A Trudinger–Moser inequality of Adimurthi–Druet type involving higher order eigenvalues

Abstract

Let $\Omega$ be a smooth bounded domain in ${\mathbb {R}}^2$ and $W_0^{1, 2}(\Omega )$ be the usual Sobolev space. Assume that $0<\lambda _1(\Omega )<\lambda _2(\Omega )<\cdots$ are all distinct eigenvalues of the Laplace–Beltrami operator $-\Delta$. Denote the eigenfunction space with respect to $\lambda _j(\Omega )$ by $E_{\lambda _j(\Omega )}$ for any integer $j\ge 1$. For any $\ell \ge 1$, we define

$$\begin{aligned} E_{\ell }^\perp =\left\{ u\in W_0^{1, 2}(\Omega ): \mathop \int \limits _\Omega uvdx=0, \, \, \forall v\in E_{\lambda _1(\Omega )}\oplus \cdots \oplus E_{\lambda _\ell (\Omega )}\right\} . \end{aligned}$$

We shall prove that there exists some constant $\alpha _0$ with $0<\alpha _0<\lambda _{\ell +1}(\Omega )$ such that when $0\le \alpha \le \alpha _0$, the supremum

$$\begin{aligned} \Lambda _{\ell , \alpha }=\sup _{u\in E_\ell ^\perp , \, \Vert \nabla u\Vert _2\le 1}\mathop \int \limits _\Omega e^{4\pi u^2(1+\alpha \Vert u\Vert _2^2)}dx \end{aligned}$$

can be attained. This complements the result of Lu and Yang (Discrete Contin Dyn Syst 25:963–979, 2009).