Semilinear problems with right-hand sides singular at u = 0 which change sign

Available online at www.sciencedirect.com ScienceDirect Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 www.elsevier.com/locate/anihpc Semilinear problems with right-hand sides singular at u =0which change sign Juan Casado-Díaz a,∗, François Murat b aDpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain bLaboratoire Jacques-Louis Lions, Sorbonne Université & CNRS, France Received 23 November 2019; received in revised form 10 August 2020; accepted 7 September 2020 Available online 15 September 2020 Abstract The present paper is devoted to the study of the existence of a solution ufor a quasilinear second order differential equation with homogeneous Dirichlet conditions, where the right-hand side tends to infinity at u =0. The problem has been considered by several authors since the 70’s. Mainly, nonnegative right-hand sides were considered and thus only nonnegative solutions were possible. Here we consider the case where the right-hand side can change sign but is non negative (finite or infinite) at u =0, while no restriction on its growth at u =0is assumed on its positive part. We show that there exists a nonnegative solution in a sense introduced in the paper; moreover, this solution is stable with respect to the right-hand side and is unique if the right-hand side is nonincreasing in u. We also show that if the right-hand side goes to infinity at zero faster than 1/|u|, then only nonnegative solutions are possible. We finally prove by means of the study of a one-dimensional example that nonnegative solutions and even many solutions which change sign can exist if the growth of the right-hand side is 1/|u|γwith 0 <γ <1. ©2020 L’Association Publications de l’Institut Henri Poincaré. Published by Elsevier B.V. All rights reserved. MSC: 35J25 Keywords: Singular equations; Monotone operators; Existence; Uniqueness; Positive and nonpositive solutions 1. Introduction The existence of a nonnegative solution for a singular semilinear second order partial differential problem with homogeneous Dirichlet conditions such as −div(A(x)∇u) =F(x,u) in  u=0on∂, (1.1) *Corresponding author. E-mail addresses: [email protected] (J. Casado-Díaz), [email protected] (F. Murat). https://doi.org/10.1016/j.anihpc.2020.09.001 0294-1449/©2020 L’Association Publications de l’Institut Henri Poincaré. Published by Elsevier B.V. All rights reserved. J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 is a classical problem which has been first considered by several authors. Here is a bounded open set of RN, A ∈L∞()N×Nis uniformly elliptic and the right-hand side F: ×(0, ∞) →Rsatisfies lim s→0F(x,s)=+∞,a.e. x∈. (1.2) In [8], it is shown the existence and uniqueness of a nonnegative solution assuming that the limit in (1.1)is uniform in and F=F(x, s) decreasing in s. Indeed, in [8]the equation is not written in a divergence form and the solution is a classical solution, i.e. it is in C2() ∩C0(¯ ), which is strictly positive in . The authors also study the behavior of the solution at the boundary and give some partial results about the existence of solution when Fis not necessarily decreasing in s. In [7]it is considered the case where Ais the identity and F=F(x, s) =1/sγ+(λs)qwith γ, λ, q> 0. It is shown that for every γ>0 and every q>1, there exists ˜ λ∈(0, ∞)such that a solution exists if λ <˜ λand there is not solution if λ ≥˜ λ. The existence and uniqueness of classical solution for every λ ≥0 and p<1was before proved in [19]. Looking for solutions in a Sobolev space, the problem has been considered in [3], where it is studied the case F(x, s) =f(x)/sγ, with γ>0, f≥0 not identically zero, f∈Lr(), for some r≥1. It is proved the existence of a solution in H1 loc() ∩W1,1 0(), which is strictly positive in . The authors also study the integrability of uand ∇u depending on γand r. In that paper the function uis a solution in the distributional sense, that is, taking test functions in D(). It is proved in [1] that more general test functions can be considered and that there is just one solution which can be obtained as the limit of the solutions corresponding to replacing Fby Fn(x, s) =fn(x)/(s +1/n) where fn≥0 increases to f. Moreover it is studied the homogenization result corresponding to replacing Ain (1.1) by a sequence of matrix functions An. In [13], it is considered the case where Fsatisfies 0≤F(x,s)≤h(x) 1 sγ+1,a.e. x∈, ∀s≥0,(1.3) with 0 <γ ≤1 and hin a certain space Lr(), r>1. In this case, the authors provide a new definition of nonnegative solution which does not need the use of the strong maximum principle. It is shown that this solution is stable when we replace the function Fby a sequence Fnconverging pointwise to Fand satisfying (1.3) with hand γindependent on F. In particular, this shows that these are the solutions we find by approximating the singular function Fby nonsingular functions. It is also proved the uniqueness of solution if Fis decreasing in s. The results are used to carry out the homogenization when the open set is replaced by a sequence of open sets εsatisfying similar conditions to those which appear in [6]. The extension of the existence, stability and uniqueness results to the “strong singular”case where Fsatisfies the weaker assumption 0≤F(x,s)≤h(x) (s) ,a.e. x∈, ∀s≥0,(1.4) with has above and a Lipschitz function, strictly increasing, and such that (0) =0is carried out in [14](see also [12] and [16]). The corresponding extension of the homogenization result for varying domains is considered in [15]. Excepting [8] where F(x, s) decreasing in simplies that F(x, s) can take negative values for sbigger enough, the rest of the papers we have mentioned above consider a function F: ×[0, ∞) →[0, ∞)in (1.1). Thus, by the maximum principle, if there exists a solution for (1.1), it must be nonnegative. However, taking Fwhich can take nonnegative values introduces several questions such as: Is there still a positive solution? Can we find nonnegative solutions or even solutions changing sign? To give an answer to these questions is the first motivation of the present paper. Indeed, with respect to the existence of nonpositive solutions we must refer for example to [9], [10], where it is considered the existence of nonnegative solutions for ⎧ ⎨ ⎩ −u =λG(x, u) −1 |u|βin  u=0on∂, (1.5) with λ >0, Gnonsingular and strictly positive (some other conditions are needed) and β∈(0, 1). Then, it is proved that a nonnegative solution exists if and only if λis bigger than a certain ˜ λ>0. Replacing uby −u, this provides an example of a singular function 878 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 F(x,s)=1 |s|β−λG(x, s) satisfying (1.2)for which problem (1.1) has a negative solution. In [11], it is also considered the case F(x, s) =g(x, s) +f(x), with |g(x, s)| ≤λ|s|p−1, 0 <p<1, f∈Lm(), m >N/2. Assuming Asymmetric, the solution is defined as a minimum point for the functional 1 2  A∇u·∇udx−  G(x, u) dx, G(x, s) = s  0 F(x,t)dt, and can change sign. The definition of solution given by the authors uses test functions which vanish at u =0 and thus the equation is satisfied in  \{u =0}. It is proved the uniqueness for nonnegative solutions and g(x, .) decreasing. In the present paper, more generally than (1.1)we deal with the problem −div a(x,u,∇u) =F(x,u) in  u=0on∂, (1.6) with a: ×R ×RN→RNa Carathéodory function satisfying usual assumptions in such way that the operator v∈W1,p 0() →−div a(x,v,∇v) ∈W−1,p(), (1.7) is pseudo-monotone in the sense of Leray-Lions ([17], [18]), for some p>1 and it is such that a(x, 0, 0) =0 (see (2.6), (2.7), (2.8) and (2.9) below for the exact assumptions on a). In the first part of the paper we are interested in the existence of nonnegative solutions for (1.6) although Fcan take nonnegative values. We assume F=F(x, s) a Carathéodory function from  ×[0, ∞)into R ∪{∞}. It is nonnegative at s=0 (being able to take an infinite value) and has a growth at s→+∞at most of order p−1. Namely, there exists ν>0 not too large such that for every δ>0, there exists kδ∈Lp(), kδ≥0, such that F(x,s)≤kδ(x) +νsp−1,a.e. x∈, ∀s≥δ. Observe that this assumption is guaranteed if Fsatisfies (1.4). Since we do not assume Fto be nonnegative, we also introduce a below estimate assuring that F−(x, v) is in W−1,p() for every v∈W1,p 0() (see (2.15)). With these general hypotheses we show that a nonnegative solution still exists. In fact, similarly to [13] and [14]we introduce a suitable definition of solution (it is in particular a solution in the distribution sense) for which we show existence, stability with respect to the right-hand side and uniqueness for F(x, s) non increasing in sand aindependent of s. The estimates we use to get the result are inspired by [13] and [14] although we use more general test functions. In particular, we observe that due to the nonlinearity of the operator given by (1.7), some duality arguments used in the choice of the test functions in [13] and [14] cannot be used in the present setting. In the second part of the paper we consider the question relative to the existence of nonpositive solutions or even solutions changing sign. Thus, the function Fis now assumed to be defined in  ×R. In Section 4we show that if there exists δ, τ>0 such that F(x,s)≥τ |s|,∀s∈(−δ,0), (1.8) then every solution of (1.6)is necessarily nonnegative. That is, although, now Fcan take nonnegative values, the fact that it is very large near s=0is enough to preclude the existence of solutions taking negative values in a portion of . Taking in particular Fof the form F(x,s)=f(x) |s|γ−g(x), (1.9) with f(x) ≥ρa.e. in for some ρ>0, and γ≥1, we get that all the solutions for (1.6)are nonnegatives. Now, the question is What happens if γ<1? As we said above, references [9], and [10]show that it is possible to get nonpositive solutions. Here we give in Section 5a different example in the simple case  =(0, l) ⊂R, a(x, s, ξ) =ξ and Fgiven by (1.9) with fand gpositive constants and γ∈(0, 1). By a suitable change of variables, the problem is transformed into 879 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 ⎧ ⎨ ⎩ −w =1 |w|γ−1in(0,l) w(0)=w(l) =0. (1.10) We show that although there is a unique strictly positive solution, there also exist nonpositive solutions and even solutions changing sign, but only if lis large enough. In fact the number of solutions increases with land tends to infinity when ltends to infinity. The paper is organized as follows: •In Section 2we show the existence of nonnegative solutions for (1.6)in the singular case and the stability and uniqueness of these solutions. •In Section 3we prove the results corresponding to Section 2. •In Section 4we show that (1.8) implies that only nonnegative solutions are possible. The proof of these results is given at the end of the Section. •In Subsection 5.1 we state the results corresponding to the one-dimensional example. The corresponding proofs are given in Subsection 5.2. 2. Existence, stability and uniqueness of nonnegative solutions for the singular problem The present section is devoted to the existence, in a sense which we describe below, of nonnegative solutions for problem −div a(x,u,∇u) =F(x,u) in  u=0on∂, (2.1) where F=F(x, s) can take the value +∞ at s=0. We also give a stability result with respect to Ffor this kind of solutions. To finish, we show a uniqueness result when Fis decreasing in sand a=a(x, s, ξ) does not depend on s and is strictly monotone in ξ. The proof of these results will be given in Section 3. We assume the following conditions on , p, a=a(x, s, ξ) and F=F(x, s): •For and pwe assume: is a bounded open set in RN,N≥1.(2.2) p∈(1,∞). (2.3) Moreover, we define p∗∈(1, ∞], p∈(1, ∞)and (p∗)∈[1, ∞)by ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ p∗=Np N−pif p<N p∗is a fixed number in (1,∞)if p=N p∗=+∞ if p>N, (2.4) p=p p−1,(p ∗)=p∗ p∗−1.(2.5) •For the function a: ×[0, ∞) ×RN→RNwe assume: ◦The function ais a Carathéodory function, i.e. a(.,s,ξ) is measurable in , ∀(s, ξ) ∈[0,∞)×RN a(x, ., .) is continuous in [0,∞)×RN,a.e. x∈. (2.6) ◦There exist α, γ, a0such that α>0,γ≥0,a 0∈L(p∗)(), a0≥0 a(x,s,ξ) ·ξ≥α|ξ|p−γsp−a0(x)s, ∀(s, ξ ) ∈[0,∞)×RN,a.e. x∈. (2.7) ◦There exist β, bsuch that 880 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 β>0,b∈Lp(), b ≥0, |a(x,s,ξ)|≤β(ξ|+s+b(x)p−1,∀(s, ξ) ∈[0,∞)×RN,a.e. x∈. (2.8) ◦The function ais monotone in the second variable, i.e. a(x,s,ξ) −a(x,s,η)·(ξ −η) ≥0,∀s∈[0,∞), ∀ξ,η ∈RN,a.e. x∈. (2.9) Moreover, we define λby λ=lim inf R→∞ min ⎧ ⎨ ⎩ R |v|pdx :v∈W1,p 0() \{0}, a(x,v,∇v) ·∇v+γ|v|pdx =R⎫ ⎬ ⎭ .(2.10) •For the function F: ×[0, ∞) →R ∪{+∞}, we assume: ◦The function Fis a Carathéodory function, i.e. F(.,s) is measurable in , ∀s∈[0,∞), F (x, .) is continuous in [0,∞), a.e. x∈. (2.11) ◦ F(x,0)≥0,a.e. x∈. (2.12) ◦There exists ν≥0 such that for every δ>0, there exists kδsatisfying kδ∈L(p∗)(), kδ≥0,F(x,s)≤kδ(x) +νsp−1,∀s≥δ, a.e. x∈. (2.13) ◦The constants γin (2.7), νin (2.13), and λdefined by (2.10)are related by γ+ν<λ. (2.14) ◦ If p≤N, ∃˜ν≥0,˜ k∈L(p∗)(), ˜ k≥0 F(x,s)≥−˜ k(x) −˜νsp∗,∀s≥0,a.e. x∈ If p>N, ∀m∈N,∃˜ km∈L1(), ˜ km≥0, F(x,s)≥−˜ km(x), ∀s∈[0,m],a.e. x∈. (2.15) Remark 2.1. The minimum in (2.10)is well defined, thanks to (2.7). If adoes not depend on sand satisfies the homogeneity condition a(x,tξ) =tp−1a(x,ξ), ∀ξ∈RN,∀t≥0,a.e. x∈, then λagrees with the first eigenvalue of the operator v∈W1,p 0() →−div a(x, ∇v), defined by λ=min v∈W1,p 0() v=0a(x,∇v) ·∇vdx |v|pdx . Moreover, if λpdenotes the first eigenvalue of the p-Laplacian operator, then, we have αλp≤λ. (2.16) This inequality can be proved as follows: We take a sequence Rntending to infinity and a sequence un∈H1 0() such that Rn= a(x,un,∇un)·∇un+γ|un|pdx, lim n→∞ Rn |un|pdx =λ. (2.17) In particular unLp() tends to infinity while the first equality, (2.7) and the definition of λpgives 881 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 λ≥lim sup n→∞ (α|∇un|p−a0|un|)dx |un|pdx ≥αλp. From (2.16)we deduce that a sufficient condition to have (2.14)is to assume γ+ν<αλ p. Remark 2.2. Condition (2.7)on aimplies a(x,0,0)=0,a.e. x∈. (2.18) In fact, taking in (2.7)s=0, replacing ξby tξ with t>0, and dividing by t, we get a(x,0,tξ)·ξ≥αtp−1|ξ|p,∀ξ∈RN,∀t>0,a.e. x∈. Letting ttend to zero, this provides a(x,0,0)·ξ≥0,∀ξ∈RN,a.e. x∈, and then (2.18). The definition of nonnegative solution for the singular problem (2.1)is as follows Definition 2.3. For a bounded open set  ⊂RNand functions a, Fsatisfying (2.6), ..., (2.9), (2.11), ..., (2.15), we say that u : →Ris a nonnegative solution of (2.1)if it satisfies u∈Lp∗() (2.19) u≥0a.e.in(2.20) (u −δ)+∈W1,p 0(), ∀δ>0 (2.21) ϕ∇u∈Lp()N,∀ϕ∈W1,p 0() ∩L∞() (2.22) F(x,u) +ϕp∈L1(), ∀ϕ∈W1,p 0() ∩L∞(), ϕ ≥0 (2.23) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩   a(x,u,∇u) ·∇h(u)ϕp)dx =  F (x, u)h(u)ϕpdx ∀ϕ∈W1,p 0() ∩L∞(), ϕ ≥0,∀h∈W1,∞(0,∞). (2.24) Remark 2.4. Assumption (2.15) on Fand (2.19) implies that F(x, u)−belongs to L1() for every nonnegative solution uof (2.1). By (2.23), this proves F(x,u)ϕp∈L1(), ∀ϕ∈W1,p 0() ∩L∞(), ϕ ≥0.(2.25) Moreover, using ∇(h(u)ϕp)=h(u)∇uϕp+p h(u)ϕp−1∇ϕ, and assumptions (2.8), (2.22), we also have that a(x, ∇u)∇(h(u)ϕ) is in L1() for every ϕand hin the conditions of (2.24). Thus, the two integrals which appear in (2.24)have a sense. Assumption (2.21)in combination with (2.20)gives 0≤u<δ on ∂, ∀δ>0, which gives a sense to the boundary condition u =0on ∂. 882 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 The following proposition shows that in (2.24)we can enlarge the set of test functions by introducing W=w∈W1,p(), ∃ϕ∈W1,p 0() ∩L∞(), ϕ ≥0a.e.in, |w|≤ϕpa.e. in , |∇w| ϕp−1χ{ϕ=0}∈Lp(). (2.26) Observe that contrarily to the test functions used in (2.24), this set does not depend on the solution u. We have preferred to give the definition of solution taking test functions of the form h(u)ϕpwith ϕ∈W1,p 0() ∩L∞(), ϕ∈W1,p 0(), h ∈W1,∞(0, ∞), instead of functions in W, in order to compare with the definition of solution given in [13] and [14]. Proposition 2.5. The space Wis a vectorial subspace of W1,p 0(). It contains the space of functions in W1,p() ∩ L∞() with compact support in and satisfies wv ∈W, ∀w∈W, ∀v∈W1,p() ∩L∞(). (2.27) If a function u : →Ris a nonnegative solution of (2.1)in the sense established in (2.3), then it also satisfies F(x,u)=0,a.e. in {u=0}(2.28) |∇u|p−1|∇w|∈L1(), ∀w∈W(2.29)   a(x,u,∇u) ·∇wdx=  F(x,u)wdx, ∀w∈W. (2.30) Remark 2.6. Proposition 2.5 proves in particular that every nonnegative solution of (2.1)in the sense given by Definition 2.3 is a solution in the distribution sense, i.e. we have −div a(x,u,∇u) =F(x,u) in D(). (2.31) By (2.22), we also know that uis in W1,p loc () and then, for every open set ωstrictly contained in , we have that div a(x, u, ∇u) is in W−1,p(ω) while by (2.23)F(x, u) is in L1(ω). It is well known that then (2.31) implies that F(x, u)v belongs to L1() for every v∈W1,p() with compact support and   a(x,u,∇u) ·∇vdx=  F(x,u)vdx, ∀v∈W1,p() with compact support in . (2.32) The following theorem provides a stability result for the nonnegative solutions of (2.1) when the right-hand side varies. Theorem 2.7. We consider a bounded open set  ⊂RN, a function a: ×[0, ∞) ×RN→RN, satisfying (2.6), (2.7), (2.8) and (2.9), and a sequence of functions Fn: ×R →R ∪{∞}such that (2.11), (2.12), (2.13)(2.14) and (2.15) hold, with constants ν, ˜νand functions kδ, ˜ kand ˜ kmindependent of n. We also assume the existence of F: ×R →R ∪{∞}such that For a.e. x∈, we have: tn≥0,t n→t=⇒ Fn(x, tn)→F(x,t). (2.33) Then, if unis a sequence of nonnegative solutions of −div a(x,un,∇un)=Fn(x, un)in  un=0in , (2.34) there exists a nonnegative solution uof (2.1)such that for a subsequence of n, still denoted by n, we have un→uin Lq(), ∀q<p ∗(2.35) unu in Lp∗() (2.36) (un−δ)+(u−δ)+in W1,p 0(), ∀δ>0.(2.37) 883 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 Thanks to Theorem 2.7 we will deduce the existence of nonnegative solutions for (2.1) which is given in the following theorem. Theorem 2.8. Let be a bounded open set of RN. Then, for every function a: ×[0, ∞) ×RN→RNwhich satisfies (2.6), (2.7), (2.8) and (2.9), and every function Fwhich satisfies (2.11), (2.12), (2.13), (2.14) and (2.15), there exists a nonnegative solution of problem (2.1)in the sense of Definition 2.3. We finish this section with the following comparison result for the nonnegative solutions of (2.1) when the operator a=a(x, s, ξ) does not depend on s. It proves in particular the uniqueness of nonnegative solutions when ais strictly monotone in ξand F=F(x, s) is nondecreasing in s. In the case when adepends on sit is also possible to extend some uniqueness results for pseudomonotone problems (see e.g. [2], [4], [5]), but the corresponding proofs are more complicated. Theorem 2.9. Let be a bounded open set of RNand let a: ×[0, ∞) ×RN→RNbe independent on sand satisfying (2.6), (2.7), (2.8) and (a(x, ξ1)−a(x,ξ2)) ·(ξ1−ξ2)>0,∀ξ1,ξ 2∈RN,a.e. x∈. (2.38) We consider two functions F1and F2which satisfy (2.11), (2.12), (2.13) and (2.15). We also assume ∃i∈{1,2}such that Fi(x, ·)is nonincreasing, a.e. x∈(2.39) F1(x, s) ≤F2(x, s), ∀s∈[0,∞), a.e. x∈. (2.40) Then, if u1, u2are nonnegative solutions of −div a(x,∇ui)=Fi(x, ui)in  u=0in , i=1,2,(2.41) we have u1≤u2a.e. in . (2.42) 3. Proof of the stability, existence and uniqueness results Let us prove in this section the different results exposed in Section 2relative to the existence and properties of the nonnegative solutions for the singular problem (2.1). Proof of Proposition 2.5.In order to show that Wis contained in W1,p 0(), we consider w∈Wand ϕ∈W1,p 0() such that w∈W, ϕ ∈W1,p 0() ∩L∞(), ϕ ≥0,|w|≤ϕpa.e. in , |∇w| ϕp−1χ{w=0}∈Lp(). (3.1) Then, taking ϕn∈W1,p(), ϕn≥0, with compact support which converges to ϕin W1,p 0() and is bounded in L∞(), it is immediate to show that the sequence wn=w∨(−ϕp n)∧ϕp n∈W1,p 0(), converges to win W1,p(). This proves that wis in W1,p 0(). Now, we consider w∈W1,p 0() ∩L∞() with compact support. Let us prove that wis in W. It is enough to observe that for every ϕ∈C1 c() with ϕ≥wL∞() in supp w, the functions wand ϕsatisfy (3.1). Let us prove that Wis stable by addition. For this purpose, we take w1, ϕ1and w2, ϕ2which are related as in (3.1), then taking into account that for i=1, 2∇wi=0a.e. in {wi=0} ⊂{ϕi=0}it is simple to check that w1+w2, ϕ1+ϕ2also satisfy (3.1). To show (2.27), we take w∈Wand v∈W1,p() ∩L∞(), which we can assume not identically zero, then we observe that for ϕsatisfying (3.1), we have 884 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 |wv|≤ϕv 1 p L∞()p,|∇(wv)| v p−1 p L∞()ϕp−1 χ{ϕ=0}≤⎛ ⎜ ⎝|∇w| ϕp−1v 1 p L∞() +|∇v|ϕ v p−1 p L∞() ⎞ ⎟ ⎠χ{ϕ=0}∈Lp(). Assume now ua nonnegative solution of (2.1)in the sense of Definition 2.3 and consider a function ϕ∈W1,p 0() ∩ L∞(), ϕ≥0a.e. in . For Sε:[0, ∞) →Rdefined by Sε(s) =1−s ε∨0,∀s≥0,(3.2) we take h =Sεin (2.24)to deduce −1 ε {u<ε} a(x,u,∇u) ·∇uϕpdx +p  a(x,u,∇u) ·∇ϕS ε(u)ϕp−1dx =  F(x,u)S ε(u)ϕpdx. (3.3) In the first term of this equality we use (2.7) and (2.18)to deduce 1 ε {u<ε} a(x,u,∇u) ·∇uϕpdx ≥α ε {0<u<ε} |∇u|pϕpdx − {0<u<ε} (γ εp−1+a0)ϕpdx and then lim sup ε→0 1 ε {0<u<ε} a(x,u,∇u) ·∇uϕpdx ≥αlim sup ε→0 1 ε {0<u<ε} |∇u|pϕpdx. For the second term, we observe that (2.8), (2.22) and Hölder’s inequality imply that the function a(x, u, ∇u) · ∇ϕϕ p−1belongs to L1(). Using then that Sε(u) converges a.e. to χ{u=0}and (2.18), and the Lebesgue dominated convergence theorem we deduce lim ε→0  a(x,u,∇u) ·∇ϕS ε(u)ϕp−1dx =0.(3.4) For the third term in (3.3), we use (2.25), which allows us to apply Lebesgue dominated convergence theorem to get lim ε→0  F(x,u)S ε(u)ϕpdx = {u=0} F(x,0)ϕpdx. Therefore, taking the limit in (3.5) when εtends to zero we obtain 0≥ {u=0} F(x,0)ϕpdx +αlim sup ε→0 1 ε {0<u<ε} |∇u|pϕpdx, which by (2.12)shows (2.28) and lim ε→∞ 1 ε {0<u<ε} |∇u|pϕpdx =0,∀ϕ∈W1,p 0() ∩L∞(), ϕ ≥0.(3.5) Let us now prove (2.29) and (2.30), for w∈W. Decomposing w=w++w−, where both w+and w−belong to W, we can assume wnonnegative. We take ϕsuch that (3.1) holds. Using |∇u|p−1|∇w|≤|∇u|p−1ϕp−1|∇w| ϕp−1χ{ϕ=0},a.e. in  and taking into account (2.22) and (3.1)we conclude that (2.29)is a simple consequence of Hölder’s inequality. 885 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ (u +δ)−∈W1,p 0(), ∀δ>0   a(x,u,∇u) ·∇vdx=  F(x,u)vdx ∀v∈W1,p 0() ∩L∞(), such that ∃δ>0with v=0in {|u|<δ}, (4.13) satisfies u≥0a.e. in . (4.14) Remark 4.3. Problem (4.12)is non standard since the equation takes place in the set  \{u =0}which depends on the solution u. In general this set is not an open set since uhas not reason to be continuous. The fact that the equation takes place in  \{u =0}is reflected in the (mathematically correct) formulation (4.13) where the test functions vhave to vanish on the set {u =0}. Remark 4.4. Definition 2.3 in Section 2was concerned with nonnegative solutions. Thus, the functions a=a(x, s, ξ) and F=F(x, s) were only defined for snonnegative. It is clear that Definition 2.3 could be extended in a natural way to the case where Fis defined for s∈Rand where the solution can take negative values. In this new setting, any such solution to (2.1) would have to be nonnegative in view of Theorem 4.2 when the function Fsatisfies assumption (4.10), reinforcing in this case the uniqueness result of Theorem 2.9. Remark 4.5. If in Lemma 4.1 we replace assumptions (4.4), (4.5), (4.6) and (4.7)by (u −δ)+∈W1,p 0(), ∀δ>0,H∈L1({u>δ}), ∀δ>0, ∃τ,δ0>0 such that H≤−τ |u|a.e. in {0<u<δ} ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩   a(x,u,∇u) ·∇v=  Hvdx ∀v∈W1,p 0() ∩L∞(), such that ∃δ>0 with v=0in{u<δ}. Then, instead of (4.8)we have that u ≤0a.e. in . The proof of this result just follows by applying Lemma 4.1 to the functions ˜a, ˜uand ˜ Hdefined by ˜a(x,s,ξ) =−a(x,−s,−ξ), ˜u=−u, ˜ H=−H. Using this result one can also modify Theorem 4.2 to prove that the solution uof a nonlinear problem with a singular right-hand side term F(x, u) such that F(x,u)≤−τ u,if 0 <u<δ, for some δ>0, which is nonpositive on the boundary is necessarily nonpositive in the whole of . The results stated in Lemma 4.1 and Theorem 4.2 are based on the fact that the sign of uon the boundary is known. When no boundary conditions are given it is still possible to show Lemma 4.6 below. It proves that the solution of a nonlinear problem with a singular term, which is sufficiently large for uclose to zero, cannot change sign. Lemma 4.6. Assume  ⊂RN, open, a: ×R ×RN→Rsatisfying (4.1), (4.2) and (4.3). We assume there exist u ∈W1,p loc () and H∈L1 loc() such that there exist τ, δ0>0satisfying one of the following conditions: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ |H|≥ τ |u|a.e. in {0<u<δ 0}   a(x,u,∇u) ·∇vdx=  Hvdx ∀v∈W1,p() ∩L∞(), spt(v) ⊂compact, ∃δ>0with v=0in {v<δ}, (4.15) 892 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 or ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ |H|≥ τ |u|a.e. in {−δ0<u<0}   a(x,u,∇u) ·∇vdx=  Hvdx ∀v∈W1,p() ∩L∞(), spt(v) ⊂compact, ∃δ>0with v=0in {v>−δ}. (4.16) Then, one of the following assertions holds: u≥0a.e. in or u≤0a.e. in . (4.17) Proof of Lemma 4.1.For ε<δ 0/2, we take vε:=1+u ε−∧1∈W1,p 0() ∩L∞(), ∀ε>0,(4.18) as test function in (4.7). We have −1 ε {−2ε<u<−ε} a(x,u,∇u) ·∇udx =  Hvεdx = {u≤−δ0} Hdx+ {−δ0<u} Hvεdx, and thus − {u≤−δ0} Hdx=1 ε {−2ε<u<−ε} a(x,u,∇u) ·∇udx+ {−δ0<u} Hvεdx. (4.19) By (4.6), we know that H≥0a.e. in {−δ0<u <0}. This allows us to use the monotone convergence theorem in the last term, which combined with (4.2), proves lim sup ε→0 α ε {−2ε<u<−ε} |∇u|pdx + {−δ0<u<0} Hdx≤−  {u≤−δ0} Hdx, and thus, taking into account (4.5), we get H∈L1({u<0}), lim sup ε→0 1 ε {−2ε<u<−ε} |∇u|pdx ≤−1 α {u<0} Hdx. (4.20) Now, we observe that thanks to (4.6), for 0 <2ε<δ 0, we have 1 2εx∈:−2ε<u<−ε ≤ {−2ε<u<−ε} 1 |u|dx ≤1 τ {−2ε<u<−ε} Hdx, and then, by the first assertion in (4.20) lim ε→0 1 εx∈:−2ε<u<−ε =0.(4.21) Let us now take  ∈C∞ c(RN)N. Since vεdefined by (4.18) belongs to W1,p 0(), we have   vεdiv dx=−  ∇vε·dx =1 ε {−2ε<u<−ε} ∇u·dx. (4.22) 893 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 Taking into account (4.20) and (4.21), we have on the first hand lim sup ε→0 1 ε {−2ε<u<−ε} |∇u|||dx ≤lim ε→0⎛ ⎜ ⎝ 1 ε {−2ε<u<−ε} |∇u|pdx⎞ ⎟ ⎠ 1 p 1 εx∈ω:−2ε<u<−ε 1 pL∞()N=0. On the other hand, using the definition (4.18)of vεand Lebesgue’s dominated convergence theorem, we have lim ε→0  vεdiv dx= {u<0} div dx. Thus, (4.22) provides 0=  χ{u<0}div dx, ∀∈C∞ c(RN)N=⇒ ∇χ{u<0}=0inD(RN). (4.23) This proves that χ{u<0}is a constant function in RN, but {x∈ :u(x) <0} ⊂implies that χ{u<0}vanishes outside . Therefore χ{u<0}=0a.e. in RN. Proof of Theorem 4.2.It is easily deduced from Lemma 4.1 with H(x) =F(x, u(x)). Namely, we observe that (4.13) implies (4.4) and (4.7), while Sobolev’s embedding theorem and (4.11) provide (4.5). Finally, assumption (4.6) follows from (4.10).  Proof of Lemma 4.6.To fix ideas we assume in the following that (4.15) holds. The other case is completely similar. The proof follows the ideas of Lemma 4.1. We take ϕ∈W1,p() ∩L∞(). Then, for ε>0, we define vε:=!u ε−1+∧1". Taking v=vεϕin (4.15), we get 1 ε {ε<u<2ε} a(x,u,∇u) ·∇uϕdx+  a(x,u,∇u) ·∇ϕv εdx =  Hvεϕdx. Then, using Lebesgue dominated convergence theorem we can pass to the limit when εtends to zero to deduce lim ε→0 1 ε {ε<u<2ε} a(x,u,∇u) ·∇uϕdx = {0<u}Hϕ−a(x,u,∇u) ·∇ϕdx, which combined with (4.2)shows lim sup ε→0 1 ε {ε<u<2ε} |∇u|pϕdx<+∞.(4.24) On the other hand, for ε<δ 0/2, we have τ ε {ε<u<2ε} ϕdx≤ {ε<u<2ε} |H|ϕdx, and so, lim ε→0 1 ε {ε<u<2ε} ϕdx=0.(4.25) 894 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 From (4.24) and (4.25)we then deduce similarly to (4.23) (observe that now the support of is contained in ).  {u>0} div dx=0,∀∈C∞ c()N. This shows that the distributional gradient of χ{u>0}is zero and so that χ{u>0}is constant a.e. in , i.e. u ≥0a.e. in or u ≤0a.e. in . 5. A one-dimensional example of a singular equation with many solutions which change sign In Section 2we have proved the existence of a nonnegative solution for the semilinear problem (2.1) where F= F(x, s) is nonnegative at s=0 and can take the value +∞. Later, in Section 4, we have shown that if F(x, s) is bigger than τ/|s|, τ>0, when sis negative and close to zero, then every solution of (2.1)is nonnegative. If contrarily, the right-hand side Ftends to +∞ when stends to 0 with a speed of order 1/|s|γ, 0 <γ <1, then some examples in [9] and [10]show that a nonpositive solution can also exist. This proves that condition (4.10)in Theorem 4.2 is optimal in the sense that it cannot be relaxed by F(x,s)≥τ |s|γa.e. in {δ0<s<0},with δ0,τ >0,γ<1. In the present section we explore the simple example in dimension N=1given by ⎧ ⎨ ⎩ −u =f |u|γ−gin (0,L) u(0)=u(L) =0, (5.1) with L, f, γpositive constants and g∈R. We will describe all the possible solutions. Theorems 2.8 and 2.9 applied to (5.1)show the existence and the uniqueness of a nonnegative solution for (5.1), for every f, γ>0 and g∈R. For γ≥1, Theorem 4.2 shows that every solution of the problem is nonnegative. So, the solution given by Theorem 2.8 is the unique solution to this problem. Also, if γ<1 and g≤0, the right-hand side in problem (5.1)is nonnegative and then, the classical weak maximum principle implies that every solution has also to be nonnegative. Thus, we will assume in what follows f,g > 0,0<γ <1. We first observe that a change of scale in the variables xand uallows us to reduce the number of parameters. Namely, for r, t>0to be chosen, we define the new unknown function wby w(x) =ru(tx), x ∈(0,L/t). (5.2) The differential equation in (5.1)is then transformed into −w =frγ+1t2 |w|γ−grt2in (0,L/t). Taking r=g f1 γ ,t=f 1 2γ g γ+1 2γ ,l=L t,(5.3) problem (5.1) reduces to ⎧ ⎨ ⎩ −w =1 |w|γ−1in(0,l) w(0)=w(l) =0. (5.4) The main results of this section are Theorems 5.3 and 5.7 below. They show in particular the existence of solutions to problem (5.1) which take negative values and even change its sign when lis big enough. Moreover the number of such solutions increases as lincreases. 895 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 5.1. Statements of the results concerning problem (5.4) In the introduction to this section we were speaking about solutions of problem (5.4) and (5.1), but we did not define what we understand by a solution of these singular problems. The definition that we will use is the following one: Definition 5.1. Let γbe such that 0<γ <1. A measurable function w:(0, l) →Ris a solution of (5.4)if it satisfies w∈L1(0,l) (5.5) (|w|−δ)+∈W1,1 0(0,l), ∀δ>0 (5.6) 1 |w|γ∈L1 loc(0,l) (5.7) −w =1 |w|γ−1inD(0,l). (5.8) This definition could seem to be weaker than the one given by Definition 2.3 for nonnegative solutions, but this is not the case because of the following Proposition. Proposition 5.2. Let wbe a solution of (5.4)in the sense of Definition 5.1. Then one has w∈W2,1(0,l)∩H1 0(0,l), (5.9) and then in particular 1 |w|γ∈L1(0,l), w∈C1([0,l]). (5.10) Moreover, defining c≥0by c=|w(0)|2,(5.11) one has |w|2=b(w) +cin [0,l],(5.12) where b:R →Ris the function defined by b(s) =− 2 1−γ s |s|γ+2s, ∀s∈R,(5.13) where b(0)is understood as zero. Finally, when c=0one has w∈W2,q (0,l), 1 |w|γ∈Lq(0,l), ∀q< 1 γ.(5.14) In the case where γ=1/2, we are able to describe all the solutions of problem (5.4)in the sense of Definition 5.1. This is the most simple case of the main result of this section Theorem 5.3. Let γbe such that γ=1 2. Define T0by T0=2√2π. Then, the set of the solutions of problem (5.4)in the sense of Definition 5.1 is the union of the following branches: 896 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 1. For every l>0, there exists 1 solution of problem (5.4)which is positive in (0, l). This solution is the unique nonnegative solution of (5.4). 2. For l=T0, there exists 1 solution of problem (5.4)which is negative in (0, l). This solution is the unique negative solution of (5.4). 3. For every l>T 0, there exist 4 solutions of problem (5.4)which are as follows: the first one is negative in (0, l), and is the unique solution (strictly) negative on (0, l); the 3 other ones change sign at every zero and are negative on 1 nonempty subinterval of (0, l) and positive on 1 or 2 nonempty subintervals of (0, l). 4. For l=2T0, there exists 1 solution of problem (5.4)which is negative in (0, T0) ∪(T0, 2T0)but which vanishes at T0. 5. For every l>2T0, there exist 4 solutions to problem (5.4)which change sign at every zero and are negative on 2 nonempty subintervals of (0, l) and positive on 1, 2 or 3 nonempty subintervals of (0, l). 6. Similarly, for every k∈N, with k≥3, for every l=kT0, there exists 1 solution of problem (5.4)which is negative on #k−1 j=0(jT0, (j +1)T0)but which vanishes at t=jT0, j∈{1, ..., k−1}. 7. Similarly, for every k∈N, with k≥3, for every l>kT 0, there exist 4 solutions to problem (5.4)which change sign at every zero and are negative on knonempty subintervals of (0, l) and positive on k−1, kor k+1nonempty subintervals of (0, l). Therefore when γ=1/2, problem (5.4) has exactly 1+4ksolutions if kT0<l<(k+1)T0,∀k≥0,(5.15) 4k−2solutions if l=kT0,∀k≥1.(5.16) Remark 5.4. Let us denote by Bj k, with k∈N, k≥1, and j=1, 2, 3, 4, the 4 functions which to any l>kassociate the 4 solutions of problem (5.4)in (0, l), which are negative on ksubintervals of (0, l), described in the latest point of Theorem 5.3, or more exactly their extensions by zero to (0, ∞)of these 4 solutions. Since these 4 solutions are uniquely defined by formulas (5.38) and (5.42) below, it is not difficult to prove that the functions Bj kare continuous on (kT0, +∞)with values in L1(0, ∞)or even in W1,p 0(0, ∞)for any p<∞. This proves that in the case where γ=1/2, the set of all the solutions of (5.4) coincide with a set made of a countable number of bundles of 4 continuous branches originating at every point l=kT0, with k≥1, to which one has to add the branch originating at l=0 consisting of the nonnegative solution. A representation of these branches with the type of solutions corresponding to each branch is given in Fig. 2. Remark 5.5. As already said in the introduction of this section, problems (5.1) and (5.4)are equivalent through the change of (independent and dependent) variables (5.2). In the case γ=1/2, since (5.3) implies that g=fl L2 3 , setting M=fT 0 L2 3 , Theorem 5.3 implies that problem (5.1) has a unique solution (which is positive) when 0 <g<M, and a unique negative solution when g≥M. Moreover at every value g=Mk2 3, with k∈N, a bundle of 4 branches of solutions appears, so that problem (5.1) has exactly 1 +4ksolutions for Mk2 3<g<M(k+1)2 3, k≥0 and 4k−2 solutions for g=Mk2 3, k≥1. Theorem 5.3 refers to the case γ=1/2. In the case γ∈(0, 1/2) ∪(1/2, 1)we do not know the exact number of solutions of problem (5.4) and the structure of such solutions. However we can still show some related results. For this purpose consider the function bdefined by (5.13), whose graph is given in Fig. 1. The following proposition gives some properties of this function bwe will need later to state and prove the results corresponding to the existence of solutions for problem (5.4). 897 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 Fig. 1. Graph of the function b. Fig. 2. Branches of solutions for γ=1/2. Proposition 5.6. For γ∈(0, 1), the function bdefined by (5.13) satisfies ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ bis strictly increasing in (−∞,−1)and (1,∞), b is strictly decreasing in (−1,1) lim s→−∞b(s) =−∞,b(−1)=2γ 1−γ,b(1)=− 2γ 1−γ,lim s→∞b(s) =∞ bvanishes at s=− 1 (1−γ) 1 γ ,s=0,s=1 (1−γ) 1 γ . (5.17) •For 0 ≤c<2γ/(1 −γ), the equation b(s) +c=0has three solutions z1(c), z2(c) and z3(c) with z1(c) <−1, 0 ≤z2(c) <1, 1 <z 3(c). •For c=2γ/(1 −γ)the equation b(s) +c=0has two solutions z1(2γ/(1 −γ)) <−1and z2(2γ/(1 −γ)) =1. •For c>2γ/(1 −γ), equation b(s) +c=0has a unique solution z1(c) and z1(c) <1. We define T−:[0, ∞) →Rand T+:[0, 2γ/(1 −γ)) →Rby T−(c) =2 0  z1(c) dt √b(t) +c,∀c∈[0,∞). (5.18) 898 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 T+(c) =2 z2(c)  0 dt √b(t) +c,∀c∈$0,2γ 1−γ.(5.19) Then, we have T−∈C0([0,∞)), T ∗:=min c≥0T−(c) > 0,lim c→∞T−(c) =∞ (5.20) T+∈C0!0,2γ 1−γ,T +is strictly increasing,T +(0)=0,lim c→2γ 1−γ T+(c) =∞.(5.21) Using the function T−defined by (5.18)we can now state our main result about the existence of solution for problem (5.8)for γ∈(0, 1). Theorem 5.7. Let γbe in (0, 1). Define T∗by (5.20) and T0by T0=T−(0). (5.22) Then, we have: If T−is strictly increasing in [0, ∞), then all the results in the statement of Theorem 5.3 still hold true. If T−is not necessarily strictly increasing, we have: 1. For every l>0, there exists 1 solution of problem (5.4)which is positive in (0, l). This solution is the unique nonnegative solution of (5.4). 2. For every l<kT∗, k≥1, there is not any solution of problem (5.4)which is negative on at least knonempty open subintervals of (0, l) and vanishes on the boundary of these intervals. 3. For every l≥T∗, there exists at least 1 solution of problem (5.4)which is negative in (0, l). 4. For every l=kT0, k≥1, there exists 1 solution of problem (5.4)which is negative on #k−1 j=0(jT0, (j +1)T0)but which vanishes at t=jT0, j∈{1, ..., k−1}. 5. For every l>kT 0, k≥1, there exist at least 4 solutions of problem (5.4)which are negative on exactly kdisjoint open subintervals of (0, l), they vanish on the boundary of these subintervals and are positive on the rest of (0, l), which is composed by k−1, kor k+1disjoint open subintervals. In particular, for γ∈(0, l), problem (5.4) has at least 1+4ksolutions if kT0<l<(k+1)T0,∀k≥0,(5.23) 4k−2solutions if l=kT0,∀k≥1.(5.24) Remark 5.8. By Theorem 5.7, the number of solutions of (5.4) agrees with (5.15)or (5.16), when the function T− is strictly increasing. In the case where γ=1/2this follows from the next proposition which provides an explicit expression for T−. In the general case we do not know when this is true or not. A numerical computation provides Fig. 5showing the graph for T−for several values on γ. It seems to indicate the existence of ˜γ≤1/2, close to 1/2, such that T−is not strictly increasing for γ∈(0, ˜γ), while it is strictly increasing for γ≥˜γ. Proposition 5.9. Assume γ=1 2. Then, the functions z1and z2defined by Proposition 5.6 are given by z1(c) =−2+c 2+√4+2c,∀c≥0,z 2(c) =2−c 2−√4−2c, ∀c∈[0,2].(5.25) The functions T−and T+defined by (5.18)are given by T−(c) =2√2π−2arctan%−z1(c) −2√−z1(c) √−z1(c) +&−z1(c) −2%−z1(c),∀c≥0,(5.26) T+(c) =2√22 arctanh √z2(c) %2√z2(c) −z2(c) −&2%z2(c) −z2(c),∀s∈[0,2). (5.27) 899 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 The function T+is a strictly increasing function in [0, ∞). 5.2. Proof of the results corresponding to the one-dimensional example The results exposed above are essentially a consequence of Theorem 5.10 below. First of stating and proving this result, let us show Propositions 5.2 and 5.6. Proof of Proposition 5.2.We start by observing that if wis a solution of (5.4)in the sense of Definition 5.1, then, (5.7) and (5.8)imply that wis in W2,1 loc (0, l), and therefore wis in C1(0, l). Multiplying equation (5.8)by w, we get −|w|2 2=w |w|γ−win (0,l), and thus, there exists a constant c∈Rsuch that |w|2=− 2 1−γ w |w|γ+2w+cin (0,l). (5.28) Since w∈L1(0, l), we deduce from this equality that |w|2belongs to L1(0, l) and therefore w∈H1(0, l). Then, (5.6) implies that wbelongs to H1 0(0, l). In particular, wbelongs to C0([0, l])and then (5.28)shows that |w|is in C0([0, l]). Integrating equation (5.8)in (δ, l−δ), with δ>0, we also have l−δ δ dx |w|γ=−w(l −δ) +w(δ) +l−2δ. Since |w|in C0([0, l])implies win L∞(0, l), the monotone convergence theorem implies that 1 |w|γ∈L1(0,l), (5.29) and then (5.8) proves that wis in W2,1(0, l). In particular, this means that wis in C0([0, l])and by (5.28) that (5.12) holds. If c=0, then, using that w∈C1([0, l])and that |w(s)| =c2for every s∈[0, l]such that w(s) =0, we deduce that 1/|w|γbelongs to Lq(0, l) for every q<1/γ which combined with (5.4)shows (5.14).  Proof of Proposition 5.6.Statement (5.17) and the results about the number and position of the zeros of the function bfollow immediately by studying the sign of the derivative of b. The continuity of the functions T−and T+defined by (5.18) and (5.19)is simple to check. Moreover, the definition of T+and z2(0) =0imply T+(0) =0. Since for c=2γ/(1 −γ), we have z2(c) =1 and b(1) +c=b(1) =0, we get lim c→2γ 1−γ T+(c) =2 lim c→2γ 1−γ z2(c)  0 dt √b(t) +c=∞.(5.30) In order to prove that T+is strictly increasing, we recall that Theorems 2.8 and 2.9 show that problem (5.1) has a unique nonnegative solution for every l≥0. Taking into account Theorem 5.10 below (see (5.41)) this solution is necessarily obtained as the restriction to [0, l]of a periodic function of period T−(c) +T+(c), with l=T+(c) for some c∈[0, 2γ/(1 −γ)). Moreover, by (5.42), it is given by w(x) =⎧ ⎪ ⎨ ⎪ ⎩ G−1(x) if x∈!0,l 2" G−1(l −x) if x∈!l 2,l", 900 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 with Gdefined by (5.36). This provides the implicit definition of w ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x= w(x)  0 dt √b(t) +cif x∈!0,l 2" x= z2  0 dt √b(t) +c+ z2  w(x) dt √b(t) +cif x∈!l 2,l". The uniqueness of wimplies then that for every l∈[0, ∞), there exists a unique c∈[0, 2γ/(1 −γ)), which satisfies T+(c) =l. Combined with T+(0) =0, (5.30) and Tcontinuous, this shows that T+is strictly increasing. In order to study the values of T−, we consider the function Rdefined by R(z) =T−(z−1 1(−z)), ∀z∈(z−1 1(0), ∞), (5.31) which is obtained by writing the function T−in the variable z=−z1(c). Then, recalling definition (5.13)of band that z1(c) is the unique negative root of b+c, we get that Ris given by R(z) =2 z  0 dr &2 1−γr1−γ−z1−γ+2(z −r) ,∀z≥1 (1−γ) 1 γ .(5.32) Now, for z∈(0, ∞)we define ψ:[0, ∞) →Rby ψ(r)=2 1−γr1−γ−z1−γ+2 zγ(z −r), ∀r>0. Then, using ψ(z)=0,ψ (r) =21 rγ−1 zγ≥0,∀r∈(0,z], we deduce that ψ(r) ≤0for every r∈[0, z], which implies 2 1−γr1−γ−z1−γ+2(z −r)≤21−1 zγ(z −r), ∀r∈[0,z], and then, for every z>1, we have z  0 dr &2 1−γr1−γ−z1−γ+2(z −r) ≥1 &21−1 zγ z  0 dr √z−r='2zγ+1 zγ−1. Taking into account (5.32), the function z→zγ+1/(zγ−1)increasing for z>(1 +γ) 1 γand (1 −γ) −1 γ>(1 +γ) 1 γ for 0 <γ <1, we deduce R(z) ≥'2 γ(1−γ) 1 γ ,∀z≥1 (1−γ) 1 γ ,lim z→∞R(z) =∞. Returning to the variable c=z−1 1(−z), we have then proved T−(c) ≥'2 γ(1−γ) 1 γ ,∀c≥0,lim c→∞T−(c) =∞.(5.33) Theorem 5.10. Assume that wis a solution of (5.4) and define cby (5.11). •If 0 ≤c<2γ/(1 −γ), then wis the restriction to [0, l]of a function in W2,1 loc (R)(W2,q loc (R), for every q<1/γ if c>0), still denoted by w, which is a solution of 901 J. Casado-Díaz and F. Murat Annales de l’Institut Henri Poincaré – Analyse non linéaire 38 (2021) 877–909 where a primitive of r2/(1 +r2)2is given by r2 (r2+1)2dr =−1 2rd dr 1 r2+1dr =− r 2(r2+1)+1 2arctan r. Using (5.57), we then get T−(c) =2√2π−4√2arctan%η2−2η η+2&2η2−4η. (5.58) By (5.56), this provides expression (5.26)for T−. Let us now show that T−is strictly increasing in c. 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