A sharp Adams inequality in dimension four and its extremal functions

TL;DR

This paper establishes a sharp Adams inequality in four dimensions involving a weighted Sobolev norm, proves its finiteness, and demonstrates the existence of extremal functions that attain the supremum.

Contribution

It extends the Adams inequality in four dimensions with a new weighted norm and proves the existence of extremal functions, improving recent results.

Findings

The supremum of the exponential integral is finite under the new norm.

Existence of extremal functions attaining the supremum.

The inequality is strengthened compared to previous work.

Abstract

Let $\Omega$ be a smooth oriented bounded domain in $\mathbb R^4$, $H_0^2(\Omega)$ be the Sobolev space, and $\lambda_1(\Omega)= \inf \{\|\Delta u\|_2^2 : u\in H_0^2(\Omega), \|u\|_2 =1\}$ be the first eigenvalue of the bi-Laplacian operator $\Delta^2$ on $\Omega$. For $\alpha \in [0,\lambda_1(\Omega))$, we define $\|u\|_{2,\alpha}^2 = \|\Delta u\|_2^2 - \alpha \|u\|_2^2$, for $u \in H_0^2(\Omega)$. In this paper, we will prove the following inequality \[ \sup_{u\in H_0^2(\Omega),\, \|u\|_{2,\alpha} \leq 1} \int_{\Omega} e^{32 \pi^2 u(x)^2} dx < \infty. \] This strengthens a recent result of Lu and Yang \cite{LuYang}. We also show that there exists a function $u^*\in H_0^2(\Omega)\cap C^4(\overline{\Omega})$ such that $\|u^*\|_{2,\alpha} =1$ and the supremum above is attained by $u^*$. Our proofs are based on the blow-up analysis method.