Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes

In this paper we consider the homogenization problem for quasilinear elliptic equations with singularities in the gradient, whose model is the following \begin{document}$\begin{equation*}\begin{cases}\displaystyle -Δ u^\varepsilon + \frac{|\nabla u^\varepsilon|^2}{{(u^\varepsilon})^θ} = f (x)& \mbox{in} \; Ω^\varepsilon,\\u^\varepsilon = 0&\mbox{on} \; \partial Ω^\varepsilon,\\\end{cases}\end{equation*}$ \end{document} where Ω is an open bounded set of \begin{document} $\mathbb{R}^N$ \end{document} , \begin{document} $θ ∈ (0,1)$ \end{document} and \begin{document} $f$ \end{document} is positive function that belongs to a certain Lebesgue's space. The homogenization of these equations is posed in a sequence of domains \begin{document} $Ω^\varepsilon$ \end{document} obtained by removing many small holes from a fixed domain Ω. We also give a corrector result.

1. Introduction. We study a homogenization problem for a singular quasilinear elliptic problem with quadratic gradient, specifically where Ω ε is a sequence of open sets which are included in a fixed bounded open set Ω of R N , N ≥ 3, g ∈ C(0, +∞) ∩ L 1 (0, 1) is a positive function and f ∈ L 2N N +2 (Ω), We study the asymptotic behaviour, as ε goes to zero, of a sequence of problems posed in domains Ω ε obtained by removing many small holes from a fixed domain Ω, in the framework of [11] for the linear case. In such paper it has been shown (see also [21] or [12] for a more general framework) that for every f ∈ L 2 (Ω), the (unique) solution u ε of satisfies that, withũ ε denoting the extension of u ε by zero in Ω \ Ω ε ,ũ ε u in H 1 0 (Ω), where u is the (unique) solution of    u ∈ H 1 0 (Ω) ∩ L 2 (Ω; dµ), with µ is a nonnegative finite Radon measure depending only on the holes. In [11] there is an example of holes for which µ is a positive constant and u satisfies It is widely remarked the presence of the "strange term" µu (which is the asymptotic memory of the fact thatũ ε was zero on the holes) appearing in the limit equation (3).
In [12] the authors deal with the homogenization problem in a general framework. In that paper it is not essential to have an a priori control of the set where the weak limit ofũ ε vanishes. However, since we are dealing with a singularity at u = 0 we adopt the framework of [11] as in references [16,17,18] in which is studied the existence of solution and homogenization of the problem where A ∈ L ∞ (Ω) N ×N is a coercive matrix and γ > 0. See also [7] which deals with homogenization of this problem in varying matrices. In [10] was studied problem (1) in the case g(s) = −γ where γ is a real constant. The author used a suitable change of unknown function, z ε = e γu ε − 1, and he obtains a new problem −∆z ε = f γe γu ε in Ω ε , z ε = 0 on ∂Ω ε .
A careful analysis of this semilinear homogenization problem allows the author to pass to the limit as in the linear case. Undoing the change of variable he proved that, as in the linear case, a new term appears in the equation that satisfies u, but in this case the new term (e γu − 1)µ/(γe γu ) is nonlinear (µ is the same measure). Specifically the homogenized problem is or equivalently, in the case where µ is a Radon measure, the solution u satisfies for every ϕ ∈ H 1 0 (Ω) ∩ L 2 (Ω; dµ), ϕ ≥ 0. The difference is that the "new equation" is no more linear. As the author remarked, this means that the perturbation of the linear problem (2) by a nonlinear term, namely γ|∇u ε | 2 , changes the structure of the new term in the limit equation. Moreover, a corrector result was proved, is that to say, a representation of ∇u ε in the strong topology of L 2 (Ω) N . Similar results were proved in [9] in which the nonlinear perturbation of (2) is a general function of the form H(x, u, ∇u), where H has a (at most) natural growth in the gradient.
We remark that in all the cases the lower order term is bounded respect to u and, up to our knowledge, the singular problem (1) has not been considered yet. In [15] there a is first homogenization result for a singular quasilinear equation (but with the nonlinearity on the right-hand side) for a fixed domain with oscillating coefficients.
In the present paper, inspired by [10], we consider singular functions g, using the results in [1,4,6] we have the existence of solution u ε ∈ H 1 0 (Ω ε ), in the sense of distributions, to problem (1) and in [5] it has been proved the uniqueness. In [19] the authors prove the existence of solution for f bounded and a general lower order term.
Observe that g(s) ≥ 0 for every s > 0, thus it is easy to prove that u ε is bounded in H 1 0 (Ω ε ) and in L ∞ (Ω ε ) following [20] and [22] respectively. Moreover we can prove thatũ ε is a bounded sequence in H 1 0 (Ω) ∩ L ∞ (Ω). Therefore, up to a subsequence, we get thatũ ε converges to some u weakly in H 1 0 (Ω). The general questions we are concerned with are the following. Do the solutions u ε converge to a limit u when the parameter ε tends to zero? If this limit exists, can it be characterized? Will the result be the same result as in the non singular case? In principle the answer is not obvious at all since, as ε tends to zero, the number of holes becomes greater and greater and the singular set for the right-hand side (which includes at least the holes' boundary) tends to "invade" the entire Ω.
In our case, the function g may presents a singularity at u = 0. Our main result is to prove that for every in Ω, in the case µ constant, where G(s) = s 1 g(t)dt and Ψ(s) = s 0 e −G(t) dt for every s > 0, or of the problem, in the case where µ is a Radon measure, for every ϕ ∈ H 1 0 (Ω) ∩ L 2 (Ω; dµ), ϕ ≥ 0. In [8] the authors prove the existence and uniqueness of solution of the previous problem in the case where µ is a constant and the proof of the uniqueness can be adapted in the case where µ is a Radon measure.
Observe that we can not guarantee that the solution of the previous problem satisfies u > 0 in Ω. The proof of this fact is usually based on a change of variables to obtain a semilinear problem where the strong maximum principle is satisfied. In our case, this procedure leads to a problem of the form in Ω, and in [18] the authors give an explicit counterexample which shows that in general the strong maximum principle fails in the case where the operator involves a zerothorder term µz when µ ∈ M b (Ω), µ ≥ 0. Therefore, we do not impose 0 < u ε in the notion of solution of the singular problem (1) and we proceed here with the notion of solution in Definition 2.1. Although we prove that both concepts are equivalents for g integrable at zero we think that Definition 2.1 is more appropriate if g ∈ L 1 (0, 1). In addition, the techniques used to prove existence of solution in this sense are important in order to deal with the homogenization of the singular problem (1). The plan of the paper is the following. We firstly prove that the problem has solution in a suitable sense that we will detail in Section 2. In Section 3 we give the precise assumptions of the perforated domains, following the framework of [11], and we prove our homogenization result for the singular quasilinear problem (1). The last part of Section 3 is devoted to prove the corrector result. Let us explicitly state that we have chosen to present the results and to perform the proofs in the case N ≥ 3. However, all the results hold true also in the case N = 2 provided we replace the assumption f ∈ L 2N N +2 (Ω) with f ∈ L p (Ω) for some p > 1.
Notation. As usual, we consider the positive and negative part functions defined on R by s + = max{s, 0} and s − = min{s, 0}, respectively.
For any 1 < p < N , p * = N p N −p is the Sobolev conjugate exponent of p. As usual, S denotes the best Sobolev constant, i.e.,

S = sup
We denote by D(Ω) the space of the functions C ∞ (Ω) whose support is compact and included on Ω, by D (Ω) the space of distributions on Ω and M b (Ω) denotes the space of the finite Radon measures.

Framework of the quasilinear problem. Let Ω be an open and bounded
We consider the boundary value problem (4) with g a continuous function in (0, +∞).
u is a sub-solution of (4)) then u satisfies this inequality for every 0 ≤ ϕ ∈ H 1 0 (Ω). Remark 2. Observe that in [6] it is proved the existence of u ∈ H 1 0 (Ω) such that u(x) > 0 for a.e. x ∈ Ω, g(u)|∇u| 2 ∈ L 1 (Ω) and In particular, u is solution in the sense of Definition 2.1. (See also [1] where a stronger hypothesis on f allows to consider more general functions g).
The next result is a direct consequence of the Stampacchia method [22], we include here only a sketch of the proof. Lemma 2.2. There exists a positive constant C f,Ω such that for every g ≥ 0 and every sub-solution u ∈ H 1 0 (Ω) of (4) we have that u H 1 0 (Ω) ≤ C f,Ω . If, in addition, f ∈ L q (Ω) for some q > N/2 then u ∈ L ∞ (Ω) and Proof. Taking u as test function (see Remark 1) and neglecting the positive lower order term we have that Moreover, if f ∈ L q (Ω) for some q > N/2, neglecting again the lower order term, the standard Stampaccchia method gives us the existence of C > 0, depending only on f and Ω, such that This concludes the proof.
Remark 3. Observe that if Ω 1 ⊂ Ω 2 are open and bounded and f ∈ L q (Ω 2 ) for some q > N/2 then we can take C f,Ω1 ≤ C f,Ω2 .

Remark 4.
In the present paper we do not consider the case in which g can be not integrable at zero. However, we prove in (i) of the previous Lemma a result that improves a hypothesis used in [1], that is that we do not impose here f to be greater than a positive constant in compact subsets of Ω.
First we observe that, for 0 < δ < 1, we can take ϕ = e −G(S δ (u)) φ as test function in (4) and we obtain that Multiplying by e G(δ) we deduce Now we pass to the limit as δ tends to zero. Since G is increasing and δ ≤ S δ (u) we have that e G(δ)−G(S δ (u)) ≤ 1, in particular, using Lebesgue theorem Analogously, passing to the limit as δ tends to zero Summarizing we obtain that In the case of item (1) we have that G(0) = −∞ since g ∈ L 1 (0, 1). In this case, from (6) we deduce that which implies that either u(x) > 0 for a.e.
x ∈ Ω or f (x) = 0 for a.e. x ∈ {u = 0}, which conclude the proof in this case.
Remark 5. Taking into account Lemma 2.3 we have sufficient conditions to have (5) satisfied. This is the key point to prove the uniqueness result in [5] which, under these conditions, it is also true for solutions in the sense of Definition 2.1. Using then Remark 2 we have that both concepts of solution are equivalent in the cases where uniqueness of solution holds. This is the case when g is integrable at zero (in [3] it is proved a uniqueness result, in the case g(s) = c/s with c < 1 if ∂Ω is smooth, whose proof cannot be adapted for solutions in the sense of Definition 2.1).
We include now the proof of the existence of solution in the sense of Definition 2.1.
Taking limits as n → ∞ and then as δ → 0 we obtain, using (8)  This can be proved as in [1] writing, for every k > 0, G δ (u n ) = T k (G δ (u n )) + G k (G δ (u n )) and taking into account that, fixed k, δ, T k (G δ (u n )) strongly converges to T k (G δ (u)) in H 1 0 (Ω) and ∇G k (G δ (u n )) L 2 (Ω) N tends to 0 uniformly in n as k → ∞. Therefore, as n → ∞, and taking limit as δ → 0 and u is a solution of (4).

Homogenization for the problem (1).
3.1. The perforated domains. In this Section, we describe the geometry of the domains, following [11], in which we study our homogenization result.
Let Ω be an open and bounded set of R N (N ≥ 2). Consider for every ε, where ε takes its values in a sequence of positive numbers which tends to zero, some closed subsets T ε i of R N , 1 ≤ i ≤ n(ε), which are the holes. The domain Ω ε is defined by removing the holes T ε i from Ω, that is Hypotheses on the holes. We suppose that the sequence of domains Ω ε is such that there exist a sequence of functions w ε , a distribution µ ∈ D (Ω) and two sequences of distributions µ ε ∈ D (Ω) and λ ε ∈ D (Ω) such that 0 ≤ w ε ≤ 1 a.e. x ∈ Ω, w ε 1 in H 1 (Ω) weakly, in L ∞ (Ω) weakly-star and a.e. in Ω, µ ∈ H −1 (Ω), The meaning of assumption (11) is that while the meaning of the last statement of (14) is that the distribution λ ε only acts on the holes T ε i , i = 1, . . . , n(ε), since taking z ε ∈ D(Ω ε ) implies that −∆w ε = µ ε in D (Ω ε ).
In order to deal with the main result in the case where no estimate in L ∞ (Ω ε ) is known we need to impose a convenient behavior of g at infinity in the sense of the following definition.
Thus we can take e G(H δ (u))−G(u ε ) ϕ ε,n as test function in (16) and we get Now, using that g ∈ L 1 (0, +∞) and it is bounded at infinity, we can pass to the limit in n obtaining the desired result.
Observe that functions e −G (Sε(u1)) and e −G(Sε(u2)) are bounded and thus we can pass to the limit as ε goes to zero and we obtain that Thus (Ψ(u 1 )−Ψ(u 2 )) + = 0 and consequently u 1 ≤ u 2 (since ψ is strictly increasing). Interchanging u 1 and u 2 we get the reverse inequality.

Corrector result.
In order to prove that the solution given by Theorem 3.2 is strictly positive we assume that the measure µ is such that − ∆w + µw verifies the strong maximum principle.
Theorem 3.3. Assume that hypotheses of Theorem 3.2 are satisfied. Suppose also that µ satisfies (22). Thenũ with r ε → 0 strongly in H 1 0 (Ω). Remark 7. Observe that, due to the presence of Ψ −1 and Ψ, this is not a standard corrector result. The change arises from the nonlinear nature of the lower order term.