Biharmonic nonlinear vector field equations in $\mathbb{R}^4$

TL;DR

This paper proves the existence of ground state solutions for biharmonic nonlinear vector field equations in four-dimensional space and extends related inequalities, completing previous results for higher dimensions.

Contribution

It establishes the existence of solutions in dimension four and extends the biharmonic logarithmic Sobolev inequality to this critical dimension.

Findings

Existence of ground state solutions in $\

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Abstract

Following the approach of Brezis and Lieb, we prove the existence of a ground state solution for the biharmonic nonlinear vector field equations in the limiting case of space dimension $4$. Our results complete those obtained by Mederski and Siemianowski for dimensions $d\geq 5$. We also extend the biharmonic logarithmic Sobolev inequality to dimension $4$.