I'm trying to figure out some basic properties of the following problem.
Fix the ball $B_1 \subset \mathbb{R}^2$, and let us look at \begin{cases} \hfill -\Delta u = g\,, \qquad &x\in B_1\,,\\ \hfill u=0\,, \qquad &x\in \partial B_1\,, \end{cases} with $g$ such that $|x|^2 g(x)$ is in $L^2$ (equivalently, $g\in L^2(B_1, |x|^4\mathcal{L}^2)$).
I would expect that a solution $u$ is such that $|x|\nabla u \in L^2$, but this does not seem immediate to prove.
Can you point me to some references (maybe even some more general theory, with weigths $|x|^p$ and in general bounded Lipschitz domains $\Omega$, possibly in $\mathbb{R}^N$)?