Sharp Sobolev and Adams-Trudinger-Moser embeddings on weighted Sobolev spaces and their applications

TL;DR

This paper establishes sharp weighted Sobolev embeddings and Adams-Trudinger-Moser inequalities, then applies these results to prove existence of solutions for certain nonlinear elliptic equations.

Contribution

It introduces new sharp weighted Sobolev and Adams-Trudinger-Moser inequalities without boundary conditions, expanding the theoretical framework and applications.

Findings

Sharp weighted Sobolev embeddings derived

Sharp Adams-Trudinger-Moser inequalities established

Existence of solutions for nonlinear elliptic equations proved

Abstract

We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential class with a sharp constant. As applications, we prove that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions.