Positivity versus dead cores

Determine whether the nonnegative weak solutions obtained for the ball-mass equation are strictly positive when 1 < p < 2, or whether they can possess dead cores.

Background

The paper obtains nonnegative solutions in a critical problem and proves strict positivity by a strong maximum principle when p ≥ 2. For 1 < p < 2, the absorption term is sublinear near zero, so the cited strong maximum principle does not apply. This unresolved issue is especially relevant to the Schrödinger–Poisson–Slater applications, where the relevant parameter range has p < 2.

References

If $1 < p < 2$, the absorption term $\Phi_u(|x|)\, u{p-1}$ is sublinear near $u = 0$, the strong maximum principle of Vázquez does not apply, and we do not know whether $u$ is positive.

— A unifying zero-mass equation  (2609.26585 - Perera, 22 Sep 2026) in Remark \ref{rmk:sign}; Section 6, item 5