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

2017/09/24 by Michael Ruzhansky, Ruzhansky, Michael, Nurgissa Yessirkegenov +1
Mathematics · Medicine · #Nonlinear Partial Differential Equations #Hidradenitis Suppurativa and Treatments #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.1709.08263

Abstract

In this paper we investigate critical Gagliardo-Nirenberg, Trudinger-type and\nBrezis-Gallouet-Wainger inequalities associated with the positive Rockland\noperators on graded Lie groups, which includes the cases of mathbb Rⁿ,\nHeisenberg, and general stratified Lie groups. As an application, using the\ncritical Gagliardo-Nirenberg inequality, the existence of least energy\nsolutions of nonlinear Schr "odinger type equations is obtained. We also\nexpress the best constant in the critical Gagliardo-Nirenberg and Trudinger\ninequalities in the variational form as well as in terms of the ground state\nsolutions of the corresponding nonlinear subelliptic equations. The obtained\nresults are already new in the setting of general stratified Lie groups\n(homogeneous Carnot groups). Among new technical methods, we also extend\nFolland's analysis of H "older spaces from stratified Lie groups to general\nhomogeneous Lie groups.\n

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