A sharp Adams inequality in dimension four and its extremal functions

Abstract

Let $\Omega$ be a smooth oriented bounded domain in ${\mathbb{R}}^4$, $H_0^2(\Omega )$ be the Sobolev space, and ${\lambda }_1(\Omega )=\inf \{\Vert \Delta u{\Vert }_2^2:u\in H_0^2(\Omega ),\Vert u{\Vert }_2=1\}$ be the first eigenvalue of the bi-Laplacian operator ${\Delta }^2$ on $\Omega$. For $\alpha \in [0,{\lambda }_1(\Omega ))$, we define $\Vert u{\Vert }_{2,\alpha }^2=\Vert \Delta u{\Vert }_2^2-\alpha \Vert u{\Vert }_2^2$, for $u\in H_0^2(\Omega )$. In this paper, we will prove the following inequality \[ \sup_{u\in H_0^2(Ω),\, \|u\|_{2,α} \leq 1} \int_Ω e^{32 π^2 u(x)^2} dx < \infty. \] This strengthens a recent result of Lu and Yang \cite{LuYang}. We also show that there exists a function $u^{\ast }\in H_0^2(\Omega )\cap C^4(\overline{\Omega })$ such that $\Vert u^{\ast }{\Vert }_{2,\alpha }=1$ and the supremum above is attained by $u^{\ast }$. Our proofs are based on the blow-up analysis method.

33 pages, comment are welcome

References in corpus (2)

• Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space
• Sharp Adams and Moser-Trudinger inequalities on R^n and other spaces of infinite measure

Cited by in corpus (2)

• The sharp Adams type inequalities in the hyperbolic spaces under the Lorentz-Sobolev norms
• An improved Moser-Trudinger inequality involving the first non-zero Neumann eigenvalue with mean value zero in ${\mathbf{R}}^2$