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

Ioannis Arkoudis; Panayotis Smyrnelis

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

Analysis of PDEs 2026-01-27 v3

Authors: Ioannis Arkoudis , Panayotis Smyrnelis

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 $4$ .

Cite

@article{arxiv.2508.14640,
  title  = {Biharmonic nonlinear vector field equations in $\mathbb{R}^4$},
  author = {Ioannis Arkoudis and Panayotis Smyrnelis},
  journal= {arXiv preprint arXiv:2508.14640},
  year   = {2026}
}

Comments

In this revised version v3, we assume that the potential satisfies an exponential growth condition in order to correct an error appearing in the previous versions (about the boundedness of functions in $\mathcal C$. Errare humanum est!

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