Strong solutions for singular Dirichlet elliptic problems

. We prove an existence result for strong solutions u ∈ W 2, q ( Ω ) of singular semilinear elliptic problems of the form − ∆ u = g ( · , u ) in Ω , u = τ on ∂ Ω , where 1 < q < ∞ , Ω is a bounded domain in R n with C 2 boundary, 0 ≤ τ ∈ W 2 − 1 q , q ( ∂ Ω ) , and with g : Ω × ( 0, ∞ ) → [ 0, ∞ ) belonging to a class of nonnegative Carathéodory functions, which may be singular at s = 0 and also at x ∈ S for some suitable subsets S ⊂ Ω . In addition, we give results concerning the uniqueness and regularity of the solutions. A related problem on punctured domains is also considered.


Introduction and statement of the main results
Our aim in this paper is to state existence and uniqueness results for strong solutions u ∈ W 2,q (Ω) of singular elliptic problems of the form where 1 < q < ∞, Ω is a bounded domain in R n with C 2 boundary, 0 ≤ τ ∈ W 2− 1 q ,q (∂Ω) , with the boundary condition understood in the sense of the trace, and where g : Ω × (0, ∞) → [0, ∞) is a suitable nonnegative Carathéodory function which may be singular at s = 0 and at x ∈ S for some suitable subsets S ⊂ Ω.
Singular elliptic problems appear in the study of nonlinear phenomena such as non-Newtonian fluids, the temperature of some electrical conductors, thin films, micro electromechanicals devices, and chemical catalysts process, (see e.g., [6,15,19,20,28] and the references therein).
In [30] it was studied the existence, uniqueness, and regularity properties of the weak solutions of problems of the form − div (A (x) ∇u) = f (x) u γ + µ in Ω, u > 0 in Ω, u = 0 on ∂Ω, in the case when A (x) is a uniformly elliptic and bounded matrix, γ > 0, 0 ≤ f ∈ L 1 (Ω) in Ω, and µ is a nonnegative bounded Radon measure.
Existence and nonexistence of solutions of problems of the form − div (A (x) ∇u) = f u −γ in Ω, u > 0 in Ω, u = 0 on ∂Ω, was studied in [4], in the case where A is a bounded elliptic matrix and f is, either a nonnegative function in a suitable L p (Ω) or a nonnegative and bounded Radon measure. The existence and uniqueness of solutions of problem of the form − div (A (x) ∇u) = H (u) µ in Ω, u > 0 in Ω, u = 0 on ∂Ω, was studied in [31] in the case when µ a bounded Radon measure, A (x) is a uniformly elliptic and bounded matrix with Lipschitz continuous coefficients, and H : (0, ∞) → (0, ∞) satisfies some suitable conditions which allow that lim s→0 + H (s) = ∞.
In [18] it was proved, via a comparison principle, the uniqueness of the weak solutions of problems of the form −∆ p u = F (·, u) in Ω, u > 0 in Ω, u = 0 on ∂Ω, in the case when F is a nonnegative Carathéodory function on Ω × (0, ∞) such that s → s 1−p F (x, s) is decreasing on (0, ∞) for a.e. x ∈ Ω. In addition, again in [18], it was proved the existence of weak solutions of problems of the form −∆ p u = f u −γ + gu q in Ω, u > 0 in Ω, u = 0 on ∂Ω, in the case when γ ≥ 0, 0 ≤ q ≤ p − 1; f and g are nonnegative functions belonging to suitable Lebesgue spaces.
Singular problems on punctured domains were studied in [3]. There it was proved that, if x 0 ∈ Ω and if a : Ω → R satisfies certain condition related to the Karamata class, then the The interested reader will find an updated account, concerning the topic of singular elliptic problems, as well as additional references, in the research books [36], and [21].
We assume, from now on, that n ≥ 2 and that Ω is a bounded domain in R n with C 2 boundary. Let q ∈ (1, ∞) , which we fix from now on. We recall that (see, e.g., [25, The-orem 2.4.2.5]), for f ∈ L q (Ω) and τ ∈ W 2− 1 q ,q (∂Ω) , there exists a unique strong solution u ∈ W 2,q (Ω) of the problem −∆u = f in Ω, with the boundary condition understood in the sense of the trace, and that u satisfies , where c is a positive constant independent of u.
Our third result refers to the punctured domain U := Ω \ {x 0 } , where x 0 ∈ Ω, and reads as follows: Let h : Ω × (0, ∞) → R and let w ∈ W 2,q (U) . Assume that w is a strong solution of the problem (with the boundary condition understood in the sense of the trace). If either w ∈ C (Ω) or lim sup δ→0 + 1 δ 2 A δ |w| = 0, then w ∈ W 2,q (Ω) and w is a strong solution of the problem We have also the following: Theorem 1.5. Assume the hypothesis of Theorem 1.3. Let x 0 ∈ Ω, U := Ω \ {x 0 } , and let w ∈ W 2,q (U) . If w is a strong solution of the problem Then: i) If lim sup x→x 0 |x − x 0 | n−2 w (x) = 0 then, after redefining w in a set with zero measure, it hold that w ∈ W 2,q (Ω) ∩ C Ω ∩ C 1 (Ω) and w is the unique solution of problem (1.1) The paper is organized as follows: in Section 2 we study, for M ≥ 1 and ε ∈ (0, 1] , the approximated problems where g M (x, s) := min {M, g (x, s)}. By using Schauder's fixed point theorem, we show that this problem has a unique solution u M,ε ∈ ∩ 1<p<∞ W 2,p (Ω) (see Lemmas 2.2 and 2.4). Lemma 2.6 states that ε → u M,ε is nonincreasing, M → u M,ε is nondecreasing, and that in Ω, with c 0 a positive constant independent of M and ε, and where τ * is the strong solution of (1.4). Lemma 2.7 shows that if u M := lim ε→0 + u M,ε , then u M ∈ W 2,q (Ω) and u M is a strong solution of the problem The main results are proved in Section 3. To prove Theorem 1.1 we define u := lim M→∞ u M and we show that u is a strong solution of problem (1.1) with the desired properties. This is achieved from thanks to Lemma 2.7 by showing that g (·, u) := lim M→∞ g M (·, u M ) with convergence in L q (Ω). To prove Theorem 1.3 we show that, for any strong solution u of problem which will give the continuity of u at ∂Ω, next we show, by a suitable bootstrap argument, that u ∈ W 2,n loc (Ω) , which gives that u ∈ C 1 (Ω) . Proved that u ∈ W 2,n loc (Ω) ∩ C Ω , the uniqueness assertion of Theorem 1.3 will follow from the fact that s → g (x, s) is nonincreasing, combined with the application of an appropriate maximum principle. Finally, Theorem 1.4 is proved by showing that, if w ∈ W 2,q (Ω \ {x 0 }) satisfies the conditions of Theorem 1.4, then w, viewed as a distribution on Ω, belongs to W 2,q (Ω) .

Preliminaries
Let g : Ω × (0, ∞) → R be a function satisfying the conditions H1)-H3) of Theorem 1.1 and, where τ * is the strong solution of problem (1.4), and let For v ∈ C M , since g is a Carathéodory function, g M (·, ε + v) is a measurable function. Let η be a positive and small enough number such that ηd Ω ≤ ε in Ω. Then, since g is nonincreasing in the second variable and v ≥ 0 in Ω,

Remark 2.1.
i) Let us recall the following form of the Aleksandrov maximum principle (which is a particular case of [23], Indeed, let f : R n → R be the extension by zero of f . Then 0 ≤ f ∈ L q (R n ) and so f can be approximated, in the L q (R n ) norm, by a sequence f j j∈N ⊂ C ∞ (R n ) obtained by convolving f with suitable mollifiers (see [33,Proposition 1.1.3]). Thus, for each j, 0 ≤ f j|Ω ∈ L ∞ (Ω) , and so the solution u j of the problem −∆u j = f j|Ω in Ω, follows that u j j∈N converges to u in W 2,q (Ω). Now, by i), u j ≥ 0 in Ω, and then u ≥ 0 in Ω.
iii) From ii), it follows immediately that if f and h belong to L q (Ω) and f ≤ h in Ω, then

Lemma 2.2.
Assume the conditions H1)-H3) of Theorem 1.1, let τ be a nonnegative function in Let v ∈ C M and let v j j∈N ⊂ C M be such that v j j∈N converges to v in L q (Ω). Then there exists a subsequence v j k k∈N such that v j k k∈N converges to v a.e. in Ω. Then, since g M is a Carathéodory function, g M ·, ε + v j k k∈N converges to g M (·, ε + v) a.e. in Ω. Thus To prove iv), consider a sequence v j j∈N ⊂ C M . Then v j j∈N is bounded in L q (Ω) , and Then there exists a subsequence v j k k∈N such that (−∆) −1 g M ·, ε + v j k k∈N converges in L q (Ω) , and so iv) holds.
Proof. Taking into account Lemma 2.2 and Schauder's fixed point theorem (as stated, e.g., in [23, Corollary 11.2]), T M,ε has a fixed point u M,ε ∈ C M , which, by the definition of T M,ε , belongs also to W 2,q (Ω) and that is a strong solution of problem (2.3). Clearly a function u ∈ W 2,q (Ω) is solution of (2.3) if and only if v := u − τ * is a solution of (2.4), and so (2.4) has, at least, a solution v M,ε ∈ W 2,q (Ω). Moreover, if v is a solution of (2.4), since g M (·, ε + τ * + v) ∈ L ∞ (Ω) and v = 0 on ∂Ω, it follows that v ∈ ∩ 1≤p<∞ W 2,p (Ω) . In particular v ∈ C Ω ∩ W 2,n loc (Ω) . Suppose now that v and w are two solutions of (2.4). Then v and w belong to C Ω ∩ W 2,n loc (Ω) and v = w = 0 on ∂Ω. Since s → g (x, ε + τ * (x) + s) is nonincreasing for any x ∈ Ω, the function h (x, s) := g M (x, ε + τ * (x) + s) is also nonincreasing for any x ∈ Ω. Then, by Lemma 2.3, v = w in Ω and so the solution of (2.4) is unique. Now, from the equivalence of problems

Remark 2.5.
i) Let us recall the following form of the Hopf maximum principle (see [5], Lemma 3.2): Suppose that ρ ≥ 0 belongs to L ∞ (Ω) . Let v be the solution of −∆v = ρ in Ω, v = 0 on ∂Ω. Then ρd Ω a.e.in Ω, (2.5) where c is a positive constant depending only on Ω.
ii) Suppose that ρ ≥ 0 belongs to L ∞ (Ω) . If h ∈ L q (Ω) and h ≥ ρ in Ω, then, from Remark 2.1 iii) and (2.5) it follows immediately that in Ω, where c is the constant given in (2.5).
iii) There exists a positive constant c 0 such that, for any ε ∈ (0 suppose that U ̸ = ∅. Since g is nonincreasing in the second variable, the same is true for g M and so, To prove iii), observe that by i) and ii) we have, for M ∈ [1, ∞) and ε ∈ (0 and so, taking into account Remark 2.5, there exists a positive constant c ′ , depending only on Ω, such that in Ω. Then, from (2.6), v M,ε ≥ c 0 d Ω with and so, since which completes the proof of the lemma.
For M ∈ [1, ∞) , let u M and v M be the functions, defined on Ω by Note that, by Lemma 2.6, u M (x) is well defined and finite for a.e. x ∈ Ω and so, since u M,ε = τ * + v M,ε , the same assertion holds also for v M . i) The map M → u M is nondecreasing on [1, ∞).

T. Godoy
and so, since g M is a Carathéodory function, in Ω.
Let us see that {g M (·, u M )} M∈N converges to g (·, u) with convergence in L q (Ω). From in Ω.
Also, since τ * ≥ 0, from (3.3) and (3.2) we have that u ≥ c 0 d Ω in Ω and that u M ≥ c 0 d Ω in Ω for any M ≥ 1. Then, recalling that g and g M are nonincreasing in the second variable, in Ω.

Example 3.3.
A second example of application of Theorem 1.1 is given by the function and with b : Ω → R satisfying (3.9).

Example 3.4.
A third example can be given by taking g (x, s) β , 1 γ and with b satisfying (3.9). Indeed, H1) and H2) clearly hold, and H3) follows easily from the last assertion of Remark 3.1.
If U and V ′ are domains in R n , we will write U ⊂⊂ V to mean that U ⊂ U ⊂ V.
Suppose now that u and u are solutions of problem (1.1). Then u and u belong to Thus, by Lemma 2.3, u = u in Ω.

A related problem in a punctured domain
Let x 0 ∈ Ω, let U := Ω \ {x 0 } and let w ∈ L 1 (U) . Then w ∈ L 1 (Ω) , and so w can be viewed as a distribution on U and also as a distribution on Ω. For 1 ≤ i, j ≤ n, we will denote by ∂ U i w and ∂ U i ∂ U j w (respectively by ∂ Ω i w and ∂ Ω i ∂ Ω j w) the first and the second derivatives of w considered as a distribution on U (resp. as a distribution on Ω), and, if φ ∈ C ∞ (R n ) , we will write simply ∂ i φ and ∂ i ∂ j φ for the first and the second derivatives of φ.
If w ∈ W 2,q (U) for some q ∈ (1, ∞) , then ∂ U i w and ∂ U i ∂ U j w belong to L q (U) and so they also belong to L q (Ω) . One may ask if ∂ U i w = ∂ Ω i w and ∂ U i ∂ U j w = ∂ Ω i ∂ Ω j w, i.e., if the equalities which hold for φ ∈ C ∞ c (U) , hold also for φ ∈ C ∞ c (Ω) . The next lemma provides a partial answer to this question. Lemma 4.1. Let x 0 ∈ Ω, let U := Ω \ {x 0 } , and, for δ > 0, let A δ be defined by (1.5) and let w ∈ W 2,q (U) . If either lim x→x 0 w (x) exists and is finite, or if ∂ Ω j w for each i and j, and so, in particular, w ∈ W 2,q (Ω) .
ii) follows directly from i). If ∥w∥ L ∞ (U) = ∞ and lim sup x→x 0 |x − x 0 | n−2 w (x) = 0, then, by i), after redefining w in a set with zero measure, we would have C Ω , which is impossible when ∥w∥ L ∞ (U) = ∞. On the other hand, it was proved in ( [30], Theorem 3.6) that, if µ is a bounded Radon measure in Ω, γ ≤ 1, and f ∈ L 1 (Ω) , then the problem has a solution in the sense that: i) u ∈ W 1,1 0 (Ω) and for any compact K ⊂ Ω there exists a positive constant c such that u ≥ c a.e. in K, ii) Ω ⟨∇w, ∇φ⟩ = Ω f u −γ φ + Ω φdµ for any φ ∈ C 1 c (Ω) .
By taking µ = δ x 0 (the Dirac's measure concentrated at x 0 ), and, for instance, f = 1, in [30,Theorem 3.6] it is clear that the conclusions of Theorems 1.4 and 1.5 could not hold anymore if the notion of solution is changed and the requirement that w is "nice enough" is dropped.