Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^N$ where $N\geq 2$ and let $f\in L^{\infty}(\Omega)$.

Then does there exist $w\in A_p$ (the class of Muckenhoupt weights) such that no solutions $u$ in the weighted Sobolev space $W_{0}^{1,p}(\Omega,w)$ of the equation $$ -\text{div}(w(x)|\nabla u|^{p-2}\nabla u)=f\,\text{in}\,\Omega, $$ is continuous?

As far as I know, in case of $p=2$ and $f=0$, if one choose $w\in A_2$, then all the solutions are continuous follows from the result of Fabes-Serapioni.

Therefore, it is better to choose $f\neq 0$ and construct such $w$. Even an example for $p=2$ case will be enough.

If possible, kindly help me.

Thanks in advance.


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