Extremal functions for the second-order Sobolev inequality on groups of polynomial growth

TL;DR

This paper establishes second-order Sobolev inequalities on Cayley graphs of polynomial growth groups, demonstrating the existence of extremal functions and applications to nonlinear PDEs.

Contribution

It extends Sobolev inequalities to discrete groups of polynomial growth and proves the existence of extremal functions using a discrete concentration-compactness approach.

Findings

Existence of extremal functions for second-order Sobolev inequalities.

Application to positive ground state solutions of p-biharmonic equations.

Application to Lane-Emden systems.

Abstract

In this paper, we prove the second-order Sobolev inequalities on Cayley graphs of groups of polynomial growth. We use the discrete Concentration-Compactness principle to prove the existence of extremal functions for best constants in supercritical cases. As applications, we get the existence of positive ground state solutions to the $p$-biharmonic equations and the Lane-Emden systems.