Sharp Adams inequalities with exact growth conditions on metric measure spaces and applications

TL;DR

This paper establishes sharp Adams inequalities with precise growth conditions on metric measure spaces, extending classical results and applying them to derive new Moser-Trudinger inequalities and solutions for elliptic equations.

Contribution

It generalizes Adams inequalities with exact growth conditions to metric measure spaces and applies these results to obtain new inequalities and existence results for elliptic equations.

Findings

Derived new Adams inequalities with exact growth on metric measure spaces

Extended Moser-Trudinger inequalities to various geometric settings

Proved existence of solutions for certain quasilinear elliptic equations

Abstract

Adams inequalities with exact growth conditions are derived for Riesz-like potentials on metric measure spaces. The results extend and improve those obtained recently on $\mathbb R^n$ by the second author, for Riesz-like convolution operators. As a consequence, we will obtain new sharp Moser-Trudinger inequalities with exact growth conditions on $\mathbb R^n$, the Heisenberg group, and Hadamard manifolds. On $\mathbb R^n$ such inequalities will be used to prove the existence of radial ground states solutions for a class of quasilinear elliptic equations, extending results due to Masmoudi and Sani.