Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 72, pp. 1–24. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.72 MULTIPLICITY RESULTS FOR SCHRÖDINGER TYPE FRACTIONAL p-LAPLACIAN BOUNDARY VALUE PROBLEMS EMER LOPERA, LEANDRO RECÔVA, ADOLFO RUMBOS Dedicated to Djairo G. de Figueiredo on his 90th birthday Abstract. In this work, we study the existence and multiplicity of solutions to the problem −(∆)spu+ V (x)|u|p−2u = λf(u), x ∈ Ω; u = 0, x ∈ RN\Ω, where Ω ⊂ RN is an open bounded set with Lipschitz boundary ∂Ω, N ⩾ 2, V ∈ L∞(RN ), and (−∆)sp denotes the fractional p-Laplacian with s ∈ (0, 1), 1 < p, sp < N , λ > 0, and f : R → R is a continuous function. We extend the results of Lopera et al. [22] by proving the existence of a second weak solution to this problem. We apply a variant of the mountain-pass theorem due to Hofer [15] and infinite-dimensional Morse theory to obtain the existence of at least two solutions. 1. Introduction Let Ω be an open bounded set in RN , N ⩾ 2, with Lipschitz boundary ∂Ω. In this work, we study the existence and multiplicity of solutions for the problem −(∆)spu(x) + V (x)|u(x)|p−2u(x) = λf(u(x)), x ∈ Ω; u(x) = 0, x ∈ RN\Ω, (1.1) where V ∈ L∞(RN ), f : R → R is a continuous function and (−∆)sp denotes the fractional p-Laplacian defined by (−∆)spu(x) = 2 lim ε→0+ ∫ |x−y|>ε |u(x)− u(y)|p−2(u(x)− u(y)) |x− y|N+sp dy, (1.2) for x ∈ Ω, with s ∈ (0, 1), 1 < p, sp < N , and λ > 0. As pointed out by Lindgren and Lidqvist [21, page 801], it is not sufficient to prescribe the boundary values only on ∂Ω, but instead, we have to assume that u = 0 in the whole complement RN\Ω because a change in u done outside Ω can impact the fractional p-Laplacian operator (−∆)sp. For more details, see Nezza et al. [12], Lindgren et al. [21], and references therein. 2020 Mathematics Subject Classification. 35J20. Key words and phrases. Mountain pass theorem; Morse theory; critical groups; comparison principle. ©2024. This work is licensed under a CC BY 4.0 license. Submitted July 15, 2024. Published November 11, 2024. 1 2 E. LOPERA, L. RECÔVA, A. RUMBOS EJDE-2024/72 In this work, the functions f and V satisfy the following hypotheses: (H1) Assume that p− 1 < q < p∗s − 1, where p∗s := Np N−sp is the fractional critical Sobolev exponent, and there exist A,B > 0 such that A(sq − 1) ⩽ f(s) ⩽ B(sq + 1), for s > 0, (1.3) f(s) = 0, for s ⩽ −1. (1.4) (H2) There exist θ > p and K ∈ R such that f satisfies the Ambrosetti- Rabinowitz type condition sf(s) ⩾ θF (s) +K, for all s ∈ R, (1.5) where F (s) = ∫ s 0 f(ξ) dξ, for s ∈ R, is the primitive of f . (H3) V ∈ L∞(RN ) and V (x) ⩾ −cV , for a.e. x ∈ RN , where 0 < cV < λ1 and λ1 is the first eigenvalue of ((−∆)sp,W s,p 0 (Ω)). The first two results establish the existence and multiplicity of solutions for problem (1.1) when f(0) ̸= 0. Theorem 1.1. Assume that Ω is a bounded domain with a Lipschitz boundary ∂Ω and (H1)–(H3) are satisfied with f(0) ̸= 0. Then there exists λ0 > 0 such tha, for all λ ∈ (0, λ0), problem (1.1) has at least two solutions. To obtain a positive solution, we need to assume that p ⩾ 2 to get enough regularity of solutions up to the boundary of Ω and V (x) ⩾ 0, for a.e x ∈ Ω. In this case, we obtain the following multiplicity result. Theorem 1.2. In addition to the hypotheses of Theorem 1.1, assume that V (x) ⩾ 0 for a.e. x ∈ Ω, p ⩾ 2, Ω is bounded and satisfies the interior ball condition at any x ∈ ∂Ω, and p− 1 < q < min {sp N p∗s, p ∗ s − 1 } . Then, there exists λ∗ > 0 such that, for all 0 < λ < λ∗, problem (1.1) has at least two solutions. Moreover: (a) If f(0) > 0, then both solutions are positive. (b) If f(0) < 0, then at least one of the solutions is positive. Remark 1.3. Observe that statement (b) encompasses the semipositone case. See, for example, Castro et al. [8] and references therein. When u ≡ 0 is a solution of problem (1.1), called the trivial solution, to obtain a multiplicity result, we need an additional condition on the primitive of f . Theorem 1.4. Assume that Ω is a bounded domain with a Lipschitz boundary ∂Ω and (H1)–(H3) are satisfied. Moreover, assume that f(0) = 0 and lim sup s→0 F (s) |s|p = 0. Then there exists λ0 > 0 such that for all λ ∈ (0, λ0) problem (1.1) has at least two nontrivial solutions. Problems involving the fractional p-Laplacian have been an object of intensive research in the last years in many branches of science such as in phase transition phenomena, population dynamics, and game theory (see [2, 7, 11, 12, 16, 17, 18, 19, 21, 22, 25, 29]). Valdinoci [30] presents a self-contained exposition on how a EJDE-2024/72 MULTIPLICITY RESULTS FOR FRACTIONAL p-LAPLACIAN 3 simple random walk with possibly long jumps is related to the fractional p-Laplacian operator. For more insights on the applications, we refer to Iannizzotto et al. [16] and Caffarelli [7] where the authors provide a detailed review of current applications and challenges faced when dealing with these nonlocal operators. This article was motivated by the results obtained by Castro et al. [8] for the case of the p-Laplacian operator and by Lopera et al. [22] for the fractional p- Laplacian. In those articles, the authors proved the existence of a positive solution for problem (1.1) when the potential V ≡ 0. The existence result was obtained by showing that the associated energy functional for problem (1.1) had the geometry of the mountain-pass theorem of Ambrosetti-Rabinowitz [1]. They also proved that the solution was positive by using some new regularity results and Hopf’s Lemma. The main goal of this work is to extend the results of Lopera et al. [22] by proving the existence of at least two solutions for problem (1.1). We will use a variant of the mountain-pass theorem due to Hofer [15] and infinite-dimensional Morse theory to obtain the existence of a second solution for both cases where f(0) ̸= 0 and f(0) = 0, respectively. This article is organized as follows: In Section 2 we present some preliminary results that will be used throughout this work. In Section 3, we prove that the associated energy functional to problem (1.1) has a critical point uλ of mountain- pass type. In Section 4, we apply infinite-dimensional Morse theory to compute the critical groups of the associated energy functional at infinity. In Section 5, we compute the critical groups of the associated energy functional for problem (1.1) at the origin. Finally, we prove the existence and multiplicity results in Section 6. 2. Preliminaries In this work, we will use a variational approach to study the existence and multi- plicity of solutions for problem (1.1). We start with some notation and preliminary results that will be used throughout this article. Let Ω be an open bounded subset of in RN , N ⩾ 2, with boundary ∂Ω. Denote by C(Ω) the set of continuous functions on Ω. The space of γ-Hölder continuous functions is defined by Cγ(Ω) = {u ∈ C(Ω) : [u]Cγ(Ω) < ∞}, where 0 < γ ⩽ 1 and [u]Cγ(Ω) = sup x,y∈Ω,x ̸=y |u(x)− u(y)| |x− y|γ . The space Cγ(Ω) is a Banach space endowed with the norm ∥u∥Cγ(Ω) = ∥u∥L∞(Ω) + [u]Cγ(Ω). In some of the regularity results that will be used in this article, it will be required that the domain Ω ⊂ RN , N ⩾ 2, be a Lipschitz domain. This is the content of the next definition. Definition 2.1. We will say that Ω ⊂ RN has a Lipschitz boundary, and call it a Lipschitz domain, if, for every x0 ∈ ∂Ω, there exists r > 0 and a map h : Br(x0) → B1(0) such that (i) h is a bijection, (ii) h and h−1 are both Lipschitz continuous functions, 4 E. LOPERA, L. RECÔVA, A. RUMBOS EJDE-2024/72 (iii) h(∂Ω ∩Br(x0)) = Q0, (iv) h(Ω ∩Br(x0)) = Q+, where Br(x0) denotes the n−dimensional open ball of radius r and center at x0 ∈ ∂Ω, and Q0 := {(x1, . . . , xn) ∈ B1(0) |xn = 0}, Q+ := {(x1, . . . , xn) ∈ B1(0) |xn > 0}. Next, we introduce the space of functions where the energy functional associated with problem (1.1) will be defined. Let s ∈ (0, 1) and 1 ⩽ p < ∞, and denote by W s,p 0 (Ω) = {u ∈ W s,p(RN ) : u = 0 a.e in RN\Ω} (2.1) the subset of the fractional Sobolev space W s,p(RN ), W s,p(RN ) = { u ∈ Lp(RN ) : ∫ R2N |u(x)− u(y)|p |x− y|N+sp dx dy < ∞ } , endowed with the norm ∥u∥s,p := ( ∥u∥pp + [u]ps,p )1/p , (2.2) where ∥ · ∥p denotes the norm in Lp(Ω) for 1 ⩽ p < ∞ and [u]s,p := ∫ R2N |u(x)− u(y)|p |x− y|N+sp dx dy, (2.3) is the Gagliardo seminorm. It can be shown thatW s,p(RN ), endowed with the norm ∥ · ∥s,p defined in (2.2) and (2.3), is a Banach space, and W s,p 0 (Ω) ⊂ W s,p(RN ) is a closed subspace. In the case 1 < p < ∞, W s,p(Ω) is a reflexive Banach space (see Asso et al. [2, Section 2.1]). By a Sobolev-type inequality (see [12, Theorem 6.7]), it can be shown that the space W s,p 0 (Ω) can also be endowed with the norm ∥u∥ := [u]s,p, (2.4) for s ∈ (0, 1) and 1 ⩽ p < ∞. We will denote by W̃ s,p(Ω) the Sobolev space{ u ∈ Lp loc(R N ) : ∃U ⊃⊃ Ω s.t ∥u∥W s,p(U) + ∫ RN |u(x)|p−1 (1 + |x|)N+ps dx < ∞ } , where Ω ⊂ RN is a bounded set (see [19, Definition 2.1] for more details). Since Ω is a bounded set, it follows from [11, Remark 1.1] that W s,p 0 (Ω) ⊂ W̃ s,p(Ω). We will refer to the space W̃ s,p(Ω) during the proof of a comparison principle for problem (1.1). For more details on fractional Sobolev spaces, see [12, Section 2], [6], and references therein. In this article, we shall denote by X the fractional Sobolev space W s,p 0 (Ω). We define Jλ : X → R, the energy functional associated with problem (1.1), by Jλ(u) = 1 p ∥u∥p + 1 p ∫ Ω V (x)|u|p dx− λ ∫ Ω F (u) dx, for u ∈ X, (2.5) and λ > 0 with ∥ · ∥ defined in (2.4). EJDE-2024/72 MULTIPLICITY RESULTS FOR FRACTIONAL p-LAPLACIAN 5 The functional Jλ is well-defined and Jλ ∈ C1(X,R). It can be shown that the Fréchet derivative of Jλ is given by ⟨J ′ λ(u), φ⟩ = ∫ R2N Φp(u(x)− u(y))(φ(x)− φ(y)) |x− y|N+sp dx dy + ∫ Ω V (x)|u|p−2uφdx− λ ∫ Ω f(u)φdx, (2.6) for all φ ∈ X, where Φp : R → R is given by Φp(s) = |s|p−2s, for s ∈ R. We will say that u is a weak solution of problem (1.1) if u is a critical point of Jλ; namely, ⟨J ′ λ(u), φ⟩ = 0, for all φ ∈ X. (2.7) For every 1 < p1 < p∗s, we shall denote by Cp1 the optimal constant in the Sobolev embedding theorem; namely, ∥u∥p1 ⩽ Cp1 ∥u∥, for all u ∈ X, (2.8) see [12, Theorem 6.7]. In the proof of the existence of a solution of mountain-pass type, we will need the following result due to Lindgren and Lindqvist [21]. Theorem 2.2 ([21, Thm. 5]). There exists a non-negative minimizer u inW s,p 0 (Ω), u ̸≡ 0, and u = 0 in RN\Ω of the fractional Rayleigh quotient: λ1 = inf u∈W s,p 0 (Ω)\{0} ∫ RN ∫ RN |u(u)−u(x)|p |y−x|αp dx dy∫ RN |u(x)|p dx . (2.9) It satisfies the Euler-Lagrange equation∫ R2N |u(y)− u(x)|p−2(u(y)− u(x))(φ(y)− φ(x)) |y − x|αp dx dy = λ ∫ RN |u|p−2uφdx, (2.10) with λ = λ1 whenever φ ∈ C∞ c (Ω). If αp > 2N , the minimizer is in C0,β(RN ) with β = α− 2N/p. Theorem 2.2 motivated the following definition. Definition 2.3 ([21, Definition 6]). We say that u ̸≡ 0, u ∈ W s,p 0 (Ω), s = α−n/p, is an eigenfunction of Ω, if the Euler-Lagrange equation (2.10) holds for all test functions φ ∈ C∞ c (Ω). The corresponding λ is called an eigenvalue. Remark 2.4. The minimizer found in Theorem 2.2 is called the first eigenfunction of ((−∆)sp,W s,p 0 (Ω)). To use some of the minimax theorems in the literature, we have to check that the associated energy functional also satisfies some kind of compactness condition. Definition 2.5. We will say that (un) ⊂ X is a PS-sequence for J if |J(un)| ⩽ C for all n, and J ′(un) → 0 as n → ∞, where C is a positive constant. We say that a functional J ∈ C1(X,R) satisfies the Palais-Smale condition (PS-condition) if any PS-sequence (un) ⊂ X possesses a convergent subsequence. 6 E. LOPERA, L. RECÔVA, A. RUMBOS EJDE-2024/72 To prove the existence of a second solution for (1.1) in Theorem 1.1, we will need the concept of critical groups from infinite-dimensional Morse Theory. Define Jc λ = {w ∈ X| Jλ(w) ⩽ c}, the sub-level set of Jλ at c, and set K = {u ∈ X| J ′ λ(u) = 0}, the critical set of Jλ. For an isolated critical point u0 of Jλ, the q–critical groups of Jλ at u0, with coefficients in a field F of characteristic 0, are defined by Ck(Jλ, u0) = Hk(J c0 λ ∩ U, Jc0 λ ∩ U\{u0}), for all k ∈ Z, where c0 = Jλ(u0), U is a neighborhood of u0 that contains no critical points of Jλ other than u0, and H∗ denotes the singular homology groups. The critical groups are independent of the choice of U by the excision property of homology (see Hatcher [14]). For more information on the definition of critical groups, we refer the reader to [9, 26, 24, 23]. Next, we present the concept of the critical groups at infinity introduced by Bartsch and Li in [4]. Assume that Jλ ∈ C1(X,R) satisfies the Palais-Smale con- dition. Let K = {u ∈ X : J ′ λ(u) = 0} be the set of critical points of Jλ and assume that under these assumptions the critical value set is bounded from below; that is, ao < inf Jλ(K), for some ao ∈ R. The critical groups at infinity are defined by Ck(Jλ,∞) = Hk(X, Jao λ ), for all k ∈ Z, (2.11) (see [4]). These critical groups are well-defined as a consequence of the Second Deformation Theorem (see Perera and Schechter [26, Lemma 1.3.7]). In this work, we use the concept of a critical point of a functional being of a mountain-pass type. We use the definition found in Hofer [15] and Montreanu et al. [24]. Definition 2.6 ([24, Definition 6.98]). Let X be a Banach space, J ∈ C1(X,R), and u0 ∈ K. We say that u0 is of mountain-pass type if, for any open neighborhood U of u0, the set {w ∈ U | J(w) < J(u0)} is nonempty and not path-connected. The critical groups of mountain-pass type can be described by the following proposition found in Montreanu et al. [24]: Proposition 2.7 ([24, Proposition 6.100]). Let X be a reflexive Banach space, J ∈ C1(X,R), and u0 ∈ K be isolated in J(K). If u0 is of mountain-pass type, then C1(J, u0) ̸∼= 0. Put Kd = {u ∈ X|J(u) = d, J ′(u) = 0}, the critical set at level d. One of the critical points that will be obtained in the proof of Theorem 1.1 satisfies a variant of the mountain-pass theorem due to Hofer, which we present next for the reader’s convenience. Theorem 2.8 ([15]). Assume that X is a real Banach space. Let J ∈ C1(X,R) satisfy the Palais-Smale condition and assume that e0 and e1 are distinct points in X. Define A = {a ∈ C([0, 1], X)| : a(i) = ei, for i = 0, 1}, (2.12) d = inf a∈A sup J(|a|), |a| = a([0, 1]), c = max{J(e0), J(e1)}. (2.13) EJDE-2024/72 MULTIPLICITY RESULTS FOR FRACTIONAL p-LAPLACIAN 7 If d > c, the set Kd is non-empty. Moreover, there exists at least one critical point u0 in Kd that is either a local minimum or of mountain-pass type. If all the critical points in Kd are isolated in X the set Kd contains a critical point of mountain-pass type. Remark 2.9. Once we prove that the functional Jλ defined in (2.5) satisfies the conditions of Theorem 2.8 in Section 4, for the case f(0) ̸= 0, assuming that Jλ has only one critical point uλ, it will follow from Proposition 2.7 that C1(Jλ, uλ) ̸∼= 0. (2.14) We do not have information about the other critical groups Ck(Jλ, uλ) when k ̸= 1. But, the fact that C1(Jλ, uλ) is nontrivial will be enough to prove the existence of a second critical point for the functional Jλ. A similar argument will be used in Section 6 for the case f(0) = 0. Finally, the last result we will need to prove multiplicity results for problem (1.1) for the case f(0) ̸= 0 is found in Bartsch and Li [4]. Proposition 2.10 ([4, Proposition 3.6]). Suppose that J ∈ C1(X,R) satisfies the Palais-Smale condition at level c for every c ∈ R. If K = ∅, then Ck(J,∞) ∼= 0 for all k ∈ Z. If K = {uλ}, then Ck(J,∞) ∼= Ck(J, uλ), for all k ∈ Z. We shall prove in Section 4 that Ck(Jλ,∞) ∼= 0 for all k > 0; that is, the critical groups of Jλ at infinity are all trivial for k ̸= 0. In particular, we will have C1(Jλ,∞) ∼= 0. Hence, assuming, by a way of contradiction, that Jλ has only the critical point uλ found in Section 3, we will then obtain a contradiction based on the result of Proposition 2.10 and the assertion in (2.14). In the next section, we will prove the existence of a mountain-pass type solution for problem (1.1). 3. Existence and a priori estimates 3.1. Existence of a mountain-pass type solution. In this section, we show that the functional Jλ defined in (2.5) satisfies the conditions of the variant of the mountain-pass theorem due to Hofer [15] as presented in Theorem 2.8. First, by conditions (1.3) and (1.4), it can be shown that there exists B1 > 0 such that F (s) ⩽ B1(|s|q+1 + 1), for all s ∈ R. (3.1) It also follows from (1.3) and (1.4) that, for all s ⩾ 0, there exist A1, C1 > 0 such that F (s) ⩾ A1(s q+1 − C1), for all s ⩾ 0. (3.2) In what follows, let r > 0 be the positive number r = 1 q + 1− p , (3.3) where p, q satisfy the conditions in hypothesis (H1). In the next two lemmas, we prove the geometric conditions in Theorem 2.8. Lemma 3.1. There exist τ > 0, c1 > 0 and λ̂2 ∈ (0, 1) such that if ∥u∥ = τλ−r then Jλ(u) ⩾ c1(τλ −r)p for all λ ∈ (0, λ̂2), where r is given in (3.3). 8 E. LOPERA, L. RECÔVA, A. RUMBOS EJDE-2024/72 Proof. By the Sobolev embedding theorem and hypothesis (H3), it follows from the definition of Jλ in (2.5) that Jλ(u) = 1 p ∥u∥p + 1 p ∫ Ω V (x)|u|p dx− λ ∫ Ω F (u)dx ⩾ 1 p ∥u∥p − cV λ1p ∥u∥p − λB1C q+1 q+1∥u∥q+1 − λB1|Ω|, (3.4) for all u ∈ X. Let τ > 0 be a small enough constant such that the following identity is satisfied: 1− cV λ1 = 3 2 pCq+1 q+1B1τ q+1−p. (3.5) Next, setting ∥u∥ = λ−rτ in (3.4) and using that r(q + 1) + 1 = −rp, we obtain Jλ(u) ⩾ λ−rp [τp p ( 1− cV λ1 ) −B1C q+1 q+1τ q+1 − λ1+rpB1|Ω| ] , (3.6) for all u ∈ X. Then, by (3.5), it follows from (3.6) that Jλ(u) ⩾ λ−rp (1 2 pCq+1 q+1B1τ q+1−p − λ1+rp|Ω|B1 ) , (3.7) for all u ∈ X. Finally, choose λ ∈ (0, λ̂2) with λ̂2 := τp/(1+rp)(4pB1|Ω|)−1/(1+rp). Then, for this choice of λ, we obtain from (3.7) that Jλ(u) ⩾ c1(τλ −r)p; for u ∈ X, where c1 = 1 4p . This completes the proof. □ Lemma 3.2. Let φo ∈ X be such that φo > 0 and ∥φo∥ = 1. There exists λ̂1 > 0 such that if λ ∈ (0, λ̂1) then Jλ(cλ −rφo) ⩽ 0, where r is given by (3.3). Proof. Set ℓ = cλ−r, where c, λ > 0 are positive constants to be chosen shortly. Then, by hypothesis (H1), the estimate (3.2), and the characterization of the first eigenvalue of the fractional p-Laplacian from Theorem 2.2, we obtain Jλ(ℓφo) = 1 p ∥ℓφo∥p + 1 p ∫ Ω V (x)|ℓφo|p dx− λ ∫ Ω F (ℓφo) dx ⩽ ℓp p ∥φo∥p + ℓp p ∥V ∥∞∥φo∥pp − λA1ℓ q+1 ∫ Ω φq+1 o dx+ λA1C1|Ω| ⩽ ℓp p ( 1 + 1 λ1 ∥V ∥∞∥φo∥p − λpA1ℓ q+1−p∥φo∥q+1 q+1 ) + λA1C1|Ω|. (3.8) Next, we define c > 0 such that cq+1 = 2cp pA1∥φo∥q+1 q+1 ( 1 + 1 λ1 ∥V ∥∞ ) . (3.9) Then, by (3.9) and the definition of ℓ, it follows from (3.8) that Jλ(ℓφo) ⩽ λ−rp c p p [ − ( 1 + 1 λ1 ∥V ∥∞ ) + λ1+rpA1C2|Ω| ] . (3.10) We set λ̂1 = [1 + 1 λ1 ∥V ∥∞ 2pA1C2|Ω| ] 1 1+rp . EJDE-2024/72 MULTIPLICITY RESULTS FOR FRACTIONAL p-LAPLACIAN 9 Then, it follows from (3.10) that Jλ(ℓφo) ⩽ − cp 2p λ−rp ⩽ 0, for all λ ∈ (0, λ̂1), which establishes the lemma. □ In the next lemma, we will show that the functional Jλ satisfies the Palais-Smale condition. Lemma 3.3. Assume that (H1)–(H3) are satisfied and λ ∈ (0, λ3) with λ3 := min{λ̂1, λ̂2}, where λ̂1 is given by Lemma 3.2 and λ̂2 is given by Lemma 3.1. Then, Jλ satisfies the Palais-Smale condition. Proof. Let (un) be a Palais-Smale sequence for Jλ in X; that is, |Jλ(un)| ⩽ C, for all n; (3.11) where C > 0 is a constant and there exists a sequence of positive numbers (εn) such that ⟨J ′ λ(un), φ⟩ ⩽ εn∥φ∥, for all n, (3.12) and all φ ∈ X and εn → 0 as n → ∞. In particular, setting φ = un in (3.12), we obtain that there exists N1 > 0 such that |⟨J ′ λ(un), un⟩| ⩽ ∥un∥, for all n ⩾ N1. Hence, we can write −∥un∥p − ∥un∥ ⩽ −∥un∥p + ⟨J ′(un), un⟩, for n ⩾ N1. (3.13) Thus, using the definition of the Fréchet derivative of Jλ given in (2.6), and the definition of the norm ∥ · ∥ given in (2.4) and (2.3), we obtain from (3.13) that −∥un∥p − ∥un∥ ⩽ ∫ Ω V (x)|un|pdx− λ ∫ Ω f(un)un dx, (3.14) for n ⩾ N1. On the other hand, using the estimate in (1.5) in hypothesis (H2), we have that 1 p ∥un∥p − λ θ ∫ Ω f(un)un dx+ λ θ K|Ω| ⩽ 1 p ∥un∥p − λ ∫ Ω F (un) dx for n ∈ N; so that, using the definition of Jλ in (2.5), 1 p ∥un∥p − λ θ ∫ Ω f(un)un dx+ λ θ K|Ω| ⩽ Jλ(un)− 1 p ∫ Ω V (x)|un|p, dx, (3.15) for all n ∈ N. Now, it follows from (3.15) and the hypothesis in (3.11) that 1 p ∥un∥p − λ θ ∫ Ω f(un)un dx+ λ θ K|Ω| ⩽ C − 1 p ∫ Ω V (x)|un|p dx, (3.16) for n ∈ N. Next, we multiply on both sides of the estimate in (3.14) by 1 θ and add 1 p∥un∥p on both sides of the inequality to obtain(1 p − 1 θ ) ∥un∥p − 1 θ ∥un∥ ⩽ 1 p ∥un∥p + 1 θ (∫ Ω V (x)|un|p dx− λ ∫ Ω f(un)un dx ) , (3.17) 10 E. LOPERA, L. RECÔVA, A. RUMBOS EJDE-2024/72 for n ⩾ N1. It follows from the estimate (3.16) that 1 p ∥un∥p + 1 θ (∫ Ω V (x)|un|p dx− λ ∫ Ω f(un)un dx ) ⩽ C − (1 p − 1 θ ) ∫ Ω V (x)|un|p dx, (3.18) for n ∈ N. Consequently, combining the estimates in (3.17) and (3.18),(1 p − 1 θ ) ∥un∥p − 1 θ ∥un∥ ⩽ C − (1 p − 1 θ ) ∫ Ω V (x)|un|p dx, (3.19) for n ⩾ N1. Next, use the estimate for the potential V in hypothesis (H3) to obtain from (3.19) that(1 p − 1 θ ) ∥un∥p − 1 θ ∥un∥ ⩽ C + (1 p − 1 θ )cV λ1 ∥un∥p, for n ⩾ N1, (3.20) where we have used the definition of λ1 in (2.9). Rearranging (3.20) we obtain(1 p − 1 θ )( 1− cV λ1 ) ∥un∥p − 1 θ ∥un∥ ⩽ C, for n ⩾ N1, from which we obtain that (un) is bounded in W s,p 0 (Ω). Hence, since (un) is bounded in X, we may invoke the Banach-Alaoglu theorem (see [20, Theorem 2.18]) to deduce, passing to a subsequence if necessary, that there exists u ∈ X such that un ⇀ u weakly in X as n → ∞. Furthermore, since 1 < q+1 < p∗, by the Sobolev embedding theorem, we can also assume that un → u in Lq+1(Ω) as n → ∞ un(x) → u(x) a.e. in Ω as n → ∞. (3.21) Next, we put q′1 = q+1 q ; so that, q′ > 1, and q′q = q + 1. Hence, by (3.1) we obtain |f(un)|q ′ 1 ⩽ B q′1 1 (|un|q + 1)q ′ 1 ⩽ C1(|un|qq ′ 1 + 1) ⩽ C1(|un|q+1 + 1), (3.22) for all n ∈ N, where C1 is a positive constant. Thus, applying Hölder’s inequality with exponent q′1 in (3.22) and its conjugate, we obtain λ ∫ Ω f(un)(un − u) dx ⩽ C(∥un∥q+1 + 1)∥un − u∥q+1 ⩽ C∥un − u∥q+1, where C is a positive constant. Consequently, letting n → ∞ in the previous estimate and applying (3.21) with the Lebesgue dominated convergence theorem, we obtain λ ∫ Ω f(un)(un − u)dx → 0, as n → ∞. (3.23) Next, we put p′ = p p−1 (recall that we are assuming p > 1); so that p′ > 1 and p′(p− 1) = p. Then, by Hölder’s inequality we have∫ Ω |V (x)||un|p−1|un−u| dx ⩽ ∥V ∥∞∥un∥p−1 p ∥un−u∥p ⩽ C∥un−u∥p ⩽ C∥un−u∥q+1, EJDE-2024/72 MULTIPLICITY RESULTS FOR FRACTIONAL p-LAPLACIAN 11 for all n ∈ N, where C is a positive constant. Hence, letting n → ∞ in the previ- ous estimate and applying eqrefpss13 with the Lebesgue dominated convergence theorem, we obtain∫ Ω |V (x)||un|p−1|un − u| dx → 0, as n → ∞ . (3.24) Next, since (un) is a Palais-Smale sequence in X, it follows from (3.12), (3.23), and (3.24) that lim n→∞ ∫ R2N Φp(un(x)− un(y))((un − u)(x)− (un − u)(y)) |x− y|N+sp dx = 0. (3.25) Once again, using the fact that u is the weak limit of un we have lim n→∞ ∫ R2N Φp(u(x)− u(y))((un − u)(x)− (un − u)(y)) |x− y|N+sp dx = 0. (3.26) On the other hand, it follows from appling Hölder’s inequality as in [22, Lemma 3] that∫ Ω Φp(un(x)− un(y))− Φp(u(x)− u(y)) |x− y|N+sp ((un − u)(x)− (un − u)(y)) dx dy = ∫ Ω [ |un(x)− un(y)|p |x− y|N+sp − Φp(un(x)− un(y))(u(x)− u(y)) |x− y|N+sp − Φp(u(x)− u(y))(un(x)− un(y)) |x− y|N+sp + |u(x)− u(y)|p |x− y|N+sp ] dx dy ⩾ ∥un∥p − ∥un∥p−1∥u∥ − ∥un∥∥u∥p−1 + ∥u∥p = (∥un∥p−1 − ∥u∥p−1)(∥un∥ − ∥u∥). (3.27) Then, in view of ( ∥un∥p−1 − ∥u∥p−1 ) (∥un∥ − ∥u∥ ⩾ 0, for all n, it follows from (3.25),(3.26), and (3.27) that lim n→∞ ([∥un∥p−1 − ∥u∥p−1)(∥un∥ − ∥u∥) = 0, from which we obtain lim n→∞ ∥un∥ = ∥u∥. (3.28) Finally, by (3.28) and that un ⇀ u weakly in X, we conclude that un → u strongly in X. Hence, Jλ satisfies the Palais-Smale condition. □ Next, we present the main result of this section. Theorem 3.4. Assume that (H1)–(H3) are satisfied. Then, for λ sufficiently small, the functional Jλ has a critical point uλ ∈ X of mountain-pass type. Moreover, c1λ −rp ⩽ Jλ(uλ) ⩽ c2λ −rp, (3.29) where c1 and c2 are positive constants independent of λ, and r is given in (3.3). Proof. It follows from Lemmas 3.8, 3.2, 3.3, that, for each λ ∈ (0, λ3), the functional Jλ defined in (2.5) satisfies the conditions of Theorem 2.8. Therefore, Jλ possesses a critical point, uλ, with critical value characterized by Jλ(uλ) = inf a∈A max Jλ(|a|), 12 E. LOPERA, L. RECÔVA, A. RUMBOS EJDE-2024/72 with A = {a ∈ C([0, 1], X) : a(0) = 0, a(1) = cλ−rφo}, where a(1) is obtained in Lemma 3.2 and |a| = a([0, 1]). Furthermore, by Lemma 3.2, we observe that Jλ(scλ −rφo) ⩽ c2λ −rp, for 0 ⩽ s ⩽ 1, where c0 is a positive constant independent of λ. Hence, we conclude that Jλ(uλ) ⩽ c2λ −rp. Finally, it follows from Lemma 3.1 that there exists a positive constant c1 indepen- dent of λ such that c1λ −rp ⩽ Jλ(u), for all ∥u∥ = τλ−r. Then, it follows from the characterization of the critical value that c1λ −rp ⩽ Jλ(uλ). This completes the proof. □ The next two results will be used in the proof of a comparison principle for problem (1.1). Lemma 3.5. Assume that (H1)–(H3) are satisfied and let uλ be the mountain-pass critical point of Jλ given in Theorem 3.4. There exists a constant c such that ∥uλ∥ ⩽ cλ−r. (3.30) and r is given in (3.3). Proof. Let uλ be a critical point of Jλ given by Theorem 3.4. Then, it follows from (2.7) that ⟨J ′ λ(u), φ⟩ = 0, for all φ ∈ X. (3.31) Then, setting φ = uλ in (3.31) and using (2.6), we obtain ∥uλ∥p + ∫ RN V (x)|uλ|p dx = λ ∫ Ω f(uλ)uλ dx. It then follows from the Ambrosetti-Rabinowitz type condition in (1.5) that(1 p − 1 θ ) ∥uλ∥p = 1 p ∥uλ∥p − 1 θ ( λ ∫ Ω f(uλ)uλ dx− ∫ RN V (x)|uλ|p dx ) ⩽ 1 p ∥uλ∥p − λ θ (∫ Ω θF (uλ) dx+K|Ω| ) + 1 θ ∫ RN V (x)|uλ|p dx ⩽ 1 p ∥uλ∥p − λ ∫ Ω F (uλ) dx+ 1 p ∫ RN V (x)|uλ|p dx− λK θ |Ω| ⩽ Jλ(uλ) + Cλ−rp; so that, using (3.29) in Theorem 3.4, (3.30) follows. □ Finally, we present lower and upper estimates for ∥uλ∥∞, where uλ is the critical point obtained in Theorem 3.4. These results will be used in the proof of comparison principle for problem (1.1). EJDE-2024/72 MULTIPLICITY RESULTS FOR FRACTIONAL p-LAPLACIAN 13 Lemma 3.6. Assume that (H1)–(H3) are satisfied. Let uλ be a weak solution of problem (1.1) obtained via Theorem 3.4 and λ3 be as in Lemma 3.3. Then, there exists a constant C such that, for all 0 < λ < λ3, Cλ−r ⩽ ∥uλ∥∞, (3.32) and r is given in (3.3). Proof. By estimate in (3.29) for Jλ(uλ) in Theorem 3.4, and that minF > −∞, we obtain λ ∫ Ω f(uλ)uλ dx = ∥uλ∥p + ∫ RN V (x)|u|p dx = pJλ(uλ) dx+ pλ ∫ Ω F (uλ) dx ⩾ pCλ−rp + pλ|Ω|minF ⩾ Cλ−rp. (3.33) On the other hand, by the growth of f in (1.3), we obtain λ ∫ Ω f(uλ)uλ dx ⩽ Bλ∥uλ∥q+1 ∞ .f (3.34) Combining the estimates (3.33) and (3.34), we obtain (3.32). □ In the proof of the comparison principle, we will need the following regularity result found in Mosconi et al. [25]. Lemma 3.7 ([25, Lemma 2.3]). Let g ∈ Lt(Ω), N/(sp) < t ⩽ ∞ and u ∈ W 1,p 0 (Ω) be a weak solution of (−∆)ps = g in Ω. Then ∥u∥∞ ⩽ C∥g∥1/(p−1) t . The following theorem due to Ianizotto et al. [17] establishes a sharp boundary regularity result for the fractional p-Laplacian, for p ⩾ 2. The assumption of p ⩾ 2 will allow us to obtain enough regularity up to the boundary of Ω to obtain a positive solution for (1.1). Theorem 3.8 ([17, Theorem 1.1]). Let p ⩾ 2, Ω be a bounded domain with C1,1 boundary and d(x) = dist(x, ∂Ω). There exist α ∈ (0, s) and C > 0 depending on N,Ω, p and s, such that, for all g ∈ L∞(Ω), a weak solution u ∈ W s,p 0 (Ω) of the problem (−∆)sp(u) = g; in Ω, u = 0 in RN\Ω, satisfies u/ds ∈ Cα(Ω) and ∥ u ds ∥Cα(Ω) ⩽ C∥g∥ 1 p−1 ∞ . Finally, we present the last result of this section that will be used to prove the existence of a positive solution for problem (1.1). Lemma 3.9. Assume that (H1)–(H3) are satisfied. Let λ3 > 0 be as in Lemma 3.3. Then, there exist α ∈ (0, s] and a constant C > 0 such that, for all 0 < λ < λ3, 14 E. LOPERA, L. RECÔVA, A. RUMBOS EJDE-2024/72 the solution uλ given in Theorem 3.4 of the problem (1.1) satisfies uλ/d s ∈ Cα(Ω). Furthermore ∥uλ∥∞ ⩽ Cλ−r,∥∥uλ ds ∥∥ Cα(Ω) ⩽ Cλ−r. and r is given in (3.3). Proof. It follows from the assumption Nq/(sp) < p∗s that there exists t > 1 such that N sp < t and tq < p∗s, which implies t(p− 1) < p∗s. Set g := λf ◦ uλ + V Φp(uλ). Since W s,p 0 (Ω) ↪→ Ltq(Ω) is a continuous embedding and |g| ⩽ A1λ(|uλ|q + 1) + ∥V ∥∞|uλ|p−1 we obtain∫ Ω |λf(uλ)(x) + V Φp(uλ)|t dx ⩽ λt ∫ Ω |A1(u q λ + 1)|t dx+ ∥V ∥t∞ ∫ Ω |uλ|t(p−1) dx ⩽ λtC ∫ Ω (|uλ|qt + 1) dx+ ∥V ∥t∞ ∫ Ω |uλ|t(p−1) dx. Hence, g ∈ Lt(Ω) and it follows from Lemma 3.7 that ∥uλ∥∞ ⩽ ∥g∥ 1 p−1 t . (3.35) On the other hand, by Lemma 3.5, we have ∥g∥t ⩽ C1λ∥uλ∥qtq + C2∥uλ∥p−1 t(p−1) ⩽ C1λ∥uλ∥q + C2∥uλ∥p−1 ⩽ C(λ1−rq + λ−r(p−1)). Therefore, from (3.35) and the fact that −r = (1− rq)/(p− 1) we obtain that ∥uλ∥∞ ≤ ∥g∥1/(p−1) t ≤ Cλ−r. (3.36) Thus, uλ ∈ L∞(Ω) and then g ∈ L∞(Ω). Hence, by Theorem 3.8, there exists α ∈ (0, s] and C > 0, depending only on N, p, s and Ω, such that the solution uλ satisfies uλ/d s ∈ Cα(Ω) and∥∥uλ ds ∥∥ Cα(Ω) ⩽ C∥g∥ 1 p−1 ∞ ⩽ λ−r. □ 3.2. Existence of a positive solution. To prove that the solution uλ found in Subsection 3.1 is positive, we will list two results found in Del Pezzo et al. [11] and one theorem due to Ianizzoto et al. [19], which will lead us to a comparison principle for the fractional p-Laplacian problem in (1.1). First, we recall two basic definitions that will be used in this section for the reader’s convenience. Definition 3.10. Let Ω ⊂ RN , N ⩾ 1, be an open set. We say that xo ∈ ∂Ω satisfies the interior ball condition if there is x ∈ Ω and r > 0 such that Br(x) ⊂ Ω, and xo ∈ ∂Br(x), where Br(x) = {z ∈ RN : |z − x| < r}. Next, we recall the concept of a function u ∈ W̃ s,p(Ω) being a super-solution of the fractional p-Laplacian problem (1.1). EJDE-2024/72 MULTIPLICITY RESULTS FOR FRACTIONAL p-LAPLACIAN 15 Definition 3.11. Let Ω ⊂ RN be an open bounded set with N ⩾ 1. We say that u ∈ W̃ s,p(Ω) is a super-solution of (1.1) if∫ R2N Φp(u(x)− u(y))(φ(x)− φ(y)) |x− y|N+sp dx dy + ∫ Ω V (x)|u|p−2uφdx ⩾ λ ∫ Ω f(u)φdx, for each φ ∈ W̃ s,p(Ω). The next two theorems due to Del Pezzo et al. [11] will play a special role in the main result to be discussed in this section. Theorem 3.12 ([11, Theorem 1.4]). Let c ∈ C(Ω) be a non-positive function and u ∈ W̃ s,p(Ω) ∩ C(Ω) be a weak super-solution of (−∆)spu(x) = c(x)|u(x)|p−2u(x), for x ∈ Ω. (3.37) If Ω is bounded and u ⩾ 0 a.e. in RN\Ω, then either u > 0 in Ω or u = 0 a.e. in RN . Theorem 3.13 ([11, Theorem 1.5]). Let Ω satisfy the interior ball condition at x0 ∈ ∂Ω, c ∈ C(Ω), and u ∈ W̃ s,p(Ω) ∩ C(Ω) be a weak super-solution of (3.37). Suppose that Ω is bounded, c(x) ⩽ 0 in Ω and u ⩾ 0 a.e. in RN\Ω. Then, either u = 0 a.e. in RN , or lim inf x→x0, x∈B u(x) (d(x))s > 0, (3.38) where B ⊆ Ω is an open ball in Ω, such that x0 ∈ ∂B, and d is the distance from x to RN\B. Next, we present a version of the comparison principle for problem (1.1) moti- vated by a result due to Lindgren et al. [21, Lemma 9] (see also Ianizzotto et al. [19, Proposition 2.10]). Theorem 3.14. Let Ω be a bounded subset of RN , N ⩾ 2, and u, v ∈ W̃ s,p(Ω) satisfy u ⩽ v in RN\Ω. Moreover, assume that∫ R2N Φp(u(x)− u(y))(φ(x)− φ(y)) |x− y|N+sp dx dy + ∫ RN V (x)Φp(u)φdx ⩽ ∫ R2N Φp(v(x)− v(y))(φ(x)− φ(y)) |x− y|N+sp dx dy + ∫ RN V (x)Φp(v)φdx, (3.39) for all φ ∈ W s,p 0 (Ω), φ ⩾ 0 a.e. in Ω. If V (x) ⩾ 0 for a.e. x ∈ RN , then u ⩽ v in Ω. Proof. We set φ = (u− v)+, where (u− v)+ = max{u− v, 0} denotes the positive part of the function u− v, in (3.39) to obtain∫ RN V (x)(Φp(u)− Φp(v))(v − u)+(x) dx ⩽ ∫ R2N (Φp(v(x)− v(y))− Φp(u(x)− u(y)))((v − u)+(x)− (v − u)+(y)) |x− y|N+sp dx dy. (3.40) Using an identity found in [21, page 809], Φp(b)− Φp(a) = (p− 1)(b− a) ∫ 1 0 |a+ t(b− a)|p−2 dt, 16 E. LOPERA, L. RECÔVA, A. RUMBOS EJDE-2024/72 with b = u(x) and a = v(x), we obtain the estimate 0 ⩽ (p− 1)(u(x)− v(x))(u− v)+(x) ∫ 1 0 |v(x) + t(u(x)− v(x))|p−2 dt = (Φp(u(x))− Φp(v(x)))(u− v)+(x). (3.41) for a.e. x ∈ RN . Hence, we conclude that the left-hand side of (3.40) is nonnegative. The remainder of the proof of the theorem follows the same line of reasoning as in [21, Lemma 9] and we omit the arguments here. □ Next, we show that the solution uλ found through Theorem 3.4 is positive in Ω. Theorem 3.15. Assume that p ⩾ 2 and V (x) ⩾ 0 for a.e x ∈ Ω. If p − 1 < q < min{ sp N p∗s, p ∗ s − 1}, then there exists λ∗ > 0 such that, for all 0 < λ < λ∗, problem (1.1) has at least one positive solution uλ ∈ Cα o (Ω) for some 0 < α < 1. Proof. From Lemma 3.4 we know that, for any λ ∈ (0, λ3), there exists a solution uλ ∈ X. Assume, by a way of contradiction, that there exists a sequence (λj)j∈N with 0 < λj < 1 such that λj → 0 as j → ∞ and, for all j ∈ N, we have |Ωj | > 0, (3.42) where Ωj = {x ∈ Ω|uλj (x) ⩽ 0}, for all j ∈ N, and |Ωj | denotes the Lebesgue measure of the set Ωj . We set wj = uλj ∥uλj ∥∞ . Notice that wj(x) ⩽ 0 for all x ∈ Ωj . Thus, by the regularity result in [19, Theorem 1.1], we obtain (−∆)sp(wj) = hj(x,wj), where hj(x, s) := −V (x)Φp(s)+λj∥uλj∥1−p ∞ f(∥uλj∥∞s). Using that λj∥uλj∥1−p ∞ < 1 and, by Lemma 3.9, λj∥uλj∥q+1−p ∞ < C for j large, and 1− r(1− p+ q) = 0, we obtain |hj(x, s)| ⩽ |V (x)||s|p−1 + λj∥uλj∥1−p ∞ B((∥uλj∥∞|s|)q + 1) ⩽ ∥V ∥∞|s|p−1 +Bλj∥uλj ∥1−p+q ∞ |s|q +Bλj∥uλj ∥1−p ∞ ⩽ ∥V ∥∞|s|p−1 +Bλ 1−r(1−p+q) j |s|q +B ⩽ C1|s|p ∗−1 + C2. Using the result of Theorem 3.8, there exists α ∈ (0, s] such that∥∥wj dsΩ ∥∥ Cα(Ω) ⩽ ∥hj(x,wj)∥1/(p−1) ∞ ⩽ (C1∥wj∥p ∗−1 ∞ + C2) 1/(p−1) = C3, (3.43) where C3 is a positive constant which does not depend on λj . Next, choose β such that 0 < β < α. By Arzelà-Ascoli Theorem (see [28, Theorem 40 on pg. 169]), up to a subsequence, it follows from (3.43) that lim j→∞ wj dsΩ = w dsΩ , in Cβ(Ω). The next step consists of using the comparison principle to prove that w(x) ⩾ 0. Indeed, let v0 ∈ W s,p 0 (Ω) be a solution of (−∆)spu+ V (x)Φp(u) = 1, in Ω; EJDE-2024/72 MULTIPLICITY RESULTS FOR FRACTIONAL p-LAPLACIAN 17 u = 0, in RN\Ω, obtained in Appendix 7. LetKj = λj ∥uλj ∥p−1 ∞ mint∈R f(t), and note thatKj < 0. Let vj = −(−Kj) 1/(p−1)v0. Then vj solves (−∆)spu+ V (x)Φp(u) = Kj , in Ω; u = 0, in RN\Ω. Observe that, for all φ ∈ W s,p 0 (Ω) with φ ⩾ 0, we have the estimate∫ R2N Φp(wj(x)− wj(y)) |x− y|N+sp (φ(x)− φ(y)) dx dy + ∫ Ω V (x)Φp(wj)φdx = ∫ Ω λjf(uλj )∥uλj ∥1−p ∞ φdx ⩾ ∫ Ω Kjφdx = ∫ R2N Φp(vj(x)− vj(y)) |x− y|N+sp (φ(x)− φ(y)) dx dy + ∫ Ω V (x)Φp(vj)φdx. (3.44) The above estimate implies that (−∆)sp(wj) ⩾ (−∆)sp(vj). By the comparison principle stated in Theorem 3.14, we conclude that wj ⩾ vj . Since vj → 0, as j → ∞, we obtain w(x) ⩾ 0, for x ∈ Ω. Next, let t := Npr/(N − sp) > 1. By Lemmas 3.6) and 3.9, we have that C1λ −r ⩽ ∥uλ∥∞ ⩽ C2λ −r. Then, we obtain λj |f(uλj (x))|∥uλj ∥1−p ∞ ⩽ Cλj(|uλj (x)|q + 1)∥uλj ∥1−p ∞ ⩽ Cλj(∥uλj∥q∞ + 1)∥uλj∥1−p ∞ ⩽ Cλj(λ −rq j + 1)λ r(p−1) j ⩽ Cλjλ −rq j λ r(p−1) j = Cλ 1−rq+r(p−1) j = C, where C is a positive constant and q < p∗s−1. It follows from the previous estimate that ∫ Ω (λjf(uj)∥uλj∥1−p ∞ )t dx ⩽ CΩ|Ω|. Thus, {λjf(uj)∥uλj ∥1−p ∞ }j is bounded in Lt(Ω) and we may assume that it con- verges weakly in Lt(Ω). Let z := limj⇀0 λjf(uj)∥uλj ∥1−p ∞ be its weak limit. Since f is bounded from below and limj→∞ λj∥uλj ∥1−p ∞ = 0, it follows that z ⩾ 0. We claim that (−∆)sp(w) = z. In fact, by Lemmas 3.5 and 3.6, we can follow the same line of reasoning as in the proof of [22, Theorem 1.1] to obtain lim j→∞ ∫ R2N |wj(x)− wj(y)|p−2(wj(x)− wj(y))(φ(x)− φ(y)) |x− y|N+sp dx dy = ∫ R2N |w(x)− w(y)|p−2(w(x)− w(y))(φ(x)− φ(y)) |x− y|N+sp dx dy. (3.45) 18 E. LOPERA, L. RECÔVA, A. RUMBOS EJDE-2024/72 On the other hand, since wj → w uniformly in Ω and w ∈ Lp(Ω), we also have that lim j→∞ ∫ Ω V (x)Φp(wj)φ(x) dx = ∫ Ω V (x)Φp(w)φ(x) dx. (3.46) Notice that wj → w in W s,p 0 (Ω), which implies that w ∈ W s,p 0 (Ω). Consequently, by (3.45), (3.46), and the fact that z is the weak limit of {λjf(uλj ∥uλj ∥1−p ∞ }, we have that (−∆)spw + V Φp(w) = z. That is, w is a weak supersolution of (−∆)spw+ V Φp(w) = 0. Hence, by Theorems 3.12 and 3.13, we have two alternatives: First, w = 0 cannot hold since wj → w in Cβ(Ω) and ∥wj∥∞ = 1 for all j ∈ N. Second, w > 0 in Ω and, for all x0 ∈ ∂Ω, lim inf x→x0, x∈B w(x) (d(x))s > 0, where B ⊆ Ω is an open ball in Ω, such that x0 ∈ ∂B, and d(x) is the distance from x to RN\B (see (3.38)). Therefore, there exist jo sufficiently large such that, for all j ⩾ jo, we have wj > 0. But this contradicts that wj(x) = uλj (x) ∥uλj∥∞ ⩽ 0, for x ∈ Ωj . Hence, |Ωj | = 0 for all j ∈ N and we conclude that problem (1.1) has at least one positive solution uλ ∈ Cα 0 (Ω), for some 0 < α < 1. □ 4. Computation of critical groups at infinity In this section, we will obtain the first multiplicity result for problem (1.1). The first step will consist of computing the critical groups of Jλ at infinity as defined in (2.11). This will require to use the concept of two topological spaces being homotopically equivalent. To show that two topological spaces A and B are homotopically equivalent, denoted by A ∼= B, one needs to show that there exist functions η : A → B and i : B → A such that η ◦ i ≈ idB and i ◦ η ≈ idA, where id denotes the identity function and the symbol ≈ denotes the existence of a homotopy. In particular, if B ⊂ A and i : B → A denotes the inclusion function and η : A → B is a deformation retraction from A onto B, then we have that η ◦ i ≈ idB and i ◦ η = idA. Hence, to obtain the critical groups of Jλ at infinity, we will prove the existence of a deformation retract from J−M λ onto S∞, for some M to be chosen soon, where S∞ denotes the unit sphere in X. Finally, the result will follow by using an argument with the long exact sequence of the topological pair (X, J−M ) and the fact that S∞ is contractible in X. Let S∞ = {u ∈ X : ∥u∥ = 1} be the unit sphere in X. Notice that, for u ∈ S∞, we have that lim t→∞ Jλ(tu) = −∞. (4.1) In fact, substituting (3.2) into (2.5) and applying (H3), we obtain Jλ(tu) ⩽ tp p ( 1 + ∥V ∥L∞∥u∥pp ) − λA1t q+1∥u∥q+1 q+1 + λA1C1|Ω|, (4.2) for all u ∈ S∞. Then, since p < q + 1, the result (4.1) follows by letting t → ∞ in (4.2). EJDE-2024/72 MULTIPLICITY RESULTS FOR FRACTIONAL p-LAPLACIAN 19 Lemma 4.1. Assume that (H1), (H2) are satisfied. Then, there exists M̃ > 0 such that, for all M ⩾ M̃ , J−M λ is homotopically equivalent to S∞. Proof. We will follow a line of reasoning similar to one in [31, Section 3] to show the existence of a deformation retract from J−M λ to S∞. First, notice that the critical value set Jλ(K) is bounded from below. In fact, if u0 ∈ K, then setting φ = u0 in (2.6), we obtain ∥u0∥p + ∫ Ω V (x)|u0|p dx = λ ∫ Ω f(u0)u0 dx. (4.3) Next, we substitute (4.3) into (2.5) and use (H2) to obtain Jλ(u0) = λ p ∫ Ω [f(u0)u0 − pF (u0)] dx ⩾ λ p ∫ Ω [(θ − p)F (u0) +K] dx ⩾ λ p |Ω|((θ − p)minF +K) =: −a0, for all u0 ∈ K, and therefore −ao ⩽ inf Jλ(K). (4.4) By (4.1), given u ∈ S∞ and M1 > 0, there exists t0 = t0(u) ⩾ 1 such that Jλ(tu) < −M1, for t0 ⩾ 1, u ∈ S∞. We define M̃ = min{−ao,−M1}. Choosing M2 > M̃ such that, for tu ∈ J−M2 λ , we have Jλ(tu) = tp p ( 1 + ∫ Ω V (x)|u|p dx ) − λ ∫ Ω F (tu) dx, (4.5) for t ⩾ 1. Using the chain rule, and taking into account that f(s)s is bounded from below, and p θ < 1, it follows from (4.5) and (H2) that d dt Jλ(tu) = 1 t [ pJ(tu) + λ ∫ Ω (pF (tu)− f(tu)tu) dx ] ⩽ 1 t [ − pM2 + λ ∫ Ω (p θ f(tu)(tu)− f(tu)tu− Kp θ ) dx ] ⩽ 1 t [ − pM2 + λ (p θ − 1 ) ∫ Ω f(tu)(tu)) dx− λ Kp θ |Ω| ] ⩽ 1 t [−Mo + K̂λ], (4.6) where Mo and K̂ are positive constants, for all tu ∈ J−M2 λ . Choosing λ small enough in (4.6), we obtain d dt Jλ(tu) < 0, (4.7) for tu ∈ J−M2 λ , and t ⩾ 1. Let us take M ⩾ M̃ . Then, combining (4.1) and (4.7), we can invoke the intermediate value theorem to conclude that there exists T (u) ⩾ 1 such that Jλ(T (u)u) = −M, for u ∈ S∞. 20 E. LOPERA, L. RECÔVA, A. RUMBOS EJDE-2024/72 It follows from the implicit function theorem [10, Theorem 15.1] that T ∈ C(S∞,R). Finally, let B∞ = {u ∈ X : ∥u∥ ⩽ 1} be the unit ball in X. We define η : [0, 1]× (X\B∞) → X\B∞ by η(t, u) = (1− t)u+ tT (u)u, for t ∈ [0, 1] and u ∈ X\B∞. Observe that η(0, u) = u and η(1, u) ∈ J−M λ . Thus, η is a deformation retract from X\B∞ onto J−M λ . Since X\B∞ ∼= S∞, we conclude that J−M λ ∼= X\B∞ ∼= S∞; that is, J−M λ is homotopically equivalent to S∞. □ Since J−M λ and S∞ are homotopically equivalent, as shown in the previous lemma, we conclude that the homology groups Hk(J −M λ ) and Hk(S ∞) are iso- morphic, for all k ∈ Z (see [14, Corollary 2.11]). Since S∞ is also contractible in X (see Benyamini-Sternfeld [5]), we obtain that the singular homology groups Hk(J −M λ ) have the homology type of a point for all k ∈ Z; namely, Hk(J −M λ ) ∼= δk,0F, for all k ∈ Z. Using an argument similar to that in [27, Section 3] with the long exact sequence of reduced homology groups of the topological pair (X, J−M λ ) and the fact that Jλ satisfies the Palais-Smale condition shown in Lemma 3.3, we conclude that the critical groups of Jλ at infinity are given by Ck(Jλ,∞) = Hk(X,J−M λ ) ∼= δk,0F, for all k ∈ Z. (4.8) 5. Computation of critical groups at the origin In this section, we study the questions of existence and multiplicity for the case f(0) = 0. In this case, the function u ≡ 0 is also a critical point of Jλ and we need to obtain some information about the critical groups of Jλ at the origin. To obtain another solution, we need to make an additional assumption about the behavior of F at the origin. This is the content of the next lemma. Lemma 5.1. Assume that the nonlinearity f satisfies (H1) and its primitive F satisfies lim sup s→0 F (s) |s|p = 0. (5.1) Then, the origin is a local minimizer of the functional Jλ and its critical groups are Ck(Jλ, 0) ∼= δk,0F, for all k ∈ Z. (5.2) Proof. By condition (5.1), for each given ε > 0, there exists a δ > 0 such that |s| < δ ⇒ F (s) < ε|s|p. (5.3) It follows from (3.1) that there exists a constant K1 = K1(δ) such that |F (s)| ⩽ K1|s|q+1, for all |s| ⩾ δ. (5.4) In fact, assuming s ⩾ δ and using (H1) we obtain |F (s)| ⩽ ∫ s 0 |f(ξ)| dξ ⩽ Bs+ B q + 1 sq+1; EJDE-2024/72 MULTIPLICITY RESULTS FOR FRACTIONAL p-LAPLACIAN 21 so that |F (s)| ⩽ B [ δ (s δ ) + δq+1 q + 1 (s δ )q+1] . (5.5) Since we are assuming that s ⩾ δ; so that s δ ⩾ 1, it follows from (5.5) that |F (s)| ⩽ B [ δ (s δ )q+1 + δq+1 q + 1 (s δ )q+1 ] , for s ⩾ δ, (5.6) from which we obtain that |F (s)| ⩽ B δq+1 [ δ + δq+1 ] sq+1, for s ⩾ δ, (5.7) where we have used that q + 1 > p > 1, in view of hypothesis (H1). Setting K1 = K1(δ) = B δq+1 [δ + δq+1], we see that (5.4) follows from (5.7). The case for s ⩽ −δ is analogous. Hence, estimate (5.4) is valid for all |s| ⩾ δ. Next, combine the estimates (5.3) and (5.4) to obtain F (s) ⩽ ε|s|p +K1|s|q+1, for s ∈ R. (5.8) Then, it follows from (5.8) that∫ Ω F (u) dx ⩽ ε ∫ Ω |u|p dx+K1 ∫ Ω |u|q+1 dx; so that, using the Sobolev inequality [12, Theorem 6.7], it follows from the previous estimate that ∫ Ω F (u) dx ⩽ C3 ( ε+K1∥u∥q+1−p ) ∥u∥p, (5.9) for some positive constant C3. Setting ρ = ( ε 2K1 )1/(q+1−p), we obtain from (5.9) that ∥u∥ < ρ ⇒ ∫ Ω F (u) dx ⩽ C3ε∥u∥p. (5.10) It follows from the definition of Jλ in (2.5), (5.10), and (H3) that Jλ(u) ⩾ (1 p − C3λε ) ∥u∥p − cV p ∥u∥pp. (5.11) On the other hand, it follows from [21, Section 3] that the first eigenvalue λ1 of (−∆)sp is characterized by the minimization of the Rayleigh quotient, λ1 = inf u∈X\{0} ∥u∥p ∥u∥pp , (5.12) with λ1 ∈ (0,∞); see [21, Theorem 5]. Hence, applying (5.12) in (5.11), we obtain Jλ(u) ⩾ [1 p ( 1− cV λ1 ) − C3λε ] ∥u∥p. (5.13) By (H3), cV < λ1, thus we can choose ε > 0 such that ε < 1 2pC3λ ( 1− cV λ1 ) . (5.14) Then, by (5.14), we obtain from (5.13) that Jλ(u) ⩾ 1 2pC3λ ( 1− cV λ1 ) ∥u∥p > J(0), for 0 < ∥u∥ < ρ, 22 E. LOPERA, L. RECÔVA, A. RUMBOS EJDE-2024/72 where ρ > 0 is sufficiently small. Consequently, u = 0 is a local minimum of Jλ in Bρ(0). It follows from ([9, Example 1, page 33] that Ck(Jλ, 0) ∼= δk,0F, for k ∈ Z. □ 6. Proofs of main results Proof of Theorem 1.1. Assume, by a way of contradiction, that K = {uλ} where uλ is the mountain-pass type solution found in Theorem 3.4. Then, it follows from Proposition 2.7 that C1(Jλ, uλ) ̸∼= 0. (6.1) Since we are assuming that K = {uλ}, we can invoke Proposition 2.10 to obtain Ck(Jλ,∞) ∼= Ck(Jλ, uλ), for all k ∈ Z. (6.2) In particular, if k = 1 in (6.2), we obtain from (4.8) and (6.1) that 0 ∼= C1(Jλ,∞) ∼= C1(Jλ, uλ) ̸∼= 0, which is a contradiction. Therefore, Jλ must have at least two critical points and this completes the proof. □ Proof of Theorem 1.2. By Theorem 1.1, we obtain the existence of two solutions for problem (1.1). Furthermore, one of them is of mountain pass type. Next, assume that p ⩾ 2 and V (x) ⩾ 0 for a.e. x ∈ Ω. For the case of f(0) > 0, it follows from the Comparison Theorem 3.15 that both solutions are positive. For the case of f(0) < 0, Theorem 3.15 leads us to the positivity of the mountain-pass type solution. □ Proof of Theorem 1.4. Assume, by a way of contradiction, that K = {0, uλ}, where uλ is the mountain-pass type solution found in Theorem 3.4. Then, it follows from [9, Theorem 4.2, page 35] that Hk(X, J−M λ ) ∼= Ck(Jλ, 0)⊕ Ck(Jλ, uλ), for all k ∈ Z. (6.3) In particular, setting k = 1 in (6.3) and using (2.14), (4.8), and (5.2), we obtain 0 ∼= C1(Jλ, 0)⊕ C1(Jλ, uλ) ∼= 0⊕ C1(Jλ, uλ) ̸∼= 0, which is a contradiction. Therefore, the critical set K must have at least three critical points. This completes the proof. □ 7. Appendix In this section, we prove that the problem (−∆)spu(x) + V (x)Φp(u(x)) = 1, for x ∈ Ω; u = 0, in RN\Ω, (7.1) has a positive weak solution. We will show that the associated energy functional with problem (7.1) is coercive and weakly lower semi-continuous. Then, the exis- tence result follows by a result found in Evans [13, Theorem 2, Chapter 8]. In fact, the associated functional with problem (7.1) is E(u) := 1 p ∥u∥ps,p + 1 p ∫ Ω V (x)|u|p dx− ∫ Ω u dx, u ∈ X. (7.2) To prove the coercivity of E, let (un)n be a sequence in X such that ∥un∥s,p → ∞ as n → ∞. From (2.8) we have that ∥un∥1 ⩽ C1∥un∥s,p, for all n. Moreover, EJDE-2024/72 MULTIPLICITY RESULTS FOR FRACTIONAL p-LAPLACIAN 23 ∥un∥pp ⩽ 1 λ1 ∥un∥ps,p, for all n. Therefore, applying these estimates and (H3) to (7.2) we obtain E(un) ⩾ 1 p ∥un∥ps,p − cV p ∥un∥pp − C1∥un∥s,p ⩾ 1 p ( 1− cV λ1 ) ∥un∥ps,p − C1∥un∥s,p, (7.3) for all n ∈ N. Since 1 − cV λ1 > 0 and p > 1, we obtain from (7.3) that E(un) → ∞ as n → ∞. Now, E is continuous because of its differentiability. Moreover, a simple computa- tion shows that the functional E is convex. Therefore, E is weakly lower semicon- tinuous (see for example [3, Theorem 1.5.3]). This proves that problem (7.1) has at least one solution u ∈ X, which is nontrivial. Finally, notice that u is a weak supersolution of the problem (−∆)spu(x) + V (x)Φp(u(x)) = 0, for x ∈ Ω, withu = 0, in RN\Ω. Thus, by Theorem (3.12), it follows that u > 0. Acknowledgements. The authors would like to thank the referees for careful reading of the original manuscript. Emer Lopera would like to express his grati- tude to Alfonso Castro for his guidance and support during his postdoctoral studies at Harvey Mudd College in Claremont, California, USA. In particular, Emer Lopera is indebted to him for his role in facilitating connections with the researchers in- volved in this investigation. This research was supported by Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Colombia, Sede Manizales, Facul- tad de Ciencias, Departamento de Matemáticas, Grupo de investigación: Análisis Matemático AM de la Universidad Nacional de Colombia-Sede Manizales. 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Rumbos; Multiple nontrivial solutions of a semilinear elliptic problem with asymmetric nonlinearity, Journal of Mathematical Analysis and Applications 484/2, 123720 (2020), 1–12. [28] H. Royden; Real analysis, third ed., Prentice Hall, 1988. [29] R. Servadei, E. Valdinoci; Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), no. 2, 887–898. [30] E. Valdinoci; From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA, 49 (2009), 33–44. MR 2584076 [31] Z. Q. Wang; On a superlinear elliptic equation, Annales de l’IHP, Section C 8 (1991), 43–57. Emer Lopera Universidad Nacional de Colombia, Manizales, Colombia Email address: edloperar@unal.edu.co Leandro Recôva California State Polytechnic University, 3801 West Temple Avenue, Pomona, CA 91768, USA Email address: llrecova@cpp.edu Adolfo Rumbos Pomona College, Mathematics and Statistics Department. 610 N. College Avenue, Claremont, CA 91711, USA Email address: arumbos@pomona.edu 1. Introduction 2. Preliminaries 3. Existence and a priori estimates 3.1. Existence of a mountain-pass type solution 3.2. Existence of a positive solution 4. Computation of critical groups at infinity 5. Computation of critical groups at the origin 6. Proofs of main results 7. Appendix Acknowledgements References