliptic problems related to nonlinear Schrödinger and biological pattern formation models, with energy growth properties linked to Toda system solutions.

Contribution

It introduces a novel class of solutions with linear energy growth, connecting elliptic problems to Toda system solutions in the plane.

Findings

Solutions exhibit linear energy growth with radius.

Solutions are related to Toda system solutions.

Applicable to nonlinear Schrödinger and biological models.

Abstract

We construct a new class of positive solutions for a classical semilinear elliptic problem in the plane which arise for instance as the standing-wave problem for the standard nonlinear Schr\"odinger equation or in nonlinear models in Turing's theory biological theory of pattern formation such as the Gray-Scott or Gierer-Meinhardt systems. The solutions we construct have the property that their energy over a ball of radius R grows linearly with R as R tends to infinity. These solutions are strongly related to the solutions of a Toda system.