The Allen-Cahn equation with nonlinear truncated Laplacians: description of radial solutions

TL;DR

This paper analyzes radial solutions of the Allen-Cahn equation with nonlinear truncated Laplacians, revealing their qualitative behavior, including the existence of unbounded solutions and the absence of oscillations, contrasting with classical operators.

Contribution

It provides a nearly complete description of radial solutions for this nonlinear PDE, highlighting new phenomena such as unbounded solutions and non-oscillatory behavior.

Findings

Existence of surprising unbounded radial solutions.

Radial solutions do not oscillate, unlike classical cases.

Transition between first and second order ODE regimes is characterized.

Abstract

We consider the Allen-Cahn equation with the so-called truncated Laplacians, which are fully nonlinear differential operators that depend on some eigenvalues of the Hessian matrix. By monitoring the sign of a quantity that is responsible for switches from a first order ODE regime to a second order ODE regime (and vice versa), we give a nearly complete description of radial solutions. In particular, we reveal the existence of surprising unbounded radial solutions. Also radial solutions cannot oscillate, which is in sharp contrast with the case of the Laplacian operator, or that of Pucci's operators.