A Detailed Analysis on Sharpened Singular Adams-Type Inequalities

TL;DR

This paper establishes sharp Adams-type inequalities with singular weights, explores their properties, and applies the results to prove existence of solutions for certain nonlinear elliptic equations involving singular exponential growth.

Contribution

It introduces new sharp singular Adams inequalities, a refined concentration-compactness principle, and a novel compact embedding, advancing the understanding of singular functional inequalities and their applications.

Findings

Proved sharp Adams-type inequalities with singular weights.

Established a refined concentration-compactness principle.

Applied results to demonstrate existence of solutions for nonlinear elliptic equations.

Abstract

We establish a sharp Adams-type inequality in higher-order function spaces with singular weights on $\mathbb{R}^n$. A sharp singular concentration-compactness principle, improving Lions' result, is also proved. The study distinguishes between critical and subcritical sharp singular Adams-type inequalities and shows their equivalence. Furthermore, we analyze the asymptotic behavior of the associated bounds and relate the suprema of the critical and subcritical cases. A new compact embedding, crucial to our analysis, is also derived. Moreover, as an application of these results, by employing the mountain pass theorem, we study the existence of nontrivial solutions to a class of nonhomogeneous quasilinear elliptic equations involving the $(p,\frac{n}{2})$-biharmonic operator with singular exponential growth.