The Rabinowitz continuum of subcritical Gelfand problems and free boundary-type equations arising in plasma physics

TL;DR

This paper introduces a new approach to analyze the Rabinowitz continuum of subcritical Gelfand problems by using plasma physics-inspired free boundary problems and energy-based parametrization, providing detailed descriptions beyond classical methods.

Contribution

The authors develop a novel global parametrization of the Rabinowitz continuum using plasma physics energy concepts and solve a longstanding uniqueness problem for Grad-Shafranov equations.

Findings

Established a new global parametrization of the solution branch.

Proved uniqueness of solutions for Grad-Shafranov type equations.

Described the full solution profile as bell-shaped on a ball in any dimension.

Abstract

The qualitative behavior of the Rabinowitz unbounded continuum of subcritical Gelfand problems is well known on balls in any dimension. We don't know of any such sharp and detailed description otherwise, which is our motivation to look for a new approach to the problem. The underlying idea is to describe solutions of Gelfand problems via suitably defined constrained problems of free boundary-type arising in plasma physics and to replace the usual $L^\infty$ norm of the solution with the energy of the plasma. Toward this goal, we first solve a long standing open problem of independent interest about the uniqueness of solutions of Grad-Shafranov type equations. Thus, we exploit these unique solutions to detect a curve containing both minimal and non minimal solutions of the associated Gelfand problem. In other words we come up with a new global parametrization of the Rabinowitz continuum, the monotonicity of the energy along the branch providing a meaningful generalization of the classical pointwise monotonicity property of minimal solutions, suitable to describe non minimal solutions as well. On a ball in any dimension, we come up as expected with a bell-shaped profile of the full branch of solutions of the Gelfand problem.