Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 101, pp. 1–21. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu MULTIPLE POSITIVE SOLUTIONS TO THE FRACTIONAL KIRCHHOFF PROBLEM WITH CRITICAL INDEFINITE NONLINEARITIES JIE YANG, HAIBO CHEN, ZHAOSHENG FENG Abstract. This article concerns the existence and multiplicity of positive solutions to the fractional Kirchhoff equation with critical indefinite nonlin- earities by applying the Nehari manifold approach and fibering maps. 1. Introduction and statement of results In this paper, we study the existence and multiplicity of positive solutions to the fractional Kirchhoff type problem M (∫ R2N |u(x)− u(y)|2 |x− y|N+2s dx dy ) (−∆)su = fλ(x)|u|q−2u+ g(x)|u|2 ∗ s−2u, in Ω, u = 0, in RN \ Ω, (1.1) where Ω ⊂ RN is an open bounded domain with the Lipschitz boundary ∂Ω, dimen- sion N > 2s with s ∈ (0, 1), 2∗s = 2N N−2s is the fractional critical Sobolev exponent and 0 < s < 1 < q < min{2, N N−2s} < ∞. Here, M(t) = a + btm−1 with m > 1, a, b > 0, fλ ∈ Lq ∗ (Ω), q∗ = 2∗ s 2∗ s−q , fλ = λf+− f− with λ > 0, and f± = max{±f, 0} and g ∈ L∞(Ω). Furthermore, g satisfies the condition (A1) g(x) = maxx∈Ω̄ g(x) ≡ 1 in Bρ(0) for some ρ > 0. We denote by (−∆)s the usual fractional Laplacian operator which is defined (up to normalization factors) as follows (see for instance [18] and the references therein for further details on the fractional Laplacian) by (−∆)su(x) = 2P.V. ∫ RN u(x)− u(y) |x− y|N+2s dy, (1.2) where P.V. stands for the principle value. When M(t) ≡ 1, λ = 1 and s = 1, equation (1.1) can be reduced to the semilinear elliptic problem −∆u = f(x)|u|q−2 + g(x)|u|2 ∗−2u, x ∈ Ω, u = 0, x ∈ ∂Ω, (1.3) 2010 Mathematics Subject Classification. 35A15, 35B33, 35R11. Key words and phrases. Fractional Kirchhoff equation; Nehari manifold; fibering maps. c©2020 Texas State University. Submitted June 12, 2020. Published September 28, 2020. 1 2 J. YANG, H. B. CHEN, Z. FENG EJDE-2020/101 where Ω is a smooth bounded domain in RN (N ≥ 3), 1 < q < 2, and the weight functions f , g are continuous and sign-changing. By using the Nehari manifold, fibering maps and Ljusternik-Schnirelmann category, Wu [25] proved that there existed at least three positive solutions of (1.3). Xie-Chen [26] presented a mul- tiplicity result on the Kirchhoff-type problems in the bounded domain by using a similar strategy. A number of works dealt with the fractional differential equa- tions [3, 6, 7, 11, 21] and some recent results on problem (1.3) can be seen in [4, 5, 10, 12, 13, 14, 15, 16, 22, 23, 27] and the references therein. As we know, the variational problems involving fractional and nonlocal opera- tors are much more complicated and challenging. In the last decade, considerable attention focused on the fractional Laplacian operator and nonlocal operator. We refer to [19] for the Brezis-Nirenberg type results for the following elliptic equation involving the fractional Laplacian (−∆)s(0 < s < 1) in a bounded domain, (−∆)su = λu+ |u|2 ∗ s−2u, x ∈ Ω, u = 0, x ∈ ∂Ω, where λ > 0, s ∈ (0, 1) is fixed, 2∗s = 2N N−2s , Ω ⊂ RN (N > 2s) is open, bounded and with the Lipschitz boundary, and (−∆)s is the fractional Laplace operator. The classical Brezis-Nirenberg result was generalized to the case of nonlocal fractional operators through variational techniques. The existence of multiple solutions to the fractional Laplacian equations of Kirchhoff type was considered in [17] and two positive solutions for proper selection of positive parameter λ was obtained. The main purpose of this article is to establish the existence and multiplicity of positive solutions to problem (1.1) with the critical growth and sign-changing weight functions. Our results encompass and improve the corresponding results presented in [26] for the fractional Kirchhoff type equations involving the critical growth. The energy functional associated with problem (1.1) is Iλ(u) = a 2 ∫ R2N |u(x)− u(y)|2 |x− y|N+2s dx dy + b 2m (∫ R2N |u(x)− u(y)|2 |x− y|N+2s dx dy )m − 1 q ∫ Ω fλ(x)|u|qdx− 1 2∗s ∫ Ω g(x)|u|2 ∗ sdx for u ∈ Hs 0(Ω). We can prove that Iλ ∈ C1(Hs 0(Ω),R) and a critical point of Iλ in Hs 0(Ω) corresponds to a weak solution of problem (1.1). We summarize our main results as follows. Theorem 1.1. Assume that m < N N−2s , f± 6≡ 0 and condition (A1) holds. Then there exist 0 < Λ∗ ≤ Λ0 and b̄ > 0 such that (i) for any λ ∈ (0,Λ0), problem (1.1) admits at least one positive solution u1 with Iλ(u1) < 0, and u1 is a ground state solution; (ii) for any λ ∈ (0,Λ∗) and b ∈ (0, b̄), problem (1.1) admits at least two positive solutions u1 and u2 satisfying Iλ(u1) < 0 < Iλ(u2), and u1 is a ground state solution. Theorem 1.2. Assume that m = N N−2s , f± 6≡ 0 and condition (A1) holds. Then the following two statements hold: (i) For b ≥ 1/Sm and any λ > 0, problem (1.1) admits at least one positive solution. EJDE-2020/101 MULTIPLE SOLUTIONS TO FRACTIONAL KIRCHHOFF PROBLEM 3 (ii) For b < 1/Sm, there exist 0 < Λ̃∗ ≤ Λ0 and b̃ > 0 such that (1) for any λ ∈ (0,Λ0), problem (1.1) admits at least one positive solution; (2) for any λ ∈ (0, Λ̃∗) and b ∈ (0, b̃), problem (1.1) admits at least two positive solutions u1 and u2 satisfying Iλ(u1) < 0 < Iλ(u2), and u1 is a ground state solution. Theorem 1.3. Assume that m > N N−2s , f− ≡ 0, and condition (A1) holds. Then there exist b∗,Λ∗ > 0 such that for any b ∈ (0, b∗) and λ ∈ (0,Λ∗), problem (1.1) admits at least three positive solutions ub, uλ, uλ,b with Iλ(uλ) < Iλ(ub) < 0 < Iλ(uλ,b), and uλ is a ground state solution. Note that the corresponding results in [26] are generalized to the nonlocal frac- tional Kirchhoff problem and the existence results are extended in the sense that the restriction on the Kirchhoff coefficient M is eliminated. When g(x) ≡ 1, by Theorems 1.1 and 1.2, we obtain the existence and multi- plicity of positive solutions to the problem M (∫ R2N |u(x)− u(y)|2 |x− y|N+2s dx dy ) (−∆)su = fλ(x)|u|q−2u+ |u|2 ∗ s−2u, in Ω, u = 0, in RN \ Ω, where M(t) = a+btm−1 with a, b > 0 for t ≥ 0 and m ∈ [1, 2∗s/2], which generalizes [17, Theorem 1.1]. In view of [2, 5], problem (1.1) appears more complicated because of the lack of compactness and the nonlocal nature of the fractional Laplacian. Theorems 1.1–1.3 can be regarded as generalizations of [26] for fractional Laplacian operators. The rest of this paper is organized as follows. In Section 2, we present mathe- matical notation and technical lemmas. We prove Theorems 1.1 and 1.2 in Section 3, and prove Theorem 1.3 in Section 4. 2. Preliminary results In this section, we introduce some notation, definitions and useful lemmas which will be used in the proofs of main results. We define the Hilbert space Hs(RN ) by Hs(RN ) := { u ∈ L2(RN ) : |u(x)− u(y)| |x− y|N+2s 2 ∈ L2 ( RN × RN ) } endowed with the norm ‖u‖Hs(RN ) = (∫ RN |u|2dx+ ∫ R2N |u(x)− u(y)|2 |x− y|N+2s dx dy )1/2 , (2.1) where the term [u]s = (∫ R2N |u(x)− u(y)|2 |x− y|N+2s dx dy )1/2 is the so-called Gagliardo semi-norm of u. In view of (1.2) and [18, Proposition 3.6], we have ‖(−∆)s/2u‖22 = 1 Cs ∫ R2N |u(x)− u(y)|2 |x− y|N+2s dx dy, 4 J. YANG, H. B. CHEN, Z. FENG EJDE-2020/101 where Cs is a positive constant depending on s. We define Ds,2(RN ) as the closure of C∞0 (RN ) with the norm ‖u‖Ds,2 = (∫ RN |(−∆)s/2u|2dx )1/2 . Then Ds,2(RN ) is continuously embedded into L2∗ s (RN ). As in [7, Theorem 1.1], let S be the best constant of the fractional Sobolev embedding Ds,2(RN ) ↪→ L2∗ s (RN ) defined by S = inf u∈Ds,2(RN )\{0} ∫ R2N |u(x)−u(y)|2 |x−y|N+2s dx dy( ∫ RN |u|2 ∗ sdx )2/2∗ s , (2.2) which is well-defined and strictly positive. We define E0 = {u ∈ Hs(RN ) : u = 0 a.e. in RN \ Ω} with the norm ‖u‖E0 = (∫ R2N |u(x)− u(y)|2 |x− y|N+2s dx dy )1/2 , which is equivalent to (2.1) [19, 20]. The embedding E0 ↪→ Lr(Ω) is continuous for any r ∈ [1, 2∗s] and compact whenever r ∈ [1, 2∗s). We recall that (E0, ‖ · ‖E0 ) is a Hilbert space with the inner product defined by 〈u, v〉 = ∫ R2N (u(x)− u(y))(v(x)− v(y)) |x− y|N+2s dx dy. For simplicity, we will just denote ‖·‖E0 and ‖·‖Lp(Ω) by ‖·‖ and |·|p, respectively. Throughout this paper, the letters C,Ci, i = 1, 2, . . . denote positive constants which may vary from line to line but independent of the associated terms and parameters. As we see, Iλ is of class C1 in E0 and for any v ∈ E0 it holds 〈I ′λ(u), v〉 =M(‖u‖2) ∫ R2N (u(x)− u(y))(v(x)− v(y)) |x− y|N+2s dx dy − ∫ Ω fλ(x)|u|q−2uvdx− ∫ Ω g(x)|u|2 ∗ s−2uvdx. Define the Nehari manifold associated with Iλ by Nλ = {u ∈ E0 \ {0} : 〈I ′λ(u), u〉 = 0}. It is well-known that the Nehari manifold is closely related to the behavior of the fibering map φu : t ∈ R+ → Iλ(tu) [2, 8]. Thus, we have φ′u(t) = at‖u‖2 + bt2m−1‖u‖2m − tq−1 ∫ Ω fλ(x)|u|qdx− t2 ∗ s−1 ∫ Ω g(x)|u|2 ∗ sdx, φ′′u(t) = a‖u‖2 + (2m− 1)bt2m−2‖u‖2m − (q − 1)tq−2 ∫ Ω fλ(x)|u|qdx− (2∗s − 1)t2 ∗ s−2 ∫ Ω g(x)|u|2 ∗ sdx. Then u ∈ Nλ if and only if φ′u(1) = 0. Moreover, for u ∈ Nλ we have φ′′u(1) = a(2− q)‖u‖2 + b(2m− q)‖u‖2m − (2∗s − q) ∫ Ω g(x)|u|2 ∗ sdx, (2.3) EJDE-2020/101 MULTIPLE SOLUTIONS TO FRACTIONAL KIRCHHOFF PROBLEM 5 or φ′′u(1) = a(2− 2∗s)‖u‖2 + b(2m− 2∗s)‖u‖2m − (q − 2∗s) ∫ Ω fλ(x)|u|qdx. (2.4) We split Nλ into three parts: N+ λ = {u ∈ Nλ|φ′′u(1) > 0}, N−λ = {u ∈ Nλ|φ′′u(1) < 0}, N0 λ = {u ∈ Nλ|φ′′u(1) = 0}, and define H+ = {u ∈ E0| ∫ Ω fλ(x)|u|qdx > 0}, H− = {u ∈ E0| ∫ Ω fλ(x)|u|qdx ≤ 0}, G+ = {u ∈ E0| ∫ Ω g(x)|u|2 ∗ sdx > 0}, G− = {u ∈ E0| ∫ Ω g(x)|u|2 ∗ sdx ≤ 0}. In view of m ≤ N N−2s and following [17, Lemma 3.2], we can derive the following lemma immediately. Lemma 2.1. If u is a minimizer of Iλ on Nλ such that u /∈ N0 λ, then I ′λ(u) = 0 in E−1 0 . Lemma 2.2. For any λ > 0, the functional Iλ is coercive and bounded below on Nλ. Proof. For u ∈ Nλ, from (2.2) and Hölder’s inequality, we have Iλ(u) = Iλ(u)− 1 2∗s 〈I ′λ(u), u〉 = (1 2 − 1 2∗s ) a‖u‖2 + ( 1 2m − 1 2∗s ) b‖u‖2m − (1 q − 1 2∗s )∫ Ω fλ(x)|u|qdx ≥ (1 2 − 1 2∗s ) a‖u‖2 − (1 q − 1 2∗s ) λ|f+|q∗S−q/2‖u‖q. Recalling that 1 < q < 2, we obtain that Iλ is coercive and bounded below on Nλ. � Let λ1 = [a(2− q) 2∗s − q ] 2−q 2∗s−2 a(2∗s − 2)S 2∗s−q 2∗s−2 (2∗s − q)|f+|q∗ . (2.5) Lemma 2.3. There exists λ1 > 0 such that N0 λ = ∅ for λ ∈ (0, λ1). Proof. By contradiction assume that for some λ ∈ (0, λ1), there is a function u ∈ N0 λ . Then from (2.3) and (2.4), we have a(2− q)‖u‖2 + b(2m− q)‖u‖2m − (2∗s − q) ∫ Ω g(x)|u|2 ∗ sdx = 0, (2.6) a(2− 2∗s)‖u‖2 + b(2m− 2∗s)‖u‖2m − (q − 2∗s) ∫ Ω fλ(x)|u|qdx = 0. (2.7) It follows from (A1), (2.6) and (2.2) that ‖u‖2 ≤ 2∗s − q a(2− q) |u|2 ∗ s 2∗ s ≤ 2∗s − q a(2− q) S− 2∗s 2 ‖u‖2 ∗ s . (2.8) 6 J. YANG, H. B. CHEN, Z. FENG EJDE-2020/101 Similarly, from (2.2), (2.7) and Hölder’s inequality, we can deduce that ‖u‖2 ≤ 2∗s − q a(2∗s − 2) λ ∫ Ω f+|u|qdx ≤ 2∗s − q a(2∗s − 2) λ|f+|q∗S−q/2‖u‖q. (2.9) Combining (2.8) and (2.9) yields[a(2− q) 2∗s − q S 2∗s 2 ] 1 2∗s−2 ≤ ‖u‖ ≤ [ (2∗s − q)λ|f+|q∗S−q/2 a(2∗s − 2) ] 1 2−q . Therefore, λ ≥ [a(2− q) 2∗s − q ] 2−q 2∗s−2 a(2∗s − 2)S 2∗s−q 2∗s−2 (2∗s − q)|f+|q∗ = λ1. This is a contradiction. � We define λ2 = λ1, m < N N−2s ,( 1 1−bSm ) 2−q 2∗s−2λ1, m = N N−2s , b < 1/Sm. (2.10) The lemma below shows that the component sets N+ λ and N−λ are nonempty. Lemma 2.4. Assume m < N N−2s . Then the following two statements are true. (i) For any u ∈ G+∩H+ and λ ∈ (0, λ2), there exist 0 < t+ = t+(u) < tmax < t− = t−(u) such that t+u ∈ N+ λ , t −u ∈ N−λ and Iλ(t+u) = inf 0≤t≤t− Iλ(tu), Iλ(t−u) = sup t≥tmax Iλ(tu). (ii) For any u ∈ G+ ∩H− and λ > 0, there exists a unique t− = t−(u) > tmax such that t−u ∈ N−λ and Iλ(t−u) = sup t≥0 Iλ(tu). Proof. Fix u ∈ E0 \ {0} and define ψu(t) : R+ → R by ψu(t) = at2−q‖u‖2 + bt2m−q‖u‖2m − t2 ∗ s−q ∫ Ω g(x)|u|2 ∗ sdx. (2.11) We remark that tu ∈ Nλ if and only if ψu(t) = ∫ Ω fλ|u|qdx. (i) Let u ∈ G+ ∩H+. From (2.11), it is easy to check that ψu(0) = 0, lim t→∞ ψu(t) = −∞, lim t→0+ ψ′u(t) > 0 and lim t→∞ ψ′u(t) < 0. Define ψ′u(t) = t1−qhu(t), where hu(t) = a(2− q)‖u‖2 + (2m− q)bt2m−2‖u‖2m − (2∗s − q)t2 ∗ s−2 ∫ Ω g(x)|u|2 ∗ sdx. Then, there exists a unique t0 > 0 such that h′u(t0) = 0, where t0 = ( (2m− q)(2m− 2)b‖u‖2m (2∗s − q)(2∗s − 2) ∫ Ω g(x)|u|2∗ sdx ) 1 2∗s−2m . From m < N N−2s it follows that limt→0+ hu(t) > 0 and limt→∞ hu(t) = −∞. This implies that there is a unique tmax > t0 such that hu(tmax) = 0. Hence, ψ′u(t) > EJDE-2020/101 MULTIPLE SOLUTIONS TO FRACTIONAL KIRCHHOFF PROBLEM 7 0 for t ∈ (0, tmax), ψ′u(t) < 0 for t ∈ (tmax,∞) and ψ′u(tmax) = 0. Moreover, ψu(tmax) = maxt>0 ψu(t) ≥ maxt>0 ψ̄u(t), where ψ̄u(t) = at2−q‖u‖2 − t2 ∗ s−q ∫ Ω g(x)|u|2 ∗ sdx. From (2.2) it follows that max t>0 ψ̄u(t) = ‖u‖q a(2∗s − 2) 2∗s − q ( (2− q)a‖u‖2∗ s (2∗s − q) ∫ Ω g(x)|u|2∗ sdx ) 2−q 2∗s−2 ≥ ‖u‖q a(2∗s − 2) 2∗s − q ( (2− q)aS 2∗s 2 2∗s − q ) 2−q 2∗s−2 . For u ∈ H+, it holds ψu(0) = 0 < ∫ Ω fλ(x)|u|qdx ≤ λ ∫ Ω f+|u|qdx ≤ λ|f+|q∗S−q/2‖u‖q. So, if λ < λ1 = a(2∗s − 2)S 2∗s−q 2∗s−2 (2∗s − q)|f+|q∗ ( (2− q)a 2∗s − q ) 2−q 2∗s−2 , there exist unique t+ = t+(u) < tmax and t− = t−(u) > tmax such that ψu(t+) = ∫ Ω fλ(x)|u|qdx = ψu(t−), ψ′u(t+) > 0, ψ′u(t−) < 0, which implies t+u, t−u ∈ Nλ. According to φ′′u(1) = tq+1ψ′u(t), we can deduce that t+u ∈ N+ λ and t−u ∈ N−λ . Since φ′u(t) = tq−1 ( ψu(t) − ∫ Ω fλ(x)|u|qdx ) , it is clear that φ′u(t) < 0 for t ∈ [0, t+) and φ′u(t) > 0 for t ∈ (t+, t−). This indicates that Iλ(t+u) = inf0≤t≤t− Iλ(tu). Similarly, from φ′u(t) > 0 for t ∈ (t+, t−) and φ′u(t) < 0 for t ∈ (t−,∞), we can obtain Iλ(t−u) = supt≥tmax Iλ(tu). (ii) The proof is essentially the same as that in Part (i), so we omit it. � As in Lemma 2.4, we can deduce the following two lemmas. Lemma 2.5. Assume that m = N N−2s and b ≥ 1/Sm. Then for any u ∈ H+, there exists a unique 0 < t+ < tmax such that t+u ∈ Nλ and Iλ(t+u) = inft≥0 Iλ(tu). Lemma 2.6. Assume that m = N N−2s and b < 1/Sm. Then the following two statements are true. (i) For any u ∈ H+ and λ ∈ (0, λ2), there exist 0 < t+ = t+(u) < tmax < t− = t−(u) such that t+u ∈ N+ λ , t −u ∈ N−λ and Iλ(t+u) = inf 0≤t≤t− Iλ(tu), Iλ(t−u) = sup t≥tmax Iλ(tu). (ii) For any u ∈ H− and λ > 0, there exists a unique t− = t−(u) > tmax such that t−u ∈ N−λ and Iλ(t−u) = sup t≥0 Iλ(tu). Lemma 2.7. Assume λ ∈ (0, λ1). Then for any u ∈ N+ λ and v ∈ N−λ , there exist B0 > Bλ > 0 such that ‖v‖ > B0 > Bλ > ‖u‖. 8 J. YANG, H. B. CHEN, Z. FENG EJDE-2020/101 Proof. Let u ∈ N+ λ ⊂ Nλ. In view of (2.2) and (2.4), it follows from Hölder’s inequality that a(2∗s − 2)‖u‖2 < (2∗s − q) ∫ Ω fλ(x)|u|qdx ≤ (2∗s − q)λS−q/2|f+|q∗‖u‖q. Then ‖u‖ < ( (2∗s − q)λS−q/2|f+|q∗ a(2∗s − 2) ) 1 2−q = Bλ. Similarly, if v ∈ N−λ ⊂ Nλ, from (2.3) and (A1) we have a(2− q)‖v‖2 < (2∗s − q) ∫ Ω g(x)|v|2 ∗ sdx ≤ (2∗s − q)S−2∗ s/2‖v‖2 ∗ s . Hence, we have ‖v‖ > (a(2− q)S 2∗s 2 2∗s − q ) 1 2∗s−2 = B0. By a direct calculation, we can verify that B0 > Bλ for λ ∈ (0, λ1), where λ1 is given in (2.5). � Corollary 2.8 ([11]). For any λ ∈ (0, λ1), N−λ is a closed set in E0 topology. 3. Proof of Theorems 1.1 and 1.2 In this section, we discuss the existence and multiplicity of solutions to problem (1.1) when m ≤ N N−2s . From Lemmas 2.3, 2.4 and 2.6, if m < N N−2s or m = N N−2s , and b < 1/Sm holds for any λ ∈ (0, λ1), then Nλ = N+ λ ∪N − λ . Now, we study the infimum of Iλ on the N±λ by defining c±λ = infN± λ Iλ(u) and λ3 = q 2λ1. Lemma 3.1. Assume that m < N N−2s or m = N N−2s , and b < 1/Sm. Then (i) for any λ ∈ (0, λ1), we have c+λ = infu∈N+ λ Iλ(u) < 0; (ii) for any λ ∈ (0, λ3), we have c−λ ≥ α > 0. In particular, if λ ∈ (0, λ1), then c+λ = inf u∈Nλ Iλ(u). Proof. (i) For u ∈ N+ λ , it follows from (2.4) that∫ Ω fλ(x)|u|qdx ≥ (2∗s − 2 2∗s − q ) a‖u‖2 + (2∗s − 2m 2∗s − q ) b‖u‖2m. (3.1) By (3.1), we obtain c+λ ≤ Iλ(u)− 1 2∗s 〈I ′λ(u), u〉 = (1 2 − 1 2∗s ) a‖u‖2 + ( 1 2m − 1 2∗s ) b‖u‖2m − (1 q − 1 2∗s )∫ Ω fλ(x)|u|qdx ≤ − (1 q − 1 2 )( 1− 2 2∗s ) a‖u‖2 − (1 q − 1 2m )( 1− 2m 2∗s ) b‖u‖2m < 0. (ii) For u ∈ N−λ , applying Lemma 2.7 and λ ∈ (0, λ3), we deduce Iλ(u) = (1 2 − 1 2∗s ) a‖u‖2 + ( 1 2m − 1 2∗s ) b‖u‖2m − (1 q − 1 2∗s )∫ Ω fλ(x)|u|qdx EJDE-2020/101 MULTIPLE SOLUTIONS TO FRACTIONAL KIRCHHOFF PROBLEM 9 ≥ ‖u‖q [as N (a(2− q) 2∗s − q S 2∗s 2 ) 2−q 2∗s−2 − λ (1 q − 1 2∗s ) |f+|q∗S−q/2 ] ≥ (2∗s − q)|f+|q∗‖u‖q 2∗sqS q 2 (λ3 − λ) ≥ α > 0. � Lemma 3.2. For each u ∈ N±λ and λ ∈ (0, λ1), there is a number ε and a differentiable function ζ : B(0, ε) ⊆ E → R such that ζ(0) = 1, the function ζ(v)(u− v) ∈ N±λ , and 〈ζ ′(0), v〉 = 2a〈u, v〉+ 2mb‖u‖2(m−1)〈u, v〉 − q ∫ Ω fλ|u|q−2uvdx− 2∗s ∫ Ω g|u|2∗ s−2uvdx (2− q)a‖u‖2 + (2m− q)b‖u‖2m − (2∗s − q) ∫ Ω g|u|2∗ sdx , where 〈u, v〉 = ∫ R2N (u(x)− u(y))(v(x)− v(y)) |x− y|N+2s dx dy for v ∈ Bε(0) = {v ∈ E0 : ‖v‖ ≤ ε}. The proof of the above lemma is similar to that of [11, Lemma 3.4], we omit it here. Lemma 3.3. Assume that λ ∈ (0, λ1). Then there exists a minimizing sequence {uk} ⊂ Nλ such that Iλ(uk)→ cλ and ‖I ′λ(uk)‖E−1 0 → 0 as k →∞ (3.2) with cλ = infu∈Nλ Iλ(u). Proof. It follows form Lemma 2.2 and the Ekeland’s variational principle [9] that there exists a minimizing sequence {uk} ⊂ Nλ such that cλ < Iλ(uk) < cλ + 1 k , (3.3) Iλ(uk) < Iλ(u) + 1 k ‖u− uk‖, u ∈ Nλ. (3.4) From (3.3) and Lemma 2.2, we have supk ‖uk‖ <∞. Now, we claim that ‖I ′λ(uk)‖E−1 0 → 0 as k →∞. From Lemma 3.2, we know the differentiable functions ζk : Bεk(0)→ R for some εk > 0 such that ζk(v)(uk− v) ∈ Nλ for v ∈ Bεk(0). For a fixed k, we take 0 < % < εk and define v% = %u/‖u‖ with u ∈ E0, u 6≡ 0 and ω% = ζk(v%)(uk − v%). Then it is easy to see that ω% ∈ Nλ. By (3.4), we can deduce that Iλ(ω%)− Iλ(uk) ≥ −1 k ‖ω% − uk‖, which implies 〈I ′λ(uk), ω% − uk〉+ ok (‖ω% − uk‖) ≥ − 1 k ‖ω% − uk‖. Therefore, −〈I ′λ(uk), v%〉+ (ζk(v%)− 1) 〈I ′λ(uk), uk − v%〉 ≥ − 1 k ‖ω% − uk‖+ ok (‖ω% − uk‖) . 10 J. YANG, H. B. CHEN, Z. FENG EJDE-2020/101 Then 〈I ′λ(ω%), uk − v%〉 = 0 yields − %〈I ′λ(uk), u ‖u‖ 〉+ (ζk(v%)− 1)〈I ′λ(uk)− I ′λ(ω%), uk − v%〉 ≥ −1 k ‖ω% − uk‖+ ok(‖ω% − uk‖). That is, 〈I ′λ(uk), u ‖u‖ 〉 ≤ 1 k% ‖ω% − uk‖+ ok(‖ω% − uk‖) % + (ζk(v%)− 1) % 〈I ′λ(uk)− I ′λ(ω%), uk − v%〉. (3.5) Since ‖ω% − uk‖ ≤ ρ|ζk(v%)| + |ζk(v%) − 1|‖uk‖ and lim%→0 |ζk(v%)−1| % ≤ ‖ζ ′k(0)‖, taking the limit %→ 0+ in (3.5), we obtain 〈I ′λ(uk), u ‖u‖ 〉 ≤ C k ( 1 + ‖ζ ′k(0)‖ ) for some C > 0 independent of u. It suffices to show that ‖ζ ′k(0)‖ is bounded. Assume by contradiction that 〈ζ ′(0), v〉 =∞. It follows from Lemma 3.2 and Hölder’s inequality that 〈ζ ′k(0), v〉 = C‖v‖ (2− q)a‖uk‖p + (2m− q)b‖uk‖2m − (2∗s − q) ∫ Ω g(x)|uk|2∗ sdx for some C > 0, which implies that there exists a subsequence {uk} such that (2− q)a‖uk‖2 + (2m− q)b‖uk‖2m − (2∗s − q) ∫ Ω g(x)|uk|2 ∗ sdx = ok(1). (3.6) Analogously, we can obtain a(2− 2∗s)‖uk‖2 + b(2m− 2∗s)‖uk‖2m − (q − 2∗s) ∫ Ω fλ(x)|uk|qdx = ok(1). (3.7) From (3.6) and (3.7), as in the proof of Lemma 2.3, we can see that λ ≥ λ1, which is impossible. � We define c∗λ := s N (aS) N 2s −Dλ 2 2−q , (3.8) where D = (2− q)(2∗s − q)|f+| 2 2−q q∗ 2q2∗s ( 2∗s − q (2∗s − 2)S ) q 2−q . Lemma 3.4. Assume that m ≤ N N−2s . Then Iλ satisfies the (PS) condition at the level cλ < c∗λ, where c∗λ is given in (3.8). Proof. Let {un} be a (PS)cλ sequence satisfying (3.2). It follows from Lemma 2.2 that {un} is bounded in E0. Hence, we may assume that, up to a subsequence, there exists u ∈ E0 such that un → u, a. e. in Ω, un ⇀ u, weakly in E0, un → u, strongly in Lr(Ω), 1 ≤ r < 2∗s. (3.9) EJDE-2020/101 MULTIPLE SOLUTIONS TO FRACTIONAL KIRCHHOFF PROBLEM 11 Meanwhile, there exists h̄ ∈ L2(Ω) such that |un(x)| ≤ h̄(x) a.e. in Ω. Note that limn→∞ ‖un‖ = β and M is continuous. We derive that M(‖un‖2) → M(β2) as n→∞. Set vn = un − u. We can assume that limn→∞ ‖vn‖ = d1 > 0. Otherwise, the conclusion follows. From [1, Lemma 2.7], (3.9) and condition (A1), we have ‖un‖2 = ‖un − u‖2 + ‖u‖2 + on(1),∫ Ω g(x)|un|2 ∗ sdx = ∫ Ω g(x)|un − u|2 ∗ sdx+ ∫ Ω g(x)|u|2 ∗ sdx+ on(1), (3.10) as n→∞. By (3.9)-(3.10), we obtain on(1) = 〈I ′λ(un), un〉 = M(‖un‖2)‖un‖2 − ∫ Ω fλ(x)|u|qdx− ∫ Ω g(x)|u|2 ∗ sdx − ∫ Ω g(x)|vn|2 ∗ sdx, (3.11) and on(1) = 〈I ′λ(un), u〉 = M ( ‖un‖2 ) ‖u‖2 − ∫ Ω fλ(x)|u|qdx− ∫ Ω g(x)|u|2 ∗ sdx. (3.12) As a consequence of (3.11) and (3.12), we obtain M ( ‖un‖2 ) ‖vn‖2 − ∫ Ω g(x)|vn|2 ∗ sdx = on(1). Let limn→∞ ∫ Ω g(x)|vn|2 ∗ sdx = d2. We derive( a+ bβ2(m−1) ) d2 1 = d2, (3.13) which implies d2 > 0. Moreover, from the definition of S in (2.2), we have d2 1 ≥ Sd 2/2∗ s 2 . (3.14) Combining (3.13) and (3.14), we obtain d2 1 ≥ a N−2s 2s S N 2s . (3.15) It follows from Hölder’s inequality that cλ = lim n→∞ ( Iλ(un)− 1 2∗s 〈I ′λ(un), un〉 ) = lim n→∞ {(1 2 − 1 2∗s ) a‖un‖2 + ( 1 2m − 1 2∗s ) b‖un‖2m − (1 q − 1 2∗s )∫ Ω fλ|un|qdx } ≥ (1 2 − 1 2∗s ) ad2 1 + (1 2 − 1 2∗s ) a‖u‖2 − (1 q − 1 2∗s ) λ|f+|q∗S−q/2‖u‖q. Setting Fλ(t) = (1 2 − 1 2∗s ) at2 − (1 q − 1 2∗s ) λ|f+|q∗S−q/2tq, we deduce that Fλ(t) attains its minimum as min t≥0 Fλ(t) = − (2− q)(2∗s − q)(λ|f+|q∗) 2 2−q 22∗sq ( 2∗s − q (2∗s − 2)S ) q 2−q = −Dλ 2 2−q , 12 J. YANG, H. B. CHEN, Z. FENG EJDE-2020/101 where D = (2− q)(2∗s − q)|f+| 2 2−q q∗ 2q2∗s ( 2∗s − q (2∗s − 2)S ) q 2−q . By applying (3.15), we obtain cλ ≥ s N (aS) N 2s −Dλ 2 2−q = c∗λ, which yields a contradiction with the hypothesis cλ < c∗λ. � We define λ4 := ( s N (aS) N 2s /D ) 2−q 2 and Λ0 = min{λ1, λ2, λ3, λ4}, where λ1, λ2 and λ3 are given in (2.5), (2.10) and Lemma 3.1, respectively. Proposition 3.5. Assume that m < N N−2s or m = N N−2s and b < 1/Sm. Then for λ ∈ (0,Λ0), Iλ has a minimizer u1 in Nλ, which is a positive solution to problem (1.1) with Iλ(u1) = c+λ and ‖u1‖ → 0 as λ→ 0. Proof. For λ ∈ (0,Λ0), combining the definition of c∗λ and Lemma 3.1 gives c+λ < 0 < c∗λ. In view of the Ekeland’s variational principle [9], there exists a (PS)c+λ sequence {un} ⊂ N+ λ satisfying (3.2). It follows from Lemma 3.4 that there exists u1 ∈ Nλ such that I ′λ(u1) = 0, Iλ(u1) = c+λ < 0, We now show that u1 ∈ N+ λ . Consider the case m < N N−2s , while the case m = N N−2s and b < 1/Sm follows similarly. Suppose by contradiction that u1 ∈ N−λ . Combining this with (2.3), we have u1 ∈ G+. On the other hand, from u1 ∈ Nλ and Iλ(u1) = c+λ < 0, we can see that u1 ∈ H+. Hence, from Lemma 2.4, we can infer that there exist t−(u1) > t+(u1) > 0 such that t−u1 ∈ N−λ and t+u1 ∈ N+ λ . This implies t− = 1 and t+ < 1. Therefore, there exists t̃ ∈ (t+, t−) such that Iλ(t+u1) = min 0≤t≤t− Iλ(tu1) < Iλ(t̃u1) < Iλ(t−u1) = Iλ(u1) = c+λ , which yields a contradiction. This implies u1 ∈ N+ λ . Furthermore, we show that u1 is positive. Note that Iλ(u) 6= Iλ(|u|) and ‖u‖ 6= ‖|u|‖ in E0. We consider the positive part of problem (1.1) by defining I+ λ (u) = a 2 ‖u‖2 + b 2m ‖u‖2m − 1 q ∫ Ω fλ(x)|u+|qdx− 1 2∗s ∫ Ω g(x)|u+|2 ∗ sdx. Then there exists a critical point u1 ∈ N+ λ for I+ λ . That is, for any v ∈ E0 it holds M ( ‖u1‖2 ) ∫ R2N (u1(x)− u1(y))(v(x)− v(y)) |x− y|N+2s dx dy = ∫ Ω fλ(x)|u+ 1 |q−1vdx− ∫ Ω g(x)|u+ 1 |2 ∗ s−1vdx. (3.16) Taking v = u−1 = min{u1, 0} as a test function in (3.16) and applying the inequality (u1(x)− u1(y))(u−1 (x))− u−1 (y)) = −u+ 1 (x)u−1 (y)− u−1 (x)u+ 1 (y)− [u−1 (x)− u−1 (y)]2 ≤ −[u−1 (x)− u−1 (y)]2, EJDE-2020/101 MULTIPLE SOLUTIONS TO FRACTIONAL KIRCHHOFF PROBLEM 13 we obtain (a+ b‖u−1 ‖2(m−1)) ∫ R2N |u−1 (x)− u−1 (y)|2 |x− y|N+2s dx dy = o(1), which implies ‖u−1 ‖ = 0, i.e. u1 ≥ 0 in RN . Moreover, by the strong maximum principle [3], we know that u1 is positive. Then, we prove that u1 is a local minimizer of Iλ in E0. From Lemmas 2.4 and 2.6, we have t+(u1) = 1 < tmax(u1). From continuity of u 7→ tmax(u), for the fixed ε > 0, there exists δ1 = δ1(ε) > 0 such that tmax(u1 − u) > 1 + ε for all ‖u‖ < δ1. Meanwhile, by Lemma 3.2, we can see that for a given δ2 > 0, there exists a C1 map ζ : Bδ2(0) → R+ such that ζ(u)(u1 − u) ∈ N+ λ and ζ(0) = 1. Hence, taking into account 0 < δ = min{δ1, δ2} and the uniqueness of zeros of fibering map, we have t+(u1−u) = ζ(u) < 1+ ε < tmax(u1−u) for all ‖u‖ < δ. By tmax(u1−u) > 1, we obtain Iλ(u1) ≤ Iλ(t+(u1 − u)(u1 − u)) ≤ Iλ(u1 − u), which implies that u1 is a local minimizer of Iλ in E0. By Lemma 2.1, we obtain that u1 is a positive solution to problem (1.1) . By Lemma 2.7, we arrive at the desired result. � In [19], it is shown that the infimum in (2.2) is attained by uε(x) = ε(N−2s)/2 (ε2 + |x|2)(N−2s)/2 , ε > 0, (3.17) which satisfies ∫ R2N |uε(x)− uε(y)|2 |x− y|N+2s dx dy = S|uε| 2∗ s 2∗ s . We define uε,η(x) = η(x)uε(x), (3.18) where η(x) ∈ C∞0 (Bρ(0)) satisfies 0 ≤ η ≤ 1 in Bρ(0), η ≡ 1 in Bρ/2(0) and η ≡ 0 in RN \ Bρ(0), for some ρ > 0 sufficiently small as given in condition (A1). From [19], we have ‖uε,η‖2 ≤ SN/(2s) +O ( εN−2s ) and |uε,η| 2∗ s 2∗ s = SN/(2s) +O ( εN ) . (3.19) It follows from (3.17) and (3.18) that∫ Bρ(0) |uε,η|qdx ≤ C (∫ Bε(0) 1 εq(N−2s)/2 dx+ ∫ Bρ(0)\Bε(0) εq(N−2s)/2 |x|q(N−2s) dx ) = CωN ( εN− (N−2s)q 2 + ε q(N−2s) 2 ∫ ρ ε rN−1−q(N−2s)dr ) = O ( εN− (N−2s)q 2 ) +O ( ε q(N−2s) 2 ) = O ( ε q(N−2s) 2 ) , where 1 < q < N/(N − 2s), and ωN denotes the unit sphere in RN . In view of condition (A1) and the definition of η, we have the following lemma. Lemma 3.6 ([11, 19]). For small ε > 0, the following statements are true. (i) ∫ Bρ(0) |uε,η|qdx = O ( εq(N−2s)/2 ) ; (ii) ∫ Bρ(0) |uε,η|2 ∗ s−1dx ≥ CεN−2s 2 ; (iii) ∫ Bρ(0) g(x)|uε,η|2 ∗ sdx = S N 2s +O(εN ). 14 J. YANG, H. B. CHEN, Z. FENG EJDE-2020/101 We consider the following two sets A1 = {u ∈ E \ {0} : 1 ‖u‖ t− ( u ‖u‖ ) > 1} ∪ {0}, A2 = {u ∈ E \ {0} : 1 ‖u‖ t− ( u ‖u‖ ) < 1}, where t− is given in Lemma 2.4. It follows from [25] that t−(u) is continuous for u ∈ E0\{0} and N−λ = {u ∈ E0\{0} : 1 ‖u‖ t −( u ‖u‖ ) = 1} splits E0 into two connected parts A1 and A2. It follows from Lemmas 2.4 and 2.6 that for any u ∈ N+ λ and λ ∈ (0, λ2), we have 1 < tmax(u) < t−(u). Then N+ λ ⊂ A1. Particularly, u+ λ ∈ A1. Lemma 3.7. Assume that m < N N−2s or m = N N−2s and b < 1/Sm. Then for any ε > 0, there exists t1 > 0 such that u1 + t1uε,η ∈ A2. Proof. We just prove the case of m = N N−2s and b < 1/Sm, since the case of m < N N−2s can be processed in a similar manner. We claim that there exists a constant c̃ > 0 such that 0 < t− ( u1+tuε,η ‖u1+tuε,η‖ ) < c̃ for m = N N−2s and b < 1/Sm. Otherwise, there is a sequence {tn} ⊂ R+ such that tn →∞ and t− ( u1+tnuε,η ‖u1+tnuε,η‖ ) →∞ as n→∞. Let vn = u1+tnuε,η ‖u1+tnuε,η‖ . From (3.19), we deduce that∫ Bρ(0) g|vn|2 ∗ sdx− b‖vn‖2 ∗ s = ∫ Bρ(0) g|u1 + tnuε,η|2 ∗ sdx ‖u1 + tnuε,η‖2∗ s − b → ∫ Bρ(0) g|uε,η|2 ∗ sdx ‖uε,η‖2∗ s − b ≥ 1/Sm − b+O ( εN ) > 0 for 0 < ε < ε1 with some ε1 > 0, as n→∞. Thus, Iλ(t−(vn)vn)→ −∞ as n→∞ for m = N N−2s and ε ∈ (0, ε1), which contradicts Lemma 2.2. According to [25, Lemma 3.6], we obtain u1 + t1uε,η ∈ A2 immediately. � Lemma 3.8. Assume that m < N N−2s or m = N N−2s and b < 1/Sm. Then there exist Λ∗ ∈ (0,Λ0] and b̄ > 0 such that for any λ ∈ (0,Λ∗) and b ∈ (0, b̄) it holds sup t≥0 Iλ(u1 + tuε,η) < c∗λ, where c∗λ is given in (3.8). Proof. For any α, β ≥ 0 and m ≥ 1, we recall the inequality (α+ β)m ≤ αm + Cm(αm + βm) +mαm−1β, where Cm > 0 is a constant depending on m. It follows from Young’s inequality that b 2m ‖u1 + tuε,η‖2m ≤ b 2m ‖u1‖2m + b‖u1‖2(m−1)t ∫ R2N (u1(x)− u1(y))(uε,η(x)− uε,η(y)) |x− y|N+2s + bCm‖u1‖2m + bDmt 2m‖uε,η‖2m, (3.20) EJDE-2020/101 MULTIPLE SOLUTIONS TO FRACTIONAL KIRCHHOFF PROBLEM 15 where Dm > 0. Since u1 is a critical point of Iλ, we obtain 〈I ′λ(u1), tuε,η〉 = 0. (3.21) In view of the inequality (α+ β)p − αp − βp − pαp−1β ≥ Cαβp−1, α, β ≥ 0, p > 2, it follows from the definition of η, (3.20), (3.21) and condition (A1) that Iλ(u1 + tuε,η) = a 2 ‖u1 + tuε,η‖2 + b 2m ‖u1 + tuε,η‖2m − 1 q ∫ Bρ(0) fλ|u1 + tuε,η|qdx− 1 2∗s ∫ Bρ(0) g|u1 + tuε,η|2 ∗ sdx ≤ Iλ(u1) + a 2 ‖tuε,η‖2 + bCm‖u1‖2m + bDm‖tuε,η‖2m − 1 q ∫ Bρ(0) fλ (∫ tuε,η 0 [|u1 + s|q−1 − |u1|q−1]ds ) dx − 1 2∗s ∫ Bρ(0) g [ |u1 + tuε,η|2 ∗ s − |u1|2 ∗ s − 2∗stuε,η|u1|2 ∗ s−1 ] dx ≤ c+λ + a 2 ‖tuε,η‖2 + bCm‖u1‖2m + bDm‖tuε,η‖2m + C|f−|∞|tuε,η|qq − 1 2∗s ∫ Bρ(0) g|tuε,η|2 ∗ sdx− C ∫ Bρ(0) |u1||tuε,η|2 ∗ s−1dx. (3.22) We now consider Jλ(tuε,η) = a 2 ‖tuε,η‖2 + bCm‖u1‖2m + bDm‖tuε,η‖2m + C|f−|∞|tuε,η|qq − 1 2∗s ∫ Bρ(0) g|tuε,η|2 ∗ sdx− C ∫ Bρ(0) |u1||tuε,η|2 ∗ s−1dx. Claim 1. There exist tε and t2 > 0 independent of ε and λ such that t2 ≤ tε ≤ t1, Jλ(tεuε,η) = sup t≥0 Jλ(tuε,η), d dt Jλ(tuε,η)|t=tε = 0, (3.23) where t1 is given in Lemma 3.7. Since λ ∈ (0, λ3), from Lemma 3.1 we have 0 < α < α− c+λ ≤ c − λ − c + λ ≤ sup t≥0 Iλ(u1 + tuε,η)− c+λ ≤ Jλ(tεuε,η), which implies t2 ≤ tε for some t2 > 0. To find the estimate of supt≥0 Jλ(tuε,η), we define h(t) = at2 2 ‖uε,η‖2 − t2 ∗ s 2∗s ∫ Bρ(0) g|uε,η|2 ∗ sdx. By (3.19) and Lemma 3.6, we obtain sup t≥0 h(t) ≤ s N (aS) N 2s +O ( εN−2s ) . (3.24) 16 J. YANG, H. B. CHEN, Z. FENG EJDE-2020/101 From (3.23), (3.24) and Lemmas 3.6 and 2.7, we deduce that Jλ(tuε,z) ≤ s N (aS) N 2s + bCm‖u1‖2m + bDmt 2m 1 ‖uε,η‖2m +O ( εN−2s ) + Ctq1|f−|∞|uε,η|qq − Ct 2∗ s−1 2 ∫ Bρ(0) |u1||uε,η|2 ∗ s−1dx ≤ s N (aS) N 2s +O(εN−2s) + Cbλ 2m 2−q + Cb+ Cε q(N−2s) 2 − Cε N−2s 2 ≤ s N (aS) N 2s + Cbλ 2m 2−q + Cb− Cε N−2s 2 0 (3.25) for some ε0 > 0. Thus, there exist two positive numbers Λ∗ ∈ (0,Λ0] and b̄ > 0 such that for any λ ∈ (0,Λ∗) and b ∈ (0, b̄) it holds Cbλ 2m 2−q + Cb+Dλ 2 2−q < Cε N−2s 2 0 . Combining this and (3.22)-(3.25) and by Lemma 3.8, we arrive at the desired result. � Proof of Theorem 1.1. Clearly, (i) follows from Proposition 3.5. (ii) Let λ ∈ (0,Λ∗). From Proposition 3.5 and Lemma 3.7, we obtain u1 ∈ A1 and u1 + t1uε,η ∈ A2. We define a path γ(s) = u1 +st1uε,η for s ∈ [0, 1]. Since γ(0) ∈ A1 and γ(1) ∈ A2, there exists s ∈ (0, 1) such that u1 + st1uε,η ∈ N−λ , which implies c−λ ≤ supt≥0 Iλ(u1 + tuε,η). According to Lemma 3.8, we obtain c−λ < c∗λ for any λ ∈ (0,Λ∗) and b ∈ (0, b̄). In view of Corollary 2.8, N−λ is a closed set. By Proposition 3.5, there exists u2 ∈ N−λ such that I ′λ(u2) = 0 and Iλ(u2) = c−λ . This indicates that u2 is also a positive solution to problem (1.1). � Proof of Theorem 1.2. (i) From Lemma 2.5, we obtain N+ λ = Nλ and define cλ = infu∈Nλ Iλ(u). It is clear that cλ < 0. By Proposition 3.5, there exists a (PS)cλ sequence {un} ⊂ N+ λ for Iλ. It follows from Lemma 2.2 that {un} is bounded in E0. Hence, up to a subsequence, there is u ∈ E0 satisfying (3.9). Denote vn = un − u, for b ≥ 1/Sm. Then ∫ Ω g(x)|u|2 ∗ sdx− b‖u‖2m < 0. Combining this (3.11) and (3.12) yields a‖vn‖2 ≤ a‖vn‖2 + b‖u‖2(m−1)‖vn‖2 + b‖vn‖2m − ∫ Ω g(x)|u|2 ∗ sdx = on(1), which implies un → u in E0. According to Proposition 3.5, we see that u is a positive solution to problem (1.1). The proof of (ii) is similar to that of Theorem 1.1, so we omit it. � 4. Proof of Theorem 1.3 In this section, we assume that m > N N−2s , f− ≡ 0 and condition (A1) holds. Let us start with a compactness result. Lemma 4.1. Iλ satisfies the (PS) condition if c < c∗λ,b := s N (aS) N 2s −D1b−D2λ 2 2−q . EJDE-2020/101 MULTIPLE SOLUTIONS TO FRACTIONAL KIRCHHOFF PROBLEM 17 Proof. Let {un} be a (PS)c sequence satisfying (3.2). We claim that {un} is bounded in E0. By way of contradiction, we assume that there is a subsequence of the original sequence such that ‖un‖ → ∞, as n → ∞. We define wn = un/‖un‖. By the Sobolev and Hölder’s inequalities, we obtain∫ Ω g(x)|wn|2 ∗ sdx ≤ S−2∗ s/2 and ∫ Ω λf(x)|wn|qdx ≤ λS−q/2|f |q∗ . Therefore, c+ on(1) ‖un‖2∗ s = a 2 ‖un‖2 ‖un‖2∗ s + b 2m ‖un‖2m ‖un‖2∗ s − 1 q ∫ Ω λf(x)|u|qdx ‖un‖2∗ s − 1 2∗s ∫ Ω g(x)|wn|2 ∗ sdx ≥ b 2m ‖un‖2m−2∗ s − S−2∗ s/2 2∗s + on(1)→∞. In view of m > N/(N − 2s), this yields a contradiction. Hence, up to a subsequence, there exists u ∈ E0 satisfying (3.9). Similar to the proof of Lemma 3.4, setting vn = un−u, we can suppose that limn→∞ ‖vn‖ = d1 > 0. Using Hölder’s inequality, (3.15) and m > N N−2s , we deduce that c = lim n→∞ ( Iλ(un)− 1 2∗s 〈I ′λ(un), un〉 ) = lim n→∞ {as N ‖un‖2 − ( 1 2∗s − 1 2m ) b‖un‖2m − (1 q − 1 2∗s ) λ ∫ Ω f |un|qdx} ≥ as N d2 1 + as N ‖u‖2 −D1b− (1 q − 1 2∗s ) λ|f |q∗‖u‖q ≥ s N (aS) N 2s −D1b−D2λ 2 2−q , where D1 = D1(N,m, s) and D2 = D2(N, q, S, a, |f |q∗). This contradicts the hy- pothesis of c < c∗λ,b.. � Lemma 4.2. There exist λ5 > 0 and r > 0 such that for any λ ∈ (0, λ5) it holds inf u∈E0,‖u‖=r Iλ(u) = α̃ > 0. In particular, when λ = 0, there exists an r0 > r such that I0(u) > 0 for all u ∈ Br0\{0}. Proof. For u ∈ E0, we have Iλ(u) = a 2 ‖u‖2 + b 2m ‖u‖2m − 1 q ∫ Ω λf(x)|u|qdx− 1 2∗s ∫ Ω g(x)|u|2 ∗ sdx ≥ a 2 ‖u‖2 − λ|f |q∗ qSq/2 ‖u‖q − 1 2∗sS 2∗ s/2 ‖u‖2 ∗ s ≥ (a 2 ‖u‖2−q − λ|f |q∗ qSq/2 − 1 2∗sS 2∗ s/2 ‖u‖2 ∗ s−q ) ‖u‖q. (4.1) We define l(t) = a 2 t2−q − 1 2∗s S−2∗ s/2t2 ∗ s−q for t ≥ 0. In view of 2 < 2∗s, for each u ∈ E0 with ‖u‖ = r := [a2∗sS 2∗ s/2(2− q) 2(2∗s − q) ]1/(2∗ s−2) , 18 J. YANG, H. B. CHEN, Z. FENG EJDE-2020/101 we obtain maxt≥0 l(t) = l(r) > 0. Thus, by taking λ < λ5 = l(r)q S−q/2|f |q∗ , we obtain Iλ(u) ≥ ( l(r)− λ |f |q ∗ qS2/q ) rq =: α̃ > 0. When λ = 0, from (4.1), there exists an r0 = [(2∗s − q)/(2 − q)]1/(2 ∗ s−2)r > r such that I0(u) > 0 for u ∈ Br0\{0}. � Lemma 4.3. Let λ5 be given in Lemma 4.2. Then there exist two positive constants b1 and 0 < λ6 ≤ λ5 such that for each b ∈ (0, b1) and λ ∈ (0, λ6), problem (1.1) admits a positive solution ub with Iλ(ub) < 0. Proof. It follows from Lemma 4.2 that there is an r > 0 such that Iλ(u) ≥ 0 for ‖u‖ = r. For u ∈ E0\{0} and t > 0 sufficiently small, we have Iλ(tu) = at2 2 ‖u‖2 + bt2m 2m ‖u‖2m − tq q ∫ Ω λf(x)|u|qdx− t2 ∗ s 2∗s ∫ Ω g(x)|u|2 ∗ sdx < 0. Thus, we obtain mλ := inf{Iλ(u) : u ∈ B̄r} < 0. (4.2) By Ekeland’s variational principle [9], there exists a minimizing sequence {un} ⊂ B̄r such as Iλ(un)→ mλ, ‖I ′λ(un)‖E−1 0 → 0, as n→∞. On the other hand, it is easy to see that there exist b1 > 0 and λ6 ≤ λ5 such that c∗λ,b > 0, b ∈ (0, b1), λ ∈ (0, λ6), where c∗λ,b is given in Lemma 4.1. It follows from (4.2) and Lemma 4.1 that there exists ub ∈ E0 such that un → ub, i.e. ub is a nontrivial solution of problem (1.1). By Proposition 3.5, we see that ub is a positive solution of (1.1). � Lemma 4.4. Let r and b1 be given in Lemmas 4.2 and 4.3, respectively. Then there exist 0 < b2 ≤ b1 and e1 ∈ E0 with ‖e1‖ > r such that Iλ(e1) < 0 for b ∈ (0, b2). Proof. If b = 0, we consider the functional Iλ denoted by Iλ,0(tuε,η) = at2 2 ‖uε,η‖2 − tq q ∫ Bρ(0) λf(x)|uε,η|qdx− t2 ∗ s 2∗s ∫ Bρ(0) g(x)|uε,η|2 ∗ sdx. Recalling Fatou’s Lemma and Lemma 3.6 (iii), we can see that for small ε > 0 it holds lim t→∞ Iλ,0(tuε,η) t2 ∗ s ≤ − 1 2∗s S N 2s + CεN < 0. Namely, there exists a large T > 0 satisfying ‖Tuε,η‖ > r and Iλ,0(Tuε,η) < 0. Since Iλ(Tuε,η) → Iλ,0(Tuε,η) as b → 0+, we deduce that there exists 0 < b2 ≤ b1 such that Iλ(Tuε,η) < 0 for any b ∈ (0, b2). � Lemma 4.5. Let λ6 and b2 be given in Lemmas 4.3 and 4.4, respectively. Then there exists αb < 0 such that for any b ∈ (0, b2) it holds αb ≤ m̄λ := inf{Iλ(u) : u ∈ E0} < 0. EJDE-2020/101 MULTIPLE SOLUTIONS TO FRACTIONAL KIRCHHOFF PROBLEM 19 Furthermore, for any b ∈ (0, b2) and λ ∈ (0, λ6), problem (1.1) admits a positive solution uλ with Iλ(uλ) = m̄λ. Proof. Note that Iλ(u) = a 2 ‖u‖2 + b 2m ‖u‖2m − 1 q ∫ Ω λf(x)|u|qdx− 1 2∗s ∫ Ω g(x)|u|2 ∗ sdx ≥ b 2m ‖u‖2m − 1 q λ|f |q∗S−q/2‖u‖q − 1 2∗s S− 2∗s 2 ‖u‖2 ∗ s . Let Ā = 2mλ|f |q∗ bqSq/2 , B̄ = 2m 2∗sbS 2∗ s/2 , ΦĀ,B̄(t) = t2m − Ātq − B̄t2 ∗ s . From [22, Lemm 2.3], for any b > 0 there exist t3, t4 > 0 such that αb = min t≥0 ΦĀ,B̄(t) = ΦĀ,B̄(t3) < 0 and ΦĀ,B̄(t) ≥ 0 for t ≥ t4. While, by Lemma 4.4, it is easy to see that for any b ∈ (0, b2) it holds m̄λ := inf{Iλ(u) : u ∈ E0} < 0. (4.3) Then using a similar strategy of Lemma 4.3, we obtain that uλ is a positive solution of problem (1.1) . � To obtain two distinct solutions to (1.1), we need to show that the infimum m̄λ < mλ. Lemma 4.6. There exists 0 < λ7 ≤ λ6 such that m̄λ < mλ for b ∈ (0, b2) and λ ∈ (0, λ7). Proof. For λ = 0, let m0 be given as in (4.3). So, for any b ∈ (0, b2), problem (1.1) admits a positive solution u0,b satisfying I0(u0,b) = m0 := inf{I0(u) : u ∈ E0} < 0. Taking into account f− = 0, we deduce m̄λ ≤ Iλ(u0,b) = I0(u0,b)− λ ∫ Ω f |u0,b|qdx = m0 − λ ∫ Ω f |u0,b|qdx ≤ m0. (4.4) In view of Lemma 4.2 and (4.2), we have mλ → 0 as λ → 0. Then, there exists 0 < λ7 ≤ λ6 such that m0 < mλ for any b ∈ (0, b2) and λ ∈ (0, λ7). Combining this and (4.4), we arrive at the desired result. � It follows from Lemmas 4.2 and 4.4 that Iλ has the mountain pass geometry. Using the mountain pass theorem [24], there exists a (PS)cλ,b sequence {un} ⊂ E0, that is Iλ(un)→ cλ,b, ‖I ′λ(un)‖E−1 0 → 0. We note that cλ,b has the characteristic property c = inf γ∈Γ max t∈[0,1] I(γ(t)), where Γ := {γ ∈ C([0, 1], E0) : γ(0) = ub, γ(1) = ub + Tuε,η}. Similar to Lemma 3.8, we can obtain the following lemma. Lemma 4.7. There exist 0 < Λ∗ ≤ λ7 and 0 < b∗ ≤ b2 such that for any λ ∈ (0,Λ∗) and b ∈ (0, b∗) it holds cλ,b ≤ sup t≥0 Iλ(ub + tuε,η) < c∗λ,b. 20 J. YANG, H. B. CHEN, Z. FENG EJDE-2020/101 Proof of Theorem 1.3. 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Chen; Nontrivial solutions for Kirchhoff-type problems with a parameter, J. Math. Anal. Appl., 433 (2016), 455-472. Jie Yang School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China. Department of Mathematics, Huaihua University, Huaihua, Hunan 418008, China Email address: dafeyang@163.com Haibo Chen School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China Email address: math chb@163.com Zhaosheng Feng School of Mathematical and Statistical Sciences, University of Texas Rio Grande Val- ley, Edinburg, TX 78539, USA Email address: zhaosheng.feng@utrgv.edu 1. Introduction and statement of results 2. Preliminary results 3. Proof of Theorems ?? and ?? 4. Proof of Theorem ?? Acknowledgments References