Adams type inequalities and related elliptic partial differential equations in dimension four

TL;DR

This paper establishes Adams type inequalities in four dimensions and applies them to prove existence and multiplicity of solutions for related elliptic PDEs using variational methods.

Contribution

It introduces new Adams inequalities in four dimensions and applies them to elliptic PDEs, extending previous work on singular Trudinger-Moser inequalities.

Findings

Existence of nontrivial weak solutions for the PDEs

Multiplicity results for solutions

New Adams inequalities in four-dimensional space

Abstract

Motivated by Ruf-Sani's recent work, we prove an Adams type inequality and a singular Adams type inequality in the whole four dimensional Euclidean space. As applications of those inequalities, a class of elliptic partial differential equations are considered. Existence of nontrivial weak solutions and multiplicity results are obtained via the mountain-pass theorem and the Ekeland's variational principle. This is a continuation of our previous work about singular Trudinger-Moser type inequality.