Extremal Functions for Trudinger–Moser Inequalities Involving Various $L^{p}$-norms in High Dimension

Abstract

In this paper, we deal with two Trudinger–Moser inequalities involving various $L^{p}$-norms on a smooth bounded domain of $\mathbb{R}^{n}$, $n \geq 3$. For any $p \gt 1$, we set \[ \lambda_{p}(\Omega) = \inf_{u \in H_{0}^{1,n}(\Omega), \; u \not\equiv 0} \frac{\|\nabla u\|_{n}^{n}}{\|u\|_{p}^{n}} \] as an eigenvalue related to the $n$-Laplacian. Based on the method of blow-up analysis, if $p_{j} \gt 1$ for all $1 \leq j \leq l$, and satisfies \[ \max_{1 \leq j \leq l} \frac{\alpha_{j}}{\lambda_{p_{j}}(\Omega)} \lt 1, \quad \sum_{j=1}^{l} \frac{\alpha_{j}}{\lambda_{p_{j}}(\Omega)} \lt 1, \] then we prove that \[ \sup_{u \in H_{0}^{1,n}(\Omega), \; \|\nabla u\|_{n} \leq 1} \int_{\Omega} e^{\alpha_{n} |u|^{\frac{n}{n-1}} \big( 1 + \sum_{j=1}^{l} \alpha_{j} \|u\|_{p_{j}}^{n} \big)^{\frac{1}{n-1}}} \, dx \] is attained, where $\alpha_{n} = n \omega_{n-1}^{1/(n-1)}$, $\omega_{n-1}$ is the surface area of the unit ball in $\mathbb{R}^{n}$. Under the same assumptions as above, we conclude that \[ \sup_{u \in H_{0}^{1,n}(\Omega), \; \|\nabla u\|_{n}^{n} - \sum_{j=1}^{l} \alpha_{j} \|u\|_{p_{j}}^{n} \leq 1} \int_{\Omega} e^{\alpha_{n} |u|^{\frac{n}{n-1}}} \, dx \] is attained.