Formulation:
Let $v\in L^1_\text{loc}(\mathbb{R}^3)$ and $f \in H^1(\mathbb{R}^3)$ such that \begin{equation} \int f^2 v_+ = \int f^2 v_- = +\infty. \end{equation} Here, $v_- = \max(0,-f)$, $v_+ = \max(0,f)$, i.e., the negative and positive parts of $v=v_+ - v_-$, respectively.
Question: Does $g\in H^1(\mathbb{R}^3)$ exist, such that \begin{equation} \int g^2 v_+ < \infty, \quad \int g^2 v_- = +\infty \quad ? \end{equation}
Some thoughts:
Let $S_\pm$ be the supports of $v_\pm$, respectively.
One can easily find $g\in L^2$ such that the last equation holds, simply multiply $f$ with the characteristic function of $S_-$. The intuitive approach is then by some smoothing of this function by a mollifier, or using a bump function to force the support of $g$ away from $S_+$.
However, the supports of $S_\pm$ can be quite complicated: for example, fat Cantor-like sets. Thus, a bump function technique or a mollifier may "accidentally" fill out any of $S_\pm$.
My motivation:
The problem comes from my original research on the mathematical foundations of Density Functional Theory (DFT) in physics and chemistry. Here, $f^2$ is proportional to the probability density of finding an electron at a space point, and $v$ is the potential energy field of the environmentn. $\int f^2 v$ is the total potential energy for the system's state. The original $f$ gives a meaningless "$\infty-\infty$" result, but for certain reasons, we are out of the woods if there is some $other$ density $g$ with the prescribed property.
Edit:
Removed claim that $S_\pm$ must be unbounded. This does not follow from the stated assumptions.