Open AccessCritical Gagliardo–Nirenberg, Trudinger, Brezis–Gallouet–Wainger inequalities on graded groups and ground states
Author(s) -
Michael Ruzhansky,
Nurgissa Yessirkegenov
Publication year - 2021
Publication title -
communications in contemporary mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.397
H-Index - 48
eISSN - 1793-6683
pISSN - 0219-1997
DOI - 10.1142/s0219199721500619
Subject(s) - mathematics , nirenberg and matthaei experiment , homogeneous , lie group , pure mathematics , nonlinear system , type (biology) , group (periodic table) , inequality , mathematical analysis , combinatorics , ecology , chemistry , physics , organic chemistry , quantum mechanics , biology
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 include the cases of [Formula: see text], Heisenberg, and general stratified Lie groups. As an application, using the critical Gagliardo–Nirenberg inequality, the existence of least energy solutions of nonlinear Schrödinger 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 Hölder spaces from stratified Lie groups to general homogeneous Lie groups.