Singular Type Trudinger–Moser Inequalities with Logarithmic Weights and the Existence of Extremals

In this paper, we study the existence of extremals for the following singular critical Trudinger–Moser inequality with logarithmic weights: \(\begin{aligned} \underset{u\in W_{0,r}^{{\small 1},n}(B,\omega _{\beta }),\left\| u\right\| _{\omega _{\beta }}\le 1}{\sup }\int _{B}\frac{\exp \big ( \alpha _{n,\beta ,\sigma }\left| u\right| ^{\frac{n}{\left( n-1\right) \left( 1-\beta \right) }}\big ) }{\left| x\right| ^{\sigma } }\textrm{d}x<\infty , \end{aligned}\) $$\begin{array}{c}\hfill \underset{u\in W_{0,r}^{1,n}(B,{\omega }_{\beta }),{∥u∥}_{{\omega }_{\beta }}\le 1}{sup}{\int }_B\frac{exp({\alpha }_{n,\beta ,\sigma }{\left|u\right|}^{\frac{n}{\left(n,-,1\right)\left(1,-,\beta \right)}})}{{\left|x\right|}^{\sigma }}\text{d}x<\infty ,\end{array}$$ where B is the unit ball in \(\mathbb {R}^{n}\) ${\mathbb{R}}^n$ , \(\beta \in [0,1)\) $\beta \in [0,1)$ , \(\sigma \in [0,n)\) $\sigma \in [0,n)$ , \(\alpha _{n,\beta ,\sigma }=\left( n-\sigma \right) \big [\omega _{n-1}^{\frac{1}{n-1}}\left( 1-\beta \right) \big ]^{\frac{1}{1-\beta }}\) ${\alpha }_{n,\beta ,\sigma }=\left(n,-,\sigma \right)[{\omega }_{n-1}^{\frac{1}{n-1}}\left(1,-,\beta \right){]}^{\frac{1}{1-\beta }}$ , \(W_{0,r}^{1,n}(B,{\omega _{\beta }})\) $W_{0,r}^{1,n}(B,{\omega }_{\beta })$ denotes the radial weighted Sobolev space with the norm \(\left\| u\right\| _{\omega _{\beta }}=\left( \int _{B}\left| \nabla u\right| ^{n}\omega _{\beta }\left( x\right) \textrm{d}x\right) ^{\frac{1}{n}}\) ${∥u∥}_{{\omega }_{\beta }}={\left({\int }_B,{\left|\mathrm{\nabla },u\right|}^n,{\omega }_{\beta },\left(x\right),\text{d},x\right)}^{\frac{1}{n}}$ , \(\omega _{\beta }(x)=\big (\log \frac{e}{\left| x\right| }\big )^{\beta (n-1)}\) ${\omega }_{\beta }(x)=(log\frac{e}{\left|x\right|}{)}^{\beta (n-1)}$ . Moreover, for \(m>0\) $m>0$ , we establish a singular supercritical Trudinger–Moser inequality with logarithmic weights \(\begin{aligned} \underset{u\in W_{0,r}^{{\small 1},n}(B,\omega _{\beta }),\left\| u\right\| _{\omega _{\beta }}\le 1}{\sup }\int _{B}\frac{\exp \big ( ( \alpha _{n,\beta ,\sigma }+\left| x\right| ^{m}) \left| u\right| ^{\frac{n}{\left( n-1\right) \left( 1-\beta \right) }}\big ) }{\left| x\right| ^{\sigma }}\textrm{d}x<\infty , \end{aligned}\) $$\begin{array}{c}\hfill \underset{u\in W_{0,r}^{1,n}(B,{\omega }_{\beta }),{∥u∥}_{{\omega }_{\beta }}\le 1}{sup}{\int }_B\frac{exp(({\alpha }_{n,\beta ,\sigma }+{\left|x\right|}^m){\left|u\right|}^{\frac{n}{\left(n,-,1\right)\left(1,-,\beta \right)}})}{{\left|x\right|}^{\sigma }}\text{d}x<\infty ,\end{array}$$ and prove the existence of its extremal functions.