Let $d \in \{1,2,3\}$ and $\Omega = B(\mathbf{0},1)$. Consider $$u(\mathbf{x}) = \sum_{k=1}^\infty \frac{1}{2^k} |\mathbf{x}-\mathbf{r}_k|^{-\alpha} \qquad \mathbf{x} \in \Omega, \alpha > 0$$ and $\{\mathbf{r}_k\}$ is a countable dense subset of $\Omega$.
When does $u\in W^{1,p}(\Omega)$?
Can someone help me find the necessary condition on $\alpha, d$ and $p$?
I think I need to show $\|u\|^p_{L^p}$ and $\|\nabla u\|^p_{L^p}$ exist, but I'm unsure on how to proceed, direct calculation of $\|u\|^p_{L^p}$ seems pretty hard or even impossible.
I have already found / proven the condition on $\lambda, d$ and $p$ to have $v(\mathbf{x}) = |\mathbf{x}|^\lambda \in W^{1,p}(\Omega)$.
- $\|v(\mathbf{x})\|_{L^p}$ exists if $\lambda p + d > 0$
- $\|\nabla v(\mathbf{x})\|_{L^p}$ exists if $\lambda p + d > p$
Where the last condition would imply $v \in W^{1,p}(\Omega)$ if $\lambda p + d > p$.