Critical Gagliardo-Nirenberg, Trudinger, Brezis-Gallouet-Wainger inequalities on graded groups and ground states
Ruzhansky M. Yessirkegenov N.
1 October 2022World Scientific
Communications in Contemporary Mathematics
2022#24Issue 8
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 n, 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 Follands analysis of Hölder spaces from stratified Lie groups to general homogeneous Lie groups.
Gagliardo-Nirenberg inequality , graded Lie group , Rockland operator , Sobolev inequality , stratified Lie group , sub-Laplacian , Trudinger inequality
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Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Belgium
School of Mathematical Sciences, Queen Mary University of London, United Kingdom
Suleyman Demirel University, Kaskelen, Kazakhstan
Institute of Mathematics and Mathematical Modeling, Kazakhstan
Department of Mathematics: Analysis
School of Mathematical Sciences
Suleyman Demirel University
Institute of Mathematics and Mathematical Modeling
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