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

A semilinear elliptic equation with a mild singularity at u= 0: Existence and homogenization

“…But when a 0 ≥ 0 does not belong to L ∞ (Ω) and is only a nonnegative element of H −1 (Ω) (this can be the case in the result of the homogenization process with many small holes that we will perform in Section 5), the strong maximum principle does not hold anymore for the operator −div A(x)Du + + a 0 u (see [9] for a counter-example due to G. Dal Maso), and therefore (3.14) does not hold anymore for such an operator. Conversely, if F (x, 0) ≡ 0, u ≡ 0 is not a solution to problem (1.1) in the sense of Definition 3.1, and Proposition 3.5 (or more exactly (3.14)) then implies that u(x) > 0 a.e.…”

“…Our method allows us to obtain results of existence, stability, uniqueness and homogenization, even if the strong maximum principle does not hold true in general in such a context (see [9] and [10]). …”

“…We consider the case γ > 1 (and more generally the case of a general singularity) in the papers [9] and [10] (see also [11]). Let us point out that in the latest case, no global energy estimate is available for the solutions when the singularity has a strong behaviour.…”

A Singular Semilinear Elliptic Equation with a Variable Exponent

“…For existence and homogenization results for this kind of problems, we refer to the papers [6][7][8].…”

“…They proved some existence, stability and homogenization results without assuming, for a more general nonlinearity F(x, s), that it is nonincreasing in the s variable and without using the strong maximum principle in the proofs of their results. They studied the case γ > in the papers [7] and [8], where the singularity has a stronger behavior and no global energy estimates are available for the solutions. This makes the problem harder, in particular from the point of view of homogenization.…”