Critical Gagliardo-Nirenberg, Trudinger, Brezis-Gallouet-Wainger inequalities on graded groups and ground states

TL;DR

This paper establishes new critical inequalities on graded Lie groups, including stratified groups, and applies them to prove existence of ground state solutions for nonlinear Schrödinger equations, extending classical analysis techniques.

Contribution

It introduces new critical inequalities on graded Lie groups and connects them to ground state solutions, extending Folland's H"older space analysis to broader groups.

Findings

Existence of least energy solutions for nonlinear Schrödinger equations.

Explicit variational formulas for best constants in inequalities.

Extension of H"older space analysis to general homogeneous Lie groups.

Abstract

In this paper we investigate critical Gagliardo-Nirenberg, Trudinger-type and Brezis-Gallouet-Wainger inequalities associated with the positive Rockland operators on graded Lie groups, which includes the cases of $\mathbb R^n$, Heisenberg, and general stratified Lie groups. As an application, using the critical Gagliardo-Nirenberg inequality, the existence of least energy solutions of nonlinear Schr\"{o}dinger type equations is obtained. We also express the best constant in the critical Gagliardo-Nirenberg and Trudinger inequalities in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The obtained results are already new in the setting of general stratified Lie groups (homogeneous Carnot groups). Among new technical methods, we also extend Folland's analysis of H\"older spaces from stratified Lie groups to general homogeneous Lie groups.