On the definition of the solution to a semilinear elliptic problem with a strong singularity at u=0

In this paper we present new results related to the ones obtained in our previous papers on the singular semilinear elliptic problem u ≥ 0 in Ω, −div A(x)Du = F(x, u) in Ω, u = 0 on ∂Ω, where F(x, s) is a Carathéodory function which can take the value +∞ when s = 0. Three new topics are investigated. First, we present definitions which we prove to be equivalent to the definition given in our paper Giachetti, Martínez-Aparicio, Murat (2018). Second, we study the set {x ∈ Ω : u(x) = 0}, which is the set where the right-hand side of the equation could be singular in Ω, and we prove that actually, at almost every point x of this set, the right-hand side is non singular since one has F(x, 0) = 0. Third, we consider the case where a zeroth order term μu, with μ a nonnegative bounded Radon measure which also belongs to H−1(Ω), is added to the left-hand side of the singular problem considered above. We explain how the definition of solution given in Giachetti, Martínez-Aparicio, Murat (2018) has to be modified in such a case, and we explicitly give the a priori estimates that every such solution satisfies (these estimates are at the basis of our existence, stability and uniqueness results). Finally we give two counterexamples which prove that when a zeroth order term μu of the above type is added to the left-hand side of the problem, the strong maximum principle in general does not hold anymore.