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

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.