# Solution to weighted $p$-Laplace equation

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.