Ricerc@Sapienza

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.