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We consider the density $X_t(x)$ of the critical$(\alpha,\beta)$-superprocess in $R^d$ with $\alpha\in (0,2)$ and $\beta<\frac\alpha d$. A recent result from PDE implies a dichotomy for the density: forfixed $x$, $X_t(x)>0$ a.s. on $\{X_t\neq 0\}$ if and only if $\beta \leq\beta^*(\alpha) = \frac{\alpha}{d+\alpha}$. We strengthen this and show that inthe continuous density regime, $\beta < \beta^*(\alpha)$ implies that thedensity function is strictly positive a.s. on $\{X_t\neq 0\}$.  We then give close to sharp conditions on a measure $\mu$ such that$\mu(X_t):=\int X_t(x)\mu(dx)>0$ a.s. on $\{X_t\neq 0 \}$. Our characterizationis based on the size of $supp(\mu)$, in the sense of Hausdorff measure anddimension. For $s \in [0,d]$, if $\beta \leq\beta^*(\alpha,s)=\frac{\alpha}{d-s+\alpha}$ and $supp(\mu)$ has positive$x^s$-Hausdorff measure, then $\mu(X_t)>0$ a.s. on $\{X_t\neq 0\}$; and when$\beta > \beta^*(\alpha,s)$, if $\mu$ satisfies a uniform lower densitycondition which implies $dim(supp(\mu)) < s$, then $P(\mu(X_t)=0|X_t\neq 0)>0$.  We also give new result for the fractional PDE $\partial_t u(t,x) =-(-\Delta)^{\alpha/2}u(t,x)-u(t,x)^{1+\beta}$ with domain $(t,x)\in(0,\infty)\times R^d$. The initial trace of a solution $u_t(x)$ is a pair$(S,\nu)$, where the singular set $S$ is a closed set around which localintegrals of $u_t(x)$ diverge as $t \to 0$, and $\nu$ is a Radon measure whichgives the limiting behaviour of $u_t(x)$ on $S^c$ as $t\to 0$. We characterizethe existence of solutions with initial trace $(S,0)$ in terms of a parametercalled the saturation dimension, $d_{sat}=d+\alpha(1-\beta^{-1})$. For $S\neqR^d$ with $dim(S)> d_{sat}$ (and in some cases $dim(S)=d_{sat}$) we prove thatno such solution exists. When $dim(S)<d_{sat}$ and $S$ is the compact supportof a measure satisfying a uniform lower density condition, we prove that asolution exists.

The density of the $(\alpha,\beta)$-superprocess and singular solutions to a fractional non-linear PDE