Extremal Functions for an Improved Trudinger-Moser Inequality Involving $L^p$-Norm in $\mathbb{R}^n$ (Journal of Partial Differential Equations, 2023)
Let $W^{1,n} (\mathbb{R}^n)$ be the standard Sobolev space. For any $τ>0$ and $p>n>2,$ wedenote $$λ_{n,p}=\inf\limits_{u\in W^{1,n}(\mathbb{R}^n),\ u ot\equiv 0}\frac{\int_{\mathbb{R}^n}(|∇u|^n+τ|u|^n)dx}{(\int_{\mathbb{R}^n}|u|^p dx)^{\frac{n}{p}}}. $$ Define a norm in $W^{1,n} (\mathbb{R}^n)$ by $$||u||_{n,p}=(\int_{\mathbb{R}^n}(|∇u|^n+τ|u|^n)dx-α(\int_{\mathbb{R}^n}|u|^pdx)^{\frac{n}{p}})^{\frac{1}{n}},$$ where $0 ≤ α < λ_{n,p}.$ Using a rearrangement argument and blow-up analysis, we will prove $$\sup\limits_{u\in W^{1,n}(\mathbb{R}^n), \ ||u||_{n,p}\le 1}\int_{\mathbb{R}^n}\left(e^{α_n |u|^{\frac{n}{n-1}}}-\sum^{n-1}_{j=0}\frac{α_n^j|u|^{\frac{jn}{n-1}}}{j!} \right)dx$$ can be attained by some function $u_0 ∈ W^{1,n} (\mathbb{R}^n )∩C^1 (\mathbb{R}^n)$ with $∥u_0∥_{n,p}= 1,$ here $α_n = nω ^{\frac{1}{ n−1}}_{n−1}$ and $ω_{n−1}$ is the measure of the unit sphere in $\mathbb{R}^n .$