Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 60, pp. 1–19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu TOPOLOGICAL STRUCTURE OF THE SOLUTION SET FOR A FRACTIONAL p-LAPLACIAN PROBLEM WITH SINGULAR NONLINEARITY MARCOS R. MARCIAL, OLIMPIO H. MIYAGAKI, GILBERTO A. PEREIRA Abstract. We establish the existence of connected components of positive solutions for the equation (−∆p)su = λf(u), under Dirichlet boundary con- ditions, where the domain is a bounded in RN and has smooth boundary, (−∆p)s is the fractional p-Laplacian operator, and f : (0,∞)→ R is a contin- uous function which may blow up to ±∞ at the origin. 1. Introduction We establish the existence of a continuum of positive solutions to the problem (−∆p) su = λf(u) in Ω, u > 0 in Ω, u = 0 on Ωc, (1.1) where Ω ⊂ RN , N > 1, is a bounded domain with smooth boundary ∂Ω, Ωc = RN\Ω, s ∈ (0, 1), λ > 0 and p ∈ (1,∞) are real numbers and f : (0,∞) → R is a continuous function which may blow up to ±∞ at the origin. We assume that the nonlinearity f satisfies (A1) f : (0,∞)→ R is continuous and limu→∞ f(u) up−1 = 0, (A2) there are positive numbers β < 1, a and A such that f(u) ≥ a uβ if u > A and lim supu→0 u β |f(u)| <∞. The above hypotheses include nonlinearities such as (i) f(u) = 1 uβ − 1 uα with 0 < β < α < 1; (ii) f(u) = uq − 1 uβ with 0 < q < p− 1 and β > 0; (iii) f(u) = lnu. There is a substantial literature on singular problems dealing with the fractional p-Laplacian operator; we refer the reader to Arora, Giacomoni and Warnault [1], Canino, Montoro, Sciunzi and Squassina [2], Diaz, Morel and Oswald [7], Gia- comoni, Mukherjee and Sreenadh [9], Lazer and McKenna [13], Mukherjee and Sreenadh [14], Ho, Perera, Sim and Squassina [10], and the references therein. See also Cui and Sun [4] for other aspects of fractional p-Laplacian problems. 2020 Mathematics Subject Classification. 35A16, 35B65, 35J75, 35J92. Key words and phrases. Monotonicity methods; singular problems; regularity; fractional p-laplacian operator. ©2022. This work is licensed under a CC BY 4.0 license. Submitted December 8, 2021. Published August 11, 2022. 1 2 M. R. MARCIAL, OLIMPIO H. MIYAGAKI, G. A. PEREIRA EJDE-2022/60 In their fundamental work, Crandall, Rabinowitz and Tartar [3], employed topo- logical methods, Schauder theory, and maximum principles to prove the existence of an unbounded connected subset in R×C0(Ω) of positive solutions u ∈ C2(Ω)∩C(Ω) of the problem −Lu = g(x, u) in Ω, u = 0 on ∂Ω, where L is a second-order uniformly elliptical operator, g is a continuous function satisfying some hypotheses, and C0(Ω) = {u ∈ C(Ω) : u = 0 on ∂Ω}. Our goal is to extend the results obtained by Crandall, Rabinowitz and Tartar[3] to the non-local fractional operator (−∆p) s. In contrast to that paper, we had to overcome the less regularity of this operator to obtain regularity up to the border of Ω. To state our main result, we introduce some notation. For a measurable function u : RN → R, we introduce the Gagliardo semi-norm [u]s,p := (∫∫ R2N |u(x)− u(y)|p |x− y|N+ps dxdy )1/p and consider the space W s,p(RN ) := {u ∈ Lp(RN ) : [u]s,p <∞}, equipped with the norm ‖u‖s,p,RN = ‖u‖Lp(RN ) + [u]s,p, where ‖ · ‖Lp(RN ) denotes the Lp(RN ) norm. We also consider the space W s,p 0 (Ω) := {u ∈W s,p(RN ) : [u]s,p <∞, u = 0 a.e. in Ωc}, which is a Banach space with respect to the norm ‖u‖ = [u]s,p. A weak solution u ∈W s,p 0 (Ω) to the problem (1.1) satisfies∫∫ RN [u(x)− u(y)]p−1(v(x)− v(y)) |x− y|N+sp dxdy = λ ∫ Ω f(u)v dx, (1.2) for every v ∈W s,p 0 (Ω), where [a− b]p−1 denotes |a− b|p−2(a− b). Let p′ and ∗ stand for the conjugate exponent of p and the dual Banach space respectively, we denote W−s,p ′(Ω) := ( W s,p 0 (Ω) )∗ , and its pairing with W s,p 0 (Ω) by 〈·, ·〉. We observe that the expression 〈(−∆p) su, v〉 := ∫∫ RN [u(x)− u(y)]p−1(v(x)− v(y)) |x− y|N+sp dxdy, u, v ∈W s,p 0 (Ω), defines a continuous, bounded and strictly monotone operator (−∆p) s : W s,p 0 (Ω)→ W−s,p ′ (Ω) given by u 7→ (−∆p) su as a consequence of Hölder’s inequality. Observe further that (−∆p) s is strictly monotone and coercive, that is 〈(−∆p) su− (−∆p) sv, u− v〉 > 0, u, v ∈W s,p 0 (Ω), u 6= v and 〈(−∆p) su, u〉 ‖u‖ → ∞ as ‖u‖ → ∞. EJDE-2022/60 TOPOLOGICAL STRUCTURE FOR A FRACTIONAL PROBLEM 3 For all α ∈ (0, 1] and all u : Ω→ R, we set [u]Cα(Ω) = sup x,y∈Ω, x 6=y |u(x)− u(y)| |x− y|α and consider the Banach space Cα(Ω) = { u ∈ C(Ω) : [u]Cα(Ω) <∞ } , endowed with the norm ‖u‖Cα(Ω) = ‖u‖L∞(Ω) + [u]Cα(Ω). The solution set of problem (1.1) is S := {(λ, u) ∈ (0,∞)× C(Ω) : u is a solution of (1.1)}. We now can state our main result. Theorem 1.1. Under assumptions (A1) and (A2), there is a number λ0 > 0 and a connected subset Σ of [λ0,∞)× C(Ω) satisfying (i) Σ ⊂ S; (ii) Σ ∩ ( {λ} × C(Ω) ) 6= ∅, λ0 ≤ λ <∞. 2. Auxiliary results We start by introducing notation and recalling some results. Let M = (M,d) be a metric space and {Σn} a sequence of connected components of M . The upper limit of {Σn} is defined by lim Σn = { u ∈M : there is (uni) ⊆ ∪Σn with uni ∈ Σni and uni → u } . Remark 2.1 ([17]). lim Σn is a closed subset of M . In the proof of Theorem 1.1 we use topological arguments to construct a suitable connected component of the solution set S of (1.1). More precisely, we apply in a nontrivial way [16, Theorem 2.1], whose proof is based on the famous Whyburn’s lemma [17, Theorem 9.3]. Theorem 2.2 (Sun and Song [16]). Let M be a metric space and {αn}, {βn} ∈ R be sequences satisfying · · · < αn < · · · < α1 < β1 < · · · < βn < . . . with αn → −∞ and βn →∞. Assume that {Σ∗n} is a sequence of connected subsets of R×M satisfying (i) Σ∗n ∩ ({αn} ×M) 6= ∅ for each n; (ii) Σ∗n ∩ ({βn} ×M) 6= ∅ for each n; (iii) for each α, β ∈ (−∞,∞) with α < β, ∪Σ∗n ∩ ([α, β]×M) is a relatively compact subset of R×M . Then there is a number λ0 > 0 and a connected component Σ∗ of lim Σ∗n such that Σ∗ ∩ ({λ} ×M) 6= ∅ for each λ ∈ (λ0,∞). Lemma 2.3 ([15]). Let p > 1. There exists a constant Cp > 0 such that( |x|p−2x− |y|p−2y, x− y ) ≥ { Cp|x− y|p, if p ≥ 2 Cp |x−y|p (1+|x|+|y|)2−p if p ≤ 2, where x, y ∈ RN and (·, ·) is the usual inner product of RN . 4 M. R. MARCIAL, OLIMPIO H. MIYAGAKI, G. A. PEREIRA EJDE-2022/60 We also recall the following Hardy-type inequality (see [10]). Lemma 2.4. For any p ∈ (1,∞) and s ∈ (0, 1),∫ Ω |u(x)|p dist(x, ∂Ω)sp dx ≤ C‖u‖p, u ∈W s,p 0 (Ω). The next lemma, which will be proved later, is an important technical result because it proves Cα-regularity up to the boundary for the weak solutions of a non-linear problem driven by the fractional p-Laplacian operator. We denote the Euclidean distance from x to ∂Ω by d(x) = dist(x, ∂Ω). Proposition 2.5. Let f ∈ L∞loc(Ω) be a nonnegative function. Assume that there are β, s ∈ (0, 1) and C > 0 such that |f(x)| ≤ C dsβ(x) , x ∈ Ω. (2.1) Then there exists a unique weak solution u ∈W s,p 0 (Ω) to the problem (−∆p) su = f in Ω u = 0 on Ωc. (2.2) Furthermore (i) u ∈ L∞(Ω). (ii) There exist constants α ∈ (0, 1) and Λ > 0 (Λ depending only on C, β,Ω) such that u ∈ Cα(Ω) and ‖u‖Cα(Ω) ≤ Λ. Proof. A weak solution u to (2.2) satisfies (1.2) for λ = 1. So, the Browder-Minty Theorem guarantees that (−∆p) s : W s,p 0 (Ω) → W−s,p ′ (Ω) is a homeomorphism. We denote Ff (u) = ∫ Ω fudx, u ∈W s,p 0 (Ω). We now prove that Ff ∈ W−s,p ′ (Ω). In fact, let V be an open neighborhood of ∂Ω such that 0 < d(x) < 1 for all x ∈ V . Thus, 1 < 1 dsβ(x) < 1 ds(x) ∀x ∈ V. Now, if v ∈W s,p 0 (Ω), for a positive constant C1 it holds |Ff (v)| ≤ ∫ Ω |f | |v|dx = ∫ V c |f | |v|dx+ ∫ V |f | |v|dx ≤ C1‖v‖+ ∫ Ω ∣∣∣ v ds ∣∣∣ dx. Applying Hölder’s inequality and Lemma 2.4 we obtain a constant C > 0 such that |Ff (v)| ≤ C‖v‖, showing that Ff ∈ W−s,p ′ (Ω). It follows that there exists a unique u ∈ W s,p 0 (Ω) such that (−∆p) su = Ff , that is, u is a weak solution to problem (2.2). To prove that u ∈ L∞(Ω), we define, for each k ∈ N, Ak := {x ∈ Ω : u(x) ≥ k}. Denoting (u − k)+ := max{u − k, 0}, we have (u − k)+ ∈ W s,p 0 (Ω). Since the inequality |v(x)− v(y)|p−2(v(x)− v(y))(v+(x)− v+(y)) ≥ |v+(x)− v+(y)|p (2.3) EJDE-2022/60 TOPOLOGICAL STRUCTURE FOR A FRACTIONAL PROBLEM 5 is valid for any measurable v, almost everywhere for x, y ∈ RN , taking v+ = (u−k)+ as a test function in (1.2) (with λ = 1), (2.3) yields∫∫ RN |v+(x)− v+(y)|p |x− y|N+sp dxdy ≤ ∫∫ RN [u(x)− u(y)]p−1(v+(x)− v+(y)) |x− y|N+sp dxdy = ∫ Ω f(x)v+ dx. Then, as a consequence of [12, Lemma 5.1, Chapter 2 ], we conclude that there exists k1 > 0, independent of u, such that u ≤ k1 a.e. in Ω. (2.4) Now, observe that the function −u satisfies (−∆p) s(−u) = −f in Ω u = 0on Ωc. Repeating the argument above we obtain k2 > 0, independent of u, such that − u ≤ k2 a.e. in Ω. (2.5) From this and (2.4) we conclude the existence of M > 0 (independent of u) such that |u(x)| ≤M a.e in Ω, proving that ‖u‖L∞(Ω) ≤M . We shall now prove the existence of α ∈ (0, 1) such that u ∈ Cα(Ω). For any x0 ∈ Ω, take R0 := d(x0) 2 . Then BR0 (x0) ⊂ B2R0 (x0) ⊂ Ω. Let u ∈ W s,p(B2R0 (x0)) ∩ L∞(B2R0 (x0)) be the weak solution of (2.2). We have (−∆p) su = f(x) ≤ C dsβ(x) ≤ C Rsβ0 in BR0 (x0). By applying [11, corollary 5.5], we infer the existence of a constant M > 0 and α ∈ (0, 1) such that [u]Cα(BR0 (x0)) ≤M [( R s(p−β) 0 ) 1 p−1 + ( Rsp0 ∫ (BR0 (x0))c |u(y)| |x− y|N+sp dx ) 1 p−1 ] R−α0 ≤ C̃. (2.6) The constant C̃ is independent of the choice of the point x0 (and R0). Because u ∈ L∞(Ω), by a covering argument for any Ω′ ⊂⊂ Ω we conclude that ‖u‖Cα(Ω′) ≤ CΩ′ , completing the proof of the interior regularity. To handle regularity up to the border, we establish a result that will also be used later. Claim 1: There exist positive constants C1 and C2 such that, for any 0 < ε < s, we have C1 ds ≤ u ≤ C2 ds−ε, in Ω. 6 M. R. MARCIAL, OLIMPIO H. MIYAGAKI, G. A. PEREIRA EJDE-2022/60 Proof. Set fn := min{n, f}. Since fn ∈ L∞(Ω), it is clear that Ffn ∈ W−s,p ′ (Ω). So, for each n ∈ N there exists un ∈W s,p 0 (Ω) ∩ L∞(Ω) satisfying (−∆p) sun = fn in Ω un = 0 on Ωc. Note also that fn →∞ as n→∞ a.e., and fn ≤ f in Ω. Let λs,p be the first eigenvalue and ϕs,p be a positive eigenfunction of the operator (−∆p) s. There exists a constant c > 0 such that 1 c ds(x) ≤ ϕs,p(x) ≤ c ds(x) for any x ∈ Ω. Indeed, the upper estimate follows from [8, Theorem 3.2] and [11, Theorem 4.4], and the lower estimate from [11, Theorem 1.1] and [5, Theorem 1.5]. Hence, choosing a constant a > 0 small enough, for any x ∈ Ω it follows that (−∆p) s(aϕs,p) ≤ fn(x) = (−∆p) sun ≤ f = (−∆p) su By applying [11, Proposition 2.10], we conclude the existence of C1 > 0 such that C1 ds(x) ≤ un(x) ≤ u(x) for any x ∈ Ω. (2.7) We now handle the upper estimate. Since sβ ∈ (0, s), we obtain (−∆p) su = f(x) ≤ Ksβ(x) = (−∆p) susβ , where usβ is the solution obtained in [1, Theorem 4.2]. Therefore, u ≤ usβ in Ω. Another application of [1, Theorem 4.2 (ii)] yields u ≤ C2 ds−ε in Ω for any ε > 0, completing the proof of our Claim. � Now, since u = 0 in Ωc, it is sufficient to prove the regularity in Ωη for η > 0 small enough, where Ωη := {x ∈ Ω : d(x) < η}. Let x, y ∈ Ωη and suppose, without loss of generality, d(x) ≥ d(y). We consider two cases. If |x−y| < d(x) 2 , set 2R0 = d(x) and y ∈ BR0 (x). Hence we apply (2.6) in BR0 (x) and obtain the regularity. However, if |x−y| ≥ d(x) 2 ≥ d(y) 2 , since Claim 2 guarantees that u ≤ C2d δ(x) for some δ, C2 > 0, we conclude that |u(x)− u(y)| |x− y|δ ≤ |u(x)| |x− y|δ + |u(y)| |x− y|δ ≤ C ( u(x) dδ(y) + u(y) dδ(y) ) ≤ C . The proof is complete. � Remark 2.6. Let us denote Mβ,∞ = { g ∈ L∞loc(Ω) : |g(x)| ≤ C dsβ(x) , x ∈ Ω } . Then the solution operator associated with (2.2) is S : Mβ,∞ →W s,p 0 (Ω) ∩ Cα(Ω), S(g) = u. Notice that ‖S(g)‖Cα(Ω) ≤M for all g ∈Mβ,∞, with M depending only on C, β,Ω. EJDE-2022/60 TOPOLOGICAL STRUCTURE FOR A FRACTIONAL PROBLEM 7 For each s ∈ R we consider fχI (s), where χI is the characteristic function of the interval I ⊂ R. Corollary 2.7. Let f, f̃ ∈ L∞loc(Ω) with f ≥ 0, f 6= 0 satisfying (2.1). Then, for each ε > 0, the problem (−∆p) suε = fχ{ ds>ε} + f̃χ{ ds<ε} in Ω; uε = 0 on ∂Ω admits a unique solution uε ∈ Cα(Ω) for some α ∈ (0, 1). In addition, for any solution u of (2.2) there exists ε0 > 0 such that uε ≥ u 2 in Ω for each ε ∈ (0, ε0). Proof. Existence and uniqueness of uε follows directly from Proposition 2.5. If u is a the solution of (2.2), there exist M > 0 and α ∈ (0, 1) such that ‖u‖Cα(Ω), ‖uε‖Cα(Ω) < M. Claim 1 yields u ≥ C1ds in Ω. Multiplying the equation (−∆p) su− (−∆p) suε = f − ( fχ[ ds(x)>ε] + f̃χ[ ds(x)<ε] ) by u− uε and integrating we have∫∫ RN ( ( [u(x)− u(y)]p−1 |x− y|N+sp − [uε(x)− uε(y)]p−1 |x− y|N+sp ) × (( u(x)− u(y) ) − ( uε(x)− uε(y) )) dy dx ≤ 2M ∫ ds(x)<ε |f − f̃ |dx. As a consequence of Lemma 2.3, we obtain ‖u− uε‖ → 0 as ε→ 0. If ν < α, the compact embedding Cα(Ω) ↪→ Cν(Ω) yields ‖u− uε‖Cν(Ω) ≤ C 2 ds. Therefore, for ε small enough, it follows from (2.7) that uε ≥ u− C 2 ds ≥ u− u 2 = u 2 in Ω. The proof is complete. � The next result is crucial for this work. Lemma 2.8. Let β ∈ (0, 1). Then the problem (−∆p) sφ = 1 φβ in Ω, φ > 0 in Ω, φ = 0 on ∂Ω, (2.8) admits a unique weak solution φ ∈ W s,p 0 (Ω). Moreover φ ≥ cϕs,p in Ω for some constant c > 0. Here ϕs,p is a positive eigenfunction for the operator (−∆p) s associated with its first eigenvalue λs,p. 8 M. R. MARCIAL, OLIMPIO H. MIYAGAKI, G. A. PEREIRA EJDE-2022/60 Proof. We consider the sequence of approximation problems (−∆p) sφn = 1 (φn + 1 n )β in Ω, φn > 0 in Ω. φn = 0 on ∂Ω, (2.9) As a consequence of [2, Proposition 2.3, Lemma 2.2, Lemma 3.1 and Lemma 3.4.], for any n ≥ 1, there exists a weak solution φn ∈W s,p 0 (Ω)∩L∞(Ω) to problem (2.9), with {φn} bounded in W s,p 0 (Ω) and φn ≤ φn+1. Then, up to a subsequence, we have φn ⇀ φ in W s,p 0 (Ω), φn → φ in Lr(Ω) for 1 ≤ r < p∗s and φn → φ a.e. in Ω. By applying [2, Theorem 3.2.] we have that φ is a weak solution to problem (2.8). Consider c > 0 such that cp−1ϕp−1 s,p ≤ 1 (‖φ1‖∞+1)β . We have (−∆p) s(cϕs,p) = cp−1ϕp−1 s,p ≤ 1 (‖φ1‖∞ + 1)β ≤ 1 (φ1 + 1)β = (−∆p) sφ1. Therefore, it follows from the comparison principle that cϕs,p ≤ φ1 ≤ · · · ≤ φn ≤ · · · ≤ φ. (2.10) Combining the left-hand side of (2.9) with (2.10), we obtain φ ≥ cϕs,p in Ω for some constant c > 0. � 3. Lower and upper solutions In this section we prove the existence of both a lower and an upper solutions to problem (1.1). For the convenience of the reader, we start by stating some definitions. Definition 3.1. A function u ∈W s,p 0 (Ω) with u > 0 in Ω such that∫∫ RN [u(x)− u(y)]p−1 (ϕ(x)− ϕ(y)) |x− y|N+sp dy dx ≤ λ ∫ Ω f(u)ϕdx, for all ϕ ∈W s,p 0 (Ω), ϕ ≥ 0 is a lower solution of (1.1). A function u ∈W s,p 0 (Ω) with u > 0 in Ω such that∫∫ RN [u(x)− u(y)]p−1(ϕ(x)− ϕ(y)) |x− y|N+sp dy dx ≥ λ ∫ Ω f(u)ϕdx, for all ϕ ∈W s,p 0 (Ω), ϕ ≥ 0 is called an upper solution of (1.1). Theorem 3.2. Assume (A1) and (A2). Then there exist λ0 > 0 and a non-negative function ψ ∈ Cα(Ω), with ψ > 0 in Ω, ψ = 0 in Ωc, α ∈ (0, 1) such that for each λ ∈ [λ0,∞), u = λrψ is a lower solution of (1.1), where r = 1/(p+ β − 1). Proof. According to (A2), there exists b > 0 such that f(t) > − b tβ if t > 0. (3.1) Applying Lemma 2.8 there exist both a function φ ∈W s,p 0 (Ω) such that (−∆p) sφ = 1 φβ in Ω, φ > 0 in Ω, φ = 0 on ∂Ω, (3.2) EJDE-2022/60 TOPOLOGICAL STRUCTURE FOR A FRACTIONAL PROBLEM 9 and a constant c > 0 such that φ ≥ cϕs,p in Ω. Thus by (2.9) we obtain φ ≥ cds in Ω. (3.3) Now, take δ = a p−1 β−1+p and γ = 2βbδ− β p−1 , where a is the constant given in (A2). According to Corollary 2.7, there exists a constant ε0 > 0 such that, for each ε ∈ (0, ε0), the problem (−∆p) sψ = δφ−βχ[ ds>ε] − γφ−βχ[ ds<ε] in Ω, ψ > 0 in Ω, ψ = 0 in Ωc (3.4) admits a solution ψ ∈ Cα(Ω) satisfying ψ ≥ (δ1/(p−1) 2 ) φ. (3.5) If λ > 0 and r = 1/(p+ β − 1), we define u = λrψ. Now, take λ0 = [ 2A (C1εδ 1 p−1 ) ]1/r , where ε ∈ (0, ε0) and A is given by (A2). Claim 2: u is a lower solution of (1.1) for any λ ≥ λ0. Indeed, take ξ ∈W s,p 0 (Ω), ξ ≥ 0. As a consequence of (3.4), we have∫∫ RN [u(x)− u(y)]p−1 (ξ(x)− ξ(y)) |x− y|N+sp dy dx = λr(p−1)δ ∫ { ds>ε} ξ φβ dx− λr(p−1)γ ∫ { ds<ε} ξ φβ dx. We consider two cases. Case 1: ds > ε. For each λ ≥ λ0, by using (3.3) and (3.4), we obtain u = λrψ≥λr δ 1 p−1 2 φ ≥ λr δ 1 p−1 2 C1d s > λr δ 1 p−1 2 C1ε > A. So, u(x) > A for each λ ≥ λ0 with ds(x) > ε. According to (3.2) and (3.3), we have (−∆p) sδ 1 p−1φ = δ φβ ≥ (−∆p) sψ. Thus, the weak comparison principle implies that δ 1 p−1φ ≥ ψ in Ω. (3.6) It follows from (A2) and (3.6) that λ ∫ ds>ε f(u)ξ dx ≥ λa ∫ ds>ε ξ uβ dx = λ1−rβa ∫ ds>ε ξ ψβ dx ≥λ p−1 p+β−1 a δ β p−1 ∫ ds>ε ξ φβ dx = λr(p−1)δ ∫ ds>ε ξ φβ dx. (3.7) 10 M. R. MARCIAL, OLIMPIO H. MIYAGAKI, G. A. PEREIRA EJDE-2022/60 Case 2: ds < ε. Applying (3.1) and (3.5) we obtain λ ∫ { d<ε} f(u)ξ dx ≥ −λb ∫ { d<ε} ξ uβ dx = −λ1−rβb ∫ d<ε ξ ψβ dx ≥ −λr(p−1)b 2β δ β p−1 ∫ d<ε ξ φβ dx = −λr(p−1)γ ∫ d<ε ξ φβ dx. (3.8) It follows from (3.7) and (3.8) that λ ∫ Ω f(u)ξ dx ≥ ∫∫ RN [u(x)− u(y)]p−1(ξ(x)− ξ(y)) |x− y|N+sp dy dx . The proof is complete. � Next, we show the existence of an upper solution. Theorem 3.3. Assume (A1) and (A2) and let Λ > λ0 with λ0 be as in Theorem 3.2. Then for each λ ∈ [λ0,Λ], (1.1) admits an upper solution u = uλ = Mφ where M > 0 is a constant and φ is given by (3.2). Proof. Choose ε > 0 such that Λε ||φ||p−1+β ∞ < 1 2 . (3.9) According to (A1) and (A2), there exist A1 > 0 and C > 0 such that |f(u)| ≤ εup−1 for u > A1, (3.10) |f(u)| ≤ C uβ for u ≤ A1. (3.11) Choose M ≥ max { Λrδ 1 p−1 , (2ΛC) 1 p+β−1 } . (3.12) Now, (3.9) and (3.12) yield Λε (M‖φ‖∞) p+β−1 + ΛC ≤ Mp+β−1 2 + Mp+β−1 2 = Mp+β−1. (3.13) Let u = Mφ. By taking λ ≤ Λ, it follows from (3.10)-(3.11) that λf(u) ≤ λ|f(u)| ≤ λ [ ε up−1χ{u>A1} + C uβ χ{u≤A1} ] ≤ λ [ ε up−1χ{u>A1} + ε up−1χ{u≤A1} + C uβ χ{u≤A1} + C uβ χ{u>A1} ] = λ[εup−1 + C uβ ]. (3.14) We conclude that λf(Mφ) ≤ λ [ε(M‖φ‖∞)p+β−1 + C [Mφ]β ] ≤ Λ ε(M‖φ‖∞)p+β−1 [Mφ]β + Λ C [Mφ]β . (3.15) EJDE-2022/60 TOPOLOGICAL STRUCTURE FOR A FRACTIONAL PROBLEM 11 Replacing (3.13) and (3.14) into (3.15), we obtain λf(Mφ) ≤ Mp+β−1 [Mφ]β = Mp−1 φβ . Thus λf(u) ≤ Mp−1 φβ . Now, taking a non-negative η ∈W s,p 0 (Ω), it follows from (3.2) that λ ∫ Ω f(u)η dx ≤Mp−1 ∫ Ω η φβ dx = Mp−1 ∫∫ RN [φ(x)− φ(y)]p−1(η(x)− η(y)) |x− y|N+sp dx = ∫∫ RN [Mφ(x)−Mφ(y)]p−1(η(x)− η(y)) |x− y|N+sp dx = ∫∫ RN [u(x)− u(y)]p−1(η(x)− η(y)) |x− y|N+sp dx, showing that u = Mφ is an upper solution of (1.1) for λ ∈ [λ0,Λ]. � Lemma 3.4. If u ∈W s,p 0 (Ω) be a weak solution of problem (1.1). Then u ∈ L∞(Ω). Proof. If u ∈W s,p 0 (Ω) solves (1.1), then 〈(−∆p) s, φ〉 = ∫∫ RN [u(x)− u(y)]p−1(v(x)− v(y)) |x− y|N+sp dxdy = ∫ Ω f(u)v dx (3.16) for any v ∈W s,p 0 (Ω). For each k ∈ N, set Ak := {x ∈ Ω : u(x) > k}. Since u ∈ W s,p 0 (Ω) and u > 0 in Ω, we have that (u− k)+ ∈W s,p 0 (Ω). Taking v = (u− k)+ in (3.16), we obtain 〈(−∆p) s, (u− k)+〉 = ∫ Ω f(u)(u− k)+ dx. (3.17) Applying the algebraic inequality |a−b|p−2(a−b)(a+−b+) ≥ |a+−b+|p to estimate the left-hand side of (3.17), we obtain(∫ Ak (u− k)p ∗ s dx ) p p∗s ≤ C ∫∫ RN |u(x)− u(y)|p |x− y|N+sp dxdy ≤ C〈(−∆p) s, (u− k)+〉 = C ∫ Ak f(u)(u− k)+ dx. Now we estimate the right hand side of (3.17). It follows from (A1) and (A2) the existence of a number M > 0 such that |f(t)| ≤M ( 1 tβ + tp−1 ) , ∀t > 0. Therefore, if k > 1, we have∫ Ak f(u)(u− k)+ dx ≤ 2M ∫ Ak up−1(u− k) dx. (3.18) 12 M. R. MARCIAL, OLIMPIO H. MIYAGAKI, G. A. PEREIRA EJDE-2022/60 Since up−1(u− k) ≤ 2p−1(u− k)p + 2p−1kp−1(u− k), it follows that∫ Ak up−1(u− k) dx ≤ 2p−1 ∫ Ak (u− k)p dx+ 2p−1kp−1 ∫ Ak (u− k) dx. Applying Hölder’s inequality, we obtain∫ Ak (u− k)p dx ≤ |Ak| p∗s−p p∗s (∫ Ak (u− k)p ∗ s dx ) p p∗s . (3.19) So, as a consequence of (3.18)-(3.19), we have∫ Ak (u− k)p dx ≤ |Ak| p∗s−p p∗s 2MC [ 2p−1 ∫ Ak (u− k)p dx+ 2p−1kp−1 ∫ Ak (u− k) dx ] . Denoting L = 2MC yields [1− 2p−1L|Ak| p∗s−p p∗s ] ∫ Ak (u− k)p dx ≤ 2p−1kp−1L|Ak| (p∗s−p) p∗s ∫ Ak (u− k) dx. If k →∞, then |Ak| → 0. Therefore, there exists k0 > 0 such that 1− 2p−1L|Ak| p∗s−p p∗s ≥ 1 2 if k ≥ k0 > 1. Thus, for such k, we conclude that 1 2 ∫ Ak (u− k)p dx ≤ 2p−1kp−1L|Ak| p∗s−p p∗s ∫ Ak (u− k) dx. (3.20) Hölder’s inequality and (3.20) yield∫ Ak (u−k)p dx ≤ |Ak|p−1 ∫ Ak (u−k)p dx ≤ |Ak|p−12p−1kp−1L|Ak| p∗s−p p∗s ∫ Ak (u−k) dx. Therefore, ∫ Ak (u− k) dx ≤ γk|Ak|1+ε, ∀k ≥ k0, (3.21) where γp−1 = 22L and ε = p∗s−p p∗s(p−1) > 0. Set g(k) := ∫ Ak (u− k) dx = ∫ ∞ k |At|dt, where the equality between integrals is a consequence of Cavaliere’s Principle. By (3.21) it follows that g(k) ≤ γk[−g′(k)]1+ε. (3.22) Taking k > k0 and integrating (3.22) from k0 to k, since g(k) > 0 it follows that 1 γ 1 1+ε [ k ε 1+ε ] ≤ { [g(k0)] ε 1+ε − [g(k)] ε 1+ε } ≤ [g(k0)] ε 1+ε . Thus k ≤ γ 1 1+γ [g(k0)] ε 1+ε − k ε 1+ε 0 . We denote Λ = 1 1+γ [g(k0)] ε 1+ε − k ε 1+ε 0 . Note that k ≤ Λ, if |Ak| > 0. Since Λ does not depend on k, we conclude that |Ak| = 0 for all k > Λ, that is, u ∈ L∞(Ω) and ‖u‖L∞(Ω) ≤ γ 1 1+γ [g(k0)] ε 1+ε − k ε 1+ε 0 . � EJDE-2022/60 TOPOLOGICAL STRUCTURE FOR A FRACTIONAL PROBLEM 13 4. Finding a solution for (1.1) Take Λ > λ0 and set IΛ := [λ0,Λ]. For each λ ∈ IΛ, according to Theorem 3.2, u = uλ = λrψ is a lower solution of (1.1). Let M = MΛ ≥ Λrδ 1 p−1 . By Theorem 3.3 we have that u = uλ = MΛφ is an upper solution of (1.1). It follows from (3.6) that u = λrψ ≤ Λrδ 1 p−1φ ≤Mφ = u. (4.1) We consider the convex, closed subset of IΛ × C(Ω) given by GΛ := { (λ, u) ∈ IΛ × C(Ω) : λ ∈ IΛ, u ≤ u ≤ u and u = 0 on Ωc } . For each u ∈ C(Ω), set fΛ(u) = χS1f(u) + χS2f(u) + χS3f(u), x ∈ Ω, where χSi denotes the characteristic function of Si, which are defined by S1 = {x ∈ Ω : u(x) < u(x)}, S2 = {x ∈ Ω : u(x) ≤ u(x) ≤ u(x)}, S3 = {x ∈ Ω : u(x) < u(x)}. Lemma 4.1. For each u ∈ C(Ω), fΛ(u) ∈ L∞loc(Ω) and there exist C > 0 and β ∈ (0, 1) such that |fΛ(u)(x)| ≤ C dsβ(x) , x ∈ Ω. (4.2) Proof. Let K ⊂ Ω be a compact subset. Then both u and u achieve a positive maximum and a positive minimum on K. Since f is continuous in (0,∞), we conclude that fΛ(u) ∈ L∞loc(Ω). Since Ω = ∪3 i=1Si, to prove (4.2) it suffices to show that |f(u(x))| ≤ C dsβ(x) , x ∈ Si, i = 1, 2, 3. According to hypothesis (A2), there are C, δ > 0 such that |f(s)| ≤ C sβ , 0 < s < δ. Let Ωδ = {x ∈ Ω : ds(x) < δ}. Recalling that u ∈ Cα(Ω) if α ∈ (0, 1), we denote D = max Ω ds(x), νδ := min Ωcδ ds(x), νδ := max Ωcδ ds(x) and observe that 0 < νδ ≤ νδ ≤ D <∞ and also that f([νδ, ν δ]) is compact. Applying Theorems 3.2 and 3.3, Lemma 2.8 and inequalities (2.5) and (4.1), we infer that 0 < λr0ψ ≤ λrψ = u ≤ u = Mφ in Ω and 1 uβ , 1 uβ ≤ 1 (λr0ψ(x))β ≤ C dsβ(x) , x ∈ Ωδ. 14 M. R. MARCIAL, OLIMPIO H. MIYAGAKI, G. A. PEREIRA EJDE-2022/60 To complete the proof, we consider three cases: (i) x ∈ S1. In this case, fΛ(u(x)) = f(u(x)). If x ∈ S1 ∩ Ωδ, we infer that |fΛ(u(x))| ≤ C uβ(x) ≤ C dsβ(x) . However, if x ∈ S1 ∩ Ωcδ, take positive numbers di (i = 1, 2) such that d1 ≤ u(x) ≤ d2, x ∈ Ωcδ. Hence |fΛ(u(x))| ≤ C dsβ(x) , x ∈ S1. (ii) x ∈ S2. In this case 0 < λr0ψ ≤ u ≤Mφ, and, as a consequence, |f(u(x))| ≤ C uβ(x) , x ∈ Ωδ. Hence, there is a positive constant C̃ such that |f(u(x))| ≤ C̃, x ∈ Ωcδ. Thus |f(u(x))| ≤ { C̃ if x ∈ Ωcδ, C dsβ(x) if x ∈ Ωδ. We also have 1 Dβ ≤ 1 dsβ(x) , x ∈ Ωcδ, and therefore there exist a constant C > 0 such that |f(u(x))| ≤ { C Dβ if x ∈ Ωcδ, C dsβ(x) if x ∈ Ωδ. Thus, |f(u(x))| ≤ C dsβ(x) , x ∈ S2. (iii) x ∈ S3. In this case fΛ(u(x)) = f(u(x)). The proof is similar to the case (i). � Remark 4.2. According to Proposition 2.5, Lemma 4.1 and Remark 2.6, for each v ∈ C(Ω) and λ ∈ IΛ, we have λfΛ(v) ∈ L∞loc(Ω) and |λfΛ(v)| ≤ CΛ dsβ(x) in Ω, (4.3) where CΛ > 0 is a constant independent of v and β ∈ (0, 1). So, for each v, (−∆p) su = λfΛ(v) in Ω, u = 0 on Ωc admits a unique solution u = S(λfΛ(v)) ∈W s,p 0 (Ω) ∩ Cα(Ω). EJDE-2022/60 TOPOLOGICAL STRUCTURE FOR A FRACTIONAL PROBLEM 15 Set FΛ(u)(x) = fΛ(u(x)), u ∈ C(Ω). and consider the operator T : IΛ × C(Ω)→W s,p 0 (Ω) ∩ Cα(Ω), defined by T (λ, u) = S(λFΛ(u)) if λ0 ≤ λ ≤ Λ, u ∈ C(Ω). Observe that, if (λ, u) ∈ IΛ×C(Ω) is such that u = T (λ, u), then u is a solution to the problem (−∆p) su = λfΛ(u) in Ω, u = 0 on Ωc. Lemma 4.3. If (λ, u) ∈ IΛ × C(Ω) and u = T (λ, u), then (λ, u) ∈ GΛ. Proof. Suppose that (λ, u) ∈ IΛ × C(Ω) satisfies T (λ, u) = u. Then∫∫ RN [u(x)− u(y)]p−1(v(x)− v(y)) |x− y|N+sp dxdy = λ ∫ Ω fΛ(u)v dx, ∀v ∈W s,p 0 (Ω). We claim that u ≥ u. Assume, by contradiction, that v := (u− u)+ 6≡ 0. Then∫∫ RN [u(x)− u(y)]p−1(v(x)− v(y)) |x− y|N+sp dx dy = ∫∫ u 0 such that GΛ ⊂ B(0, RΛ) ⊂ C(Ω) and T ( IΛ ×B(0, RΛ) ) ⊆ B(0, RΛ). Note that, by (4.3) and Lemma 4.3, if (λ, u) ∈ IΛ×C(Ω) satisfies u = T (λ, u) then (λ, u) is a solution of (Pλ). So, Remark 4.2 shows that it suffices to find a fixed point of T in order to solve (1.1). 16 M. R. MARCIAL, OLIMPIO H. MIYAGAKI, G. A. PEREIRA EJDE-2022/60 Lemma 4.6. The mapping T : IΛ ×B(0, RΛ)→ B(0, RΛ) is continuous and com- pact. Proof. Let {(λn, un)} ⊆ IΛ×B(0, RΛ) be a sequence such that λn → λ and un → u in C(Ω), as n→∞. Set vn = T (λn, un) and v = T (λ, u) so that vn = S(λnFΛ(un)) and v = S(λFΛ(u)). It follows that∫∫ RN [ [vn(x)− vn(y)]p−1 |x− y|N+sp − [v(x)− v(y)]p−1 |x− y|N+sp ] (vn(x)− v(y)) dx dy = λn ∫ Ω (fΛ(un)− fΛ(u)) (vn − v) dx ≤ C ∫ Ω |fΛ(un)− fΛ(u)|dx. Since |fΛ(un)− fΛ(u)| ≤ C dsβ(x) ∈ L1(Ω) and fΛ(un(x))→ fΛ(u(x)) a.e. x ∈ Ω, as n→∞, it follows that∫ Ω |fΛ(un)− fΛ(u)|dx→ 0, as n→∞. Therefore vn → v as n→∞ in W 1,p 0 (Ω). On the other hand, since un → u in C(Ω), as n → ∞, the proof of Lemma 4.1 shows that λnfΛ(un) ∈ L∞loc(Ω) and |λnfΛ(un)| ≤ CΛ dsβ(x) in Ω. Proposition 2.5 guarantees the existence of a constant M > 0 such that ‖vn‖Cα(Ω) ≤M, so that vn → v in C(Ω). This shows that T : IΛ × B(0, RΛ)→ B(0, RΛ) is contin- uous. The compactness of T is a consequence. � 5. Bounded connected sets of solutions of (1.1) We recall the Leray-Schauder Continuation Theorem (see [6]) for the convenience of the reader. Theorem 5.1. Let D be an open bounded subset of the Banach space X. Let a, b ∈ R with a < b and assume that T : [a, b]×D → X is compact and continuous. Consider Φ: [a, b]×D → X defined by Φ(t, u) = u− T (t, u). Assume that (i) Φ(t, u) 6= 0 for all t ∈ [a, b] and all u ∈ ∂D; (ii) deg ( Φ(t, .), D, 0 ) 6= 0 for some t ∈ [a, b] and set Sa,b = {(t, u) ∈ [a, b]×D : Φ(t, u) = 0}. Then, there exists a connected compact subset Σa,b of Sa,b such that Σa,b ∩ ({a} ×D) 6= ∅ and Σa,b ∩ ({b} ×D) 6= ∅. EJDE-2022/60 TOPOLOGICAL STRUCTURE FOR A FRACTIONAL PROBLEM 17 Consider Φ : IΛ ×B(0, R)→ B(0, R)) defined by Φ(λ, u) = u− T (λ, u). Lemma 5.2. Φ satisfies: (i) Φ(λ, u) 6= 0 ∀(λ, u) ∈ IΛ × ∂B(0, RΛ), (ii) deg(Φ(λ, .), B(0, RΛ), 0) 6= 0 for each λ ∈ IΛ, Proof. The verification of (i) is straightforward, since T ( IΛ×B(0, RΛ) ) ⊂ B(0, RΛ). To prove (ii), set R = RΛ, take λ ∈ IΛ and consider the homotopy Ψλ(t, u) = u− tT (λ, u), (t, u) ∈ [0, 1]×B(0, R). It follows that 0 /∈ Ψλ(I × ∂B(0, R)). In fact, if 0 ∈ Hλ(IΛ × ∂B(0, R)), then there exist t0 ∈ [0, 1] and u0 ∈ ∂B(0, R) such that u0 = t0T (λ, u0). Since u0 ∈ ∂B(0, R), we have t0 6= 0. And t0 6= 1 because u0 6= T (λ, u0). Therefore ‖u0‖ t0 = ‖T (λ, u0)‖ < ‖u0‖, which is a contradiction. The homotopy invariance of the Leray-Schauder degree guarantees that deg(Ψλ(t, .), B(0, R), 0) = deg(Ψλ(0, .), B(0, R), 0) = 1, t ∈ [0, 1]. Thus, deg(Φ(λ, .), B(0, R), 0) = 1, λ ∈ IΛ, completing the proof. � Theorem 5.3. There exist a number λ0 > 0 and a connected set ΣΛ ⊂ [λ0,Λ] × C(Ω) satisfying (i) ΣΛ ⊂ S; (ii) ΣΛ ∩ ( {λ0} × C(Ω) ) 6= ∅; (iii) ΣΛ ∩ ( {Λ} × C(Ω) ) 6= ∅ for each Λ > λ0. Proof. Maintaining the notation of Lemma 5.2, we apply Theorem 5.1 to the op- erator T . We have already proved that T is continuous, compact and T ( IΛ × B(0, RΛ) ) ⊂ B(0, RΛ). Set SΛ = { (λ, u) ∈ IΛ ×B(0, R) : Φ(λ, u) = 0 } ⊂ GΛ. By Theorem 5.1 there is a connected component ΣΛ ⊂ SΛ such that ΣΛ ∩ ({λ∗} ×B(0, R)) 6= ∅ and ΣΛ ∩ ({Λ} ×B(0, R)) 6= ∅. We point out that SΛ is the solution set of the auxiliary problem (−∆p) su = λfΛ(u) in Ω, u = 0 on Ωc and, since ΣΛ ⊂ SΛ ⊂ GΛ, it follows from the definition of fΛ that (−∆p) su = λf(u) in Ω, u = 0 on Ωc for (λ, u) ∈ ΣΛ, showing that ΣΛ ⊂ S. This completes the proof. � 18 M. R. MARCIAL, OLIMPIO H. MIYAGAKI, G. A. PEREIRA EJDE-2022/60 6. Proof of Theorem 1.1 Proof. Consider Λ as introduced in Section 4 and take a sequence {Λn} such that λ0 < Λ1 < Λ2 < . . . with Λn → ∞. Set βn = Λn and take a sequence {αn} ⊂ R such that αn → −∞ and · · · < αn < · · · < α1 < λ0. Keeping up the notation of Section 4, consider the sequence of intervals In = [λ0,Λn]. Set M = C(Ω) and GΛn := { (λ, u) ∈ In ×BRn : u ≤ u ≤ u, u = 0 on ∂Ω } , where Rn = RΛn . Look at the sequence of compact operators Tn : [λ0,Λn]×BRn → BRn defined by Tn(λ, u) = S(λFΛn(u))) if λ0 ≤ λ ≤ Λn, u ∈ BRn . Next, we consider the extension T̃n : R×BRn → BRn of Tn, defined by T̃n(λ, u) =  Tn(λ0, u) if λ ≤ λ0, Tn(λ, u) if λ0 ≤ λ ≤ Λn, Tn(Λn, u) if λ ≥ Λn. Observe that T̃n is continuous and compact. Applying Theorem 5.1 to T̃n : [αn, βn] × BRn → BRn we obtain a compact connected component Σ∗n of Sn = { (λ, u) ∈ [αn, βn]×BRn : Φn(λ, u) = 0 } , where Φn(λ, u) = u− T̃n(λ, u). Note that Σ∗n is also a connected subset of R ×M . According to Theorem 2.2, there exists a connected component Σ∗ of lim Σ∗n such that Σ∗ ∩ ({λ} ×M) 6= ∅ for each λ ∈ R. Set Σ = ([λ∗,∞)×M) ∩ Σ∗. Then Σ ⊂ R×M is connected and Σ ∩ ({λ} ×M) 6= ∅, λ0 ≤ λ <∞. We claim that Σ ⊂ S. Indeed, note that T̃n+1 ∣∣ [λ0,Λn]×BRn = T̃n ∣∣ [λ0,Λn]×BRn = Tn. (6.1) If (λ, u) ∈ Σ and λ > λ0, there is a sequence (λni , uni) ∈ ∪Σ∗n with (λni , uni) ∈ Σ∗ni such that λni → λ and uni → u asni →∞. Then u ∈ BRN for some integer N > 1. We can assume that (λni , uni) ∈ [λ0,ΛN ]×BRN . 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Song; A property of connected components and its applications, Topology Appl. 125 (2002), no. 3, 553–560. [17] G. T. Whyburn; Topological analysis, Princeton University Press, Princeton, 2015. Addendum posted on August 19, 2022 The authors want to insert the following lines at the end of Remark 2.1, and to add 3 references. These arguments were already used in [19, 20], one of them involving the p- Laplacian operator with singular term. Also [18] studied a nonlinear fourth-order operator with Navier boundary conditions. References [18] F. J. S. A. Correa, J. V. Gonçalvesm A. Roncalli: On a class of fourth order nonlinear elliptic equations under Navier boundary conditions, Analysis and Applications, 8, (2010), no. 2, 185–197. [19] Gonçalves,J. V., Marcial,M. R. Miyagaki, O. H.: Topological structure of the solution set of singular equations with sign changing terms under Dirichlet boundary condition. Topological Methods in Nonlinear Analysis, v. 47, p. 1-16, 2015 [20] Gonçalves, J. V.; Marcial, M. R.; Miyagaki, O. H.: Singular nonhomogeneous quasilinear elliptic equations with a convection term. Math. Nachr. 290 (2017), no. 14-15, 2280-2295. 20 M. R. MARCIAL, OLIMPIO H. MIYAGAKI, G. A. PEREIRA EJDE-2022/60 End of addendum. Marcos Roberto Marcial Universidade Federal de Ouro Preto, Departamento de Matemática, 35400-000 - Ouro Preto - MG, Brazil Email address: mrmarcial@ufop.edu.br Olimpio H. Miyagaki Departmento de Matemática, Universidade Federal de São Carlos, 13565-905 - São Car- los - SP, Brazil Email address: olimpio@ufscar.br, ohmiyagaki@gmail.com Gilberto A. Pereira Universidade Federal de Ouro Preto, Departamento de Matemática, 35400-000 - Ouro Preto - MG, Brazil Email address: gilberto.pereira@ufop.edu.br 1. Introduction 2. Auxiliary results 3. Lower and upper solutions 4. Finding a solution for (??) 5. Bounded connected sets of solutions of (??) 6. Proof of Theorem ?? Acknowledgments References Addendum posted on August 19, 2022 References