Characterization of asymptotic tails of smooth supercritical ground states

Characterize which asymptotic tails at infinity can be generated by smooth solutions of the radial supercritical nonlinear elliptic equation \(\Delta Q+Q^p=0\), \(Q>0\), in the energy-supercritical range \(p>(d+2)/(d-2)\).

Background

The paper draws an analogy with the energy-supercritical nonlinear heat equation, whose radial ground state has a smooth, fat self-similar tail and infinite energy. Such profiles motivate the search for analogous smooth fat-tailed stationary and self-similar Euler vortices.

The set of smooth solutions to the associated elliptic equation is described as poorly understood because infinite-energy solutions are not accessible through standard variational methods. The authors explicitly formulate the characterization of tails generated by smooth solutions as an open problem.