Consider the $d$-dimensional Coulomb "kernel" defined by: \begin{equation} x \in \mathbb{R}^{d} \mapsto g(x):=\left\{\begin{array}{ll} \log \frac{1}{|x|} & \text { if } d=2 \\ \frac{1}{|x|^{d-2}} & \text { if } d \geq 3 \end{array}\right. \end{equation} It is known that $g$ satisfies, in the sense of Swchartz distributions, the Poisson equation: \begin{equation} \Delta g = - c_d \delta_{0} \end{equation} where $c_d$ is some constant, $\delta_{0}$ the dirac mass at the origin and $\Delta$ the Laplace operator.

Sometimes the Coulomb kernel is "softened" because it has a singularity and one considers the "Plummer kernel" defined by: \begin{equation} p_{\epsilon}(x) = \frac{1}{\left(\sqrt{|x|^{2}+\epsilon^{2}}\right)^{d-2}} \end{equation} for $\epsilon >0$.

My question is: does this kernel also satisfy some Poisson like equation ? More generally are there some known PDE for which this kernel is the solution ?