Sharp critical and subcritical trace Trudinger-Moser and Adams inequalities on the upper half spaces

TL;DR

This paper establishes sharp trace Trudinger-Moser and Adams inequalities on upper half spaces, proves the existence of extremals and least energy solutions for related bi-harmonic equations, advancing borderline Sobolev trace inequality theory.

Contribution

It introduces new sharp inequalities on half spaces and demonstrates the existence of extremals and solutions, using Fourier rearrangement and harmonic extension methods.

Findings

Established sharp critical and subcritical trace inequalities.

Proved existence of extremals for these inequalities.

Demonstrated existence of least energy solutions for bi-harmonic equations.

Abstract

In this paper, we establish the sharp critical and subcritical trace Trudinger-Moser and Adams inequalities on the half spaces and prove the existence of their extremals through the method based on the Fourier rearrangement, harmonic extension and scaling invariance. These trace Trudinger-Moser and Adams inequalities can be considered as the borderline case of the Sobolev trace inequalities of first and higher orders. Furthermore, we show the existence of the least energy solutions for a class of bi-harmonic equations with nonlinear Neumann boundary condition associated with the trace Adams inequalities.