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

TL;DR

The paper proves the existence of ground state solutions for biharmonic vector field systems in dimension $4$ by a constrained minimization scheme, establishing a minimizer $\bar u$ with $\int_{\mathbb{R}^4}G(\bar u)=0$ that solves $\Delta^2\bar u=\frac{1}{\lambda}g(\bar u)$ for some $\lambda>0$ and decays to zero at infinity. It extends the biharmonic logarithmic Sobolev inequality to $d=4$, linking equality cases to normalized ground states, and derives sharp quantitative relations between the ground-state energy $T$ and the optimal constants in the inequality. The work relies on a careful variational framework with the class $\mathcal C$, Orlicz embeddings, Pohozaev identities, and regularity theory to obtain compactness and regularity of minimizers. These results generalize prior approaches for $d\ge5$ to the critical four-dimensional setting and highlight the role of higher-order Sobolev–Orlicz controls in attaining ground states for nonlinear biharmonic systems.

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 $4$.

Theorems & Definitions (28)

• Theorem 1.1
• Remark 1
• Theorem 1.2
• Remark 2
• Remark 3
• Lemma 1
• proof
• Lemma 2
• Remark 4
• Lemma 3