Let $\Omega \subset \mathbb{R}^3$ be a Lipschitz bounded domain and $x_0 \in \Omega$. Recall the definitions of some weighted Sobolev spaces:
\begin{align} &H^1(\omega; \Omega) := \{ v \in L^2(\omega; \Omega) : |\nabla v| \in L^2(\omega; \Omega) \} \\ &L^2(\omega; \Omega) := \{ v \in L^1_{loc}(\Omega) : \int_{\Omega} v^2 \omega < +\infty \} \\ &H^1_{0}(\omega; \Omega) := \overline{\mathcal{D}(\Omega)}^{\|\cdot\|_{1,\omega}}. \end{align}
If we choose $\alpha \in (1,3)$ and set
$$ \omega(x) := |x-x_0|^{\alpha}, $$ the space $H^1(\omega; \Omega)$ turns out to be a Hilbert space (endowed with a suitable inner product), embedded in $L^2(\Omega)$.
Consider now the following elliptic b.v.p with a Dirac source:
\begin{align} -\Delta u &= \delta_{x_0} \quad \text{in } \Omega, \\ u &= 0 \qquad \text{on } \partial \Omega \end{align} and the corresponding weak formulation: $$ u \in H^1_{0}(\omega; \Omega) : \ \ \int_{\Omega} \nabla u \cdot \nabla w = \langle \delta_{x_0} , w \rangle \qquad \forall w \in H^1_0(\omega^{-1}; \Omega). \quad (**)$$
Known results on weighted Sboloev spaces guarantee that $\delta_{x_0} \in (H^1_0(\omega^{-1}; \Omega))'$ and that $(**)$ is well-posed.
However, we can also attack the very first problem by means of a fundamental solution, taking care of the boundary condition:
$$ u = \Phi_{x_0} + \widetilde{u}, $$ where $\Phi_{x_0} = \frac{1}{4 \pi |x-x_0|}$ and $\widetilde{u}$ is the solution of
\begin{align} -\Delta \widetilde{u} &= \widetilde{f} \quad \text{in } \Omega, \\ \widetilde{u} &= - \Phi_{x_0} \qquad \text{on } \partial \Omega, \end{align} where
$$ \widetilde{f}= \begin{cases} 0 \qquad \text{in } \ B_r(x_0) \\ \Delta \Phi_{x_0} \qquad \text{in } \ \Omega \setminus B_r(x_0) \end{cases} $$ for some radius $r>0$ such that the ball is contained in $\Omega$.
I was wondering if we could prove that these two formulations are equivalent, or at least that $\Phi_{x_0} + \widetilde{u}$ is a actually a solution to $(**)$. Modulo the details, I think it is easy to prove that $\Phi_{x_0} + \widetilde{u} \in H^1_0(\omega; \Omega)$; checking the identity seems more delicate because a low-regularity lifting of $-\Phi_{x_0}$ arises, and it is not clear how $\Phi$ acts on functions in $H^1_0(\omega^{-1})$... Roughly speaking, we have:
$$ \int_{\Omega} (\nabla \Phi_{x_0} + \nabla \widetilde{u}) \cdot \nabla w = \int_{\Omega} \nabla \Phi_{x_0} \cdot \nabla w + \int_{\Omega} \nabla \widetilde{u} \cdot \nabla w ; $$ the first integral should give us $\langle \delta_{x_0} , w \rangle $, while the second should vanish. Does anybody have an idea to make this rigorous?