A Lane-Emden system of free boundary type: existence, uniqueness and monotonicity of solutions

TL;DR

This paper investigates a Hamiltonian free boundary system, establishing existence, uniqueness, and monotonicity of solutions, and explores implications for the parametrization of solution continua in superlinear elliptic systems.

Contribution

It provides new results on the existence, uniqueness, and monotonicity of solutions for a Hamiltonian free boundary system, and discusses implications for elliptic system continua.

Findings

Existence of solutions and free boundary bounds established.

Uniqueness of positive solutions proved in bounded domains.

Monotonic behavior of energies and boundary values demonstrated.

Abstract

We consider a Hamiltonian system of free boundary type, showing first uniform bounds and existence of solutions and of the free boundary. Then, for any smooth and bounded domain, we prove uniqueness of positive solutions in a suitable interval and show that the associated energies and boundary values have a monotonic behavior. Some consequences are discussed about the parametrization of the unbounded Rabinowitz continuum for a class of superlinear strongly coupled elliptic systems.