Let $\Omega\subset\mathbb{R}^N$ be a bounded and smooth domain with $N\geq 2$. Let us consider the following class of weights $$ B_s=\{w\in W_p: w^{-s}\in L^1(\Omega)\text{ for some }s\in Y\},\quad 1<p<\infty, $$ where $Y=[\frac{1}{p-1},\infty)\cap(\frac{N}{p},\infty)$. Here $W_p$ is the class of $p$-admissible weights. I want to know, if there is some function in $B_s$ for some $s\in I$ and $1<p<\infty$ such that $w\in B_s\setminus A_p$, where $A_p$ denotes the class of Muckenhoupt weights and $w$ may vanish or blow up in $\Omega$.
I tried $w(x)=|x|^{\alpha}$ type function, then I only found that $w(x)=|x|^{\alpha}\in B_s$ for $s\in Y$, for $-N<\alpha<\frac{N}{s}$. But this function also belong to $A_p$, since $|x|^{\alpha}\in A_p$ iff $-N<\alpha<N(p-1)$ and therefore, I am unable to consturct any function of type $|x|^{\alpha}$ which is in $w\in B_s\setminus A_p$.
Other examples include, for example, $w(x)=J_f(x)^{1-\frac{p}{N}}$ for $1<p<N$ belong to the $p$-admissible class $W_p$. Here, $f:\mathbb{R}^N\to\mathbb{R}^N$ is a $K$-quasiconformal mapping. But I am unable to find an example of $w\in B_s\setminus A_p$, which may vanish or blow up in $\Omega$, for example $|x|^{\alpha}$ has such property.
Can somebody please help with such an example? Thank you very much.