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

In the present paper we make some remarks on the definition of the solution that we introduced in our paper [GMM2] for the semilinear elliptic problem ---------- u ≥ 0 in Ω, ---------- −divA(x)Du = F(x,u) in Ω, ---------- u = 0 on ∂Ω, ---------- when F(x,s) is a Carathéodory function such that 0 ≤ F(x,s) ≤ h(x)/Γ(s) a.e. x ∈ Ω for every s > 0, with h in some L^r(Ω) and Γ a C^1([0, +∞[) function such that Γ(0) = 0 and Γ'(s) > 0 for every s > 0. In [GMM2] we indeed introduced a new notion of solution to this problem in the spirit of the solutions defined by transposition. This definition allowed us to prove the existence and the stability of this solution, as well as its uniqueness when F(x,s) is further assumed to be nonincreasing in s. ---------- The goal of the present paper, which is a companion to [GMM2], is to make some remarks on this definition in three directions. ---------- The first part of the present paper is concerned with equivalences of the definition given in [GMM2] with other definitions. In this direction we first prove that in the case where the singularity of F(x,s) is "mild", namely when F(x,s) satisfies 0 ≤ F(x,s) ≤ h(x) (1 + (1/s)) (a case that we studied in [GMM1), the definition of the solution given in [GMM2] is equivalent to the definition of the solution that we used in [GMM1], which is a "natural" and rather usual definition. We then prove that the definition that we gave in [GMM2], which involves some requirements "for every k", is equivalent to a similar definition which involves these requirements only "for a fixed k_0" where k_0 can be arbitrarily chosen. ---------- The second part of the present paper is concerned with the study of the set where the solution u of the above semilinear problem is equal to zero. ---------- The third part of the present paper is concerned with the adaptations of the definition given in [GMM2] which are needed in the case where the operator −divA(x)Du which constitutes the left-hand side of the above semilinear equation is replaced by the operator −divA(x)Du + µu, where µ is a bounded Radon measure which also belongs to H^{-1}(Ω). Such a "strange term" µu appears in particular when one performs the homogenization of the above semilinear problem with homogeneous Dirichlet boundary condition on the whole of the boundary for a sequence of open sets Ω^ε obtained by removing many small holes from a fixed open set Ω (see [GMM1] and [GMM3] for the study of this homogenization problem). Note that the replacement of the operator −divA(x)Du by the operator −divA(x)Du + µu in the above semilinear problem, which looks innocuous, actually is not innocuous at all, since in general the strong maximum principle does not hold any more for the latest operator. ---------- ----------[GMM1] D. Giachetti, P.J. Martinez-Aparicio and F. Murat,"A semilinear elliptic equation with a mild singularity at u = 0: existence and homogenization". J. Math. Pures Appl. 107 (2017), 41-77. ----------[GMM2] D. Giachetti, P.J. Martinez-Aparicio and F. Murat, "Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0".Ann. Sc. Norm. Sup. Pisa, accepted for publication. ----------[GMM3] D. Giachetti, P.J. Martinez-Aparicio and F. Murat,"Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes". In preparation. ---------