Advances in the Study of Singular Semilinear Elliptic Problems

Abstract

In this paper we deal with some results concerning semilinear elliptic singular problems with Dirichlet boundary conditions. The problem becomes singular where the solution u vanishes. The model of this kind of problems is

$$\displaystyle\begin{array}{rcl} \left \{\begin{array}{@{}l@{\quad }l@{}} u \geq 0 \quad &\mbox{ in }\varOmega, \\ -div\,A(x)Du = F(x,u)\quad &\mbox{ in}\;\varOmega, \\ u = 0 \quad &\mbox{ on}\;\partial \varOmega,\\ \quad \end{array} \right.& & {}\\ \end{array}$$

where Ω is a bounded open set of $\mathbb{R}^{N}$, N ≥ 1, A is a coercive matrix with coefficients in $L^{\infty }(\varOmega )$ and $F: (x,s) \in \varOmega \times [0,+\infty [\rightarrow F(x,s) \in [0,+\infty ]$ is a Carathéodory function which is singular at s = 0. Our aim is to study the meaning of the assumptions made on the singular function F(x, s) in the papers [Giachetti et al., J. Math. Pures Appl. (2016, in press); Giachetti et al., Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0 (Preprint, 2016); Giachetti et al., Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes (Preprint, 2016)], to extend some uniqueness results of the solution given in the same papers, and to prove the $L^{\infty }$-regularity of the solutions under some regularity assumption on the data.