Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 61, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GROUND STATE SOLUTIONS FOR FRACTIONAL p-KIRCHHOFF EQUATION LIXIONG WANG, HAIBO CHEN, LIU YANG Abstract. We study the fractional p-Kirchhoff equation( a+b ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy ) (−∆)spu−µ|u|p−2u = |u|q−2u, x ∈ RN , where (−∆)sp is the fractional p-Laplacian operator, a and b are strictly positive real numbers, s ∈ (0, 1), 1 < p < N s , and p < q < p∗s − 2 with p∗s = Np N−ps . By using the variational method, we prove the existence and uniqueness of global minimum or mountain pass type critical points on the Lp-normalized manifold S(c) := { u ∈W s,p(RN ) : ∫ RN |u|pdx = cp } . 1. Introduction and statement of main results In this article, we consider the fractional p-Kirchhoff equation( a+ b ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy ) (−∆)spu− µ|u|p−2u = |u|q−2u, (1.1) for x ∈ RN , where (−∆)spu is the fractional p-Laplacian operator, a and b are strictly positive real numbers, s ∈ (0, 1), 1 < p < N s , and p < q < p∗s − 2 with p∗s = Np N−ps . Equation (1.1) is related to stationary solutions of utt + ( a+ b ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy ) (−∆)spu = f(x, u), (1.2) where f(x, u) is a general nonlinearity. Kirchhoff’s equation was suggested as a model for the transverse oscillations of a stretched string of the form [13] ρhutt − ( p0 + Eh 2L ∫ L 0 |∇u|2 dx ) ∆u+ δut + f(x, u) = 0 (1.3) for 0 < x < L and t ≥ 0, where u = u(x, t) is the lateral displacement at position x and at time t, L is the length of the string, h is the cross section area, ρ is the mass density, p0 is the initial stress tension, E is the Young modulus, δ is the resistance modulus and f is the external force. Comparing with the semilinear equations , it 2020 Mathematics Subject Classification. 35J20, 35J60. Key words and phrases. Variational method; Lp-normalized critical point; fractional; p-Kirchhoff equation; uniqueness. ©2022. This work is licensed under a CC BY 4.0 license. Submitted April 11, 2022. Published August 19, 2022. 1 2 L. WANG, H. CHEN, L. YANG EJDE-2022/61 is way more challenge and fascinating to research equations (1.1) and (1.2) visible of the existence of the nonlocal term∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy(−∆)spu. In recent years, many authors have dealt with Kirchhoff-type problems in the context of classical Laplace operators and proved results concerning the existence, multiplicity and properties of the solutions by variational methods. The existence, nonexistence and multiplicity of nontrivial solutions of fractional Kirchhoff-type equation with Hardy-Littlewood-Sobolev critical exponent were presented in [20]. By using a fibering-type approach, Che and Wu [3] obtained several quantitative results for the problem − ( a+ b ∫ RN |∇u|2 dx ) ∆u+ u = k(x)|u|p−2u+m(x)|u|q−2u in RN , (1.4) where N ≥ 3, a, b > 0, 1 < q < 2 < p < min{4, 2∗}. The three positive solutions are obtained mainly by using the Ekeland variational principle and the innovative constraint method of Nehari manifolds. For the p-Laplace operators, the uniqueness of the positive solution of the p-Laplace equation with Hardy potential and the asymptotic behavior was established [8]. For instance, replacing the term |u|pu with a general nonlinearity f(x, u), there are many results on the existence of solutions for such equations, one can refer to [1, 5, 10, 15, 16] and the references therein. For the fractional Laplace equation, Feng and Su [7] establishes a generalized version of the lion-type theorem for the fractional Laplace that obtains the ground state solution. The existence of ground state solutions of fractional equations can also be found in Su and Feng [21] recent article. However, there is little literature concerned about the normalized solutions for the fractional p-Kirchhoff equation. With regard to the point, we attempt to study this kind of problem in this paper. By treating µ as an unknown Lagrange multiplier, Equation (1.1) can be viewed as an eigenvalue problem. From this perspective, we can solve it by studying some constraint variational problems (1.1) and obtain a normalized solution. Inspired by [2, 12, 23], we first consider the following minimization problem I(c) := inf u∈S(c) Ep(u) (1.5) where S(c) := { u ∈W s,p(RN ) : ∫ RN |u|pdx = cp } . and Ep(u) = a p ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy + b 2p (∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy )2 − 1 q ∫ RN |u|qdx (1.6) A normalized solution to problem (1.5) exists if u ∈ S(c) is a minimizer of problem (1.5) such that there exists µ ∈ R such that E′p(u) = µ|u|p−2u, i.e., u ∈ S(c) is a solution of (1.1) for some µ ∈ R. For the case of p = 2 and s = 1, scholars have made in-depth research. For example, Ye [23] according to the principle of concentrated compactness, it is proved that there exists c∗s > 0 such that if c > c∗s, problem (1.5) is reachable, where the EJDE-2022/61 FRACTIONAL p-KIRCHHOFF EQUATION 3 constant c is related to the ground state solution of equation (1.7) below. Zeng and Zhang [24] reproved some of the results in [22] by applying some simple energy estimates. They also showed that the minimum element of problem (1.5) (if it exists) is unique and is a telescopic translation of the ground state solution of equation (1.7). This article intends to prove the existence and uniqueness of the minimal element of problem (1.5) and extend the results of paper [11, 24] to the case of p ∈ (1, Ns ). To do this, we first study the equation (−∆)spu+ [ pqs N(q − p) − 1 ] |u|p−2u = |u|q−2u, 0 < q < p∗s. (1.7) Note that if p 6= 2 or s 6= 1, the operator (−∆)sp is no longer linear, which leads to some different properties from the case of p = 2 and s = 1. For example, when p = 2, equation (1.7) has a unique positive radial symmetric solution; it is not clear whether the positive radial solution of (1.7) with general p ∈ (1, Ns ) is unique. This brings some new difficulties to the study of problems (1.5) and (1.7). First, we introduce some known results about equation (1.7). The energy functional of (1.7) can be defined as G(u) = 1 p ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy+ 1 p [ pqs N(q − p) −1 ] ∫ RN |u|pdx− 1 q ∫ RN |u|qdx. Moreover, all nontrivial solutions of (1.7) can be expressed as W := { u ∈W s,p(RN ) \ {0} : 〈G′(u), ϕ〉 = 0,∀ϕ ∈W s,p(RN ) } . We say that Q(x) ∈W s,p(RN ) is a ground state solution of (1.7), if u satisfies Q(x) ∈ N := { u ∈ W : G(u) = inf v∈W G(v) } = { u ∈ W : G(u) = inf v∈W s N ∫ RN |v|pdx } . Combining with the Pohozaev and Nehari identity, u(x) satisfies∫ RN ∫ RN |Q(x)−Q(y)|p |x− y|N+ps dx dy = ∫ RN |Q(x)|pdx = N(q − p) pqs ∫ RN |Q(x)|qdx. (1.8) Before stating our main results, we introduce the fractional Gagliar do-Nirenberg inequality [9]∫ RN |u|qdx ≤ pqs N(q − p)‖Q‖q−pLp (∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy )N(q−p) p2s × (∫ RN |u|pdx ) q p− N(q−p) p2s , ∀u ∈W s,p(RN ). (1.9) Furthermore, Q is an optimizer of the fractional Gagliardo-Nirenberg inequality. We note that, in a similar way to the literature [6], we can prove that all optimizers of (1.9) are in fact the scaling and translations of Q(x), i.e., belong to the set {λQ(αx+ y) : α, λ ∈ R+, y ∈ R, Q ∈ W}. (1.10) Remark 1.1. If p ∈ (1, Ns ), it is known from the conclusion in [14, 6, 18] that the radially symmetric ground state solution of equation (1.7) is unique (up to trans- lations). Accordingly, Q(|x|) is the unique (up to translations) radially symmetric positive solution of the following equation (1.7) in W s,p(RN ). The following theorem discusses the existence and uniqueness of the reachable elements of equation (1.5). 4 L. WANG, H. CHEN, L. YANG EJDE-2022/61 Theorem 1.2. (i) Suppose that 0 < q < p + p2s N , problem (1.5) has a unique minimizer uc (up to translations). Moreover, the function uc satisfies uc = cλ N/p p ‖Q‖Lp Q(λpx), where λ = ( tp cp ) 1 ps with tp being the unique minimum point of the function fp(t) = a p t+ b 2p t2 − psc ( q−N(q−p) ps ) N(q − p)‖Q‖(q−p)Lp t N(q−p) p2s , t ∈ (0,+∞). (1.11) (ii) Suppose that q = p + p2s N , if c > (aN(q−p) p2s ) 1 q−p ‖Q‖Lp , then problem (1.5) has a unique minimizer uc (up to translations). Moreover, uc = cλ N/p p ‖Q‖Lp Q(λpx), (1.12) where λp = [p2sc(q−p) − aN(q − p)‖Q‖q−pLp bN(q − p)cp‖Q‖q−pLp ] 1 ps . On the contrary, problem (1.5) has no minimizer if c ≤ [aN(q−p) p2s ] 1 q−p ‖Q‖Lp . (iii) Suppose that p + p2s N < q < min{p + 2p2s N , p∗s}, if c ≥ c∗, then (1.5) has a unique minimizer uc (up to translations). Moreover, uc = cλ N/p p ‖Q‖Lp Q(λpx), (1.13) with λp = { 2 [ N(q − p)− p2s ] a[ 2p2s−N(q − p) ] cpb } 1 ps , I(c) = (c∗)q− N(q−p) ps − cq− N(q−p) ps N(q−p)‖Q‖q−p Lp ps {2[N(q − p)− p2s]a [2p2s−N(q − p)]b }N(q−p) p2s for all c ≥ c∗. Conversely, problem (1.5) has no minimizer if c < c∗, where c∗ is given by (3.6). (iv) Suppose that p + 2p2s N ≤ q < p∗s, problem (1.5) has no minimizer for all c > 0. By the above theorem, we first obtain a complete classification with respect to the exponent q with the Lp-normalized solutions of problem (1.5). Moreover, all these solutions are unique up to translations, our proof relies only on some simple energy estimates and avoids the use of the concentration-compactness principle. Theorem 1.2 shows us that the minimizer of (1.5) must be a scaling of Q(x), which extends [11, Theorem 1.1], also the existence of the minimizer of (1.5) is discussed therein. Moreover, we see that problem (1.5) has no minimizer if q ≥ p + 2p2s N . Thus, to obtain the normalized solutions for (1.1), one may search for saddle point for functional (1.6). Inspired by other studies [11, 24], we examine the mountain pass type critical point for Ep(·) on S(c). Before stating our second result, we introduce the following definition. EJDE-2022/61 FRACTIONAL p-KIRCHHOFF EQUATION 5 Definition 1.3. Functional Ep(·) is said to have the mountain pass geometry on S(c) given c > 0, if there exists K(c) > 0 such that γ(c) := inf h∈Γ(c) max t∈[0,1] Ep ( h(t) ) > max{Ep(h(0)), Ep(h(1))} (1.14) holds in the set Γ(c) = { h ∈ C ( [0, 1];S(c) ) : h(0) ∈ AK(c) and E ( h(1) ) < 0 } , where AK(c) = { u ∈ S(c) : (∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy )1/p ≤ K(c) } . Since only q ≥ p + 2p2s N is considered, we assume that q satisfies one of the conditions below: (A1) q > p+ 2p2s N ; (A2) q = p+ 2p2s N and c > c∗ := [ bN(q−p)‖Q‖q−p Lp 2p2s ] ps pqs−N(q−p) . As well as by studying some analytical properties of γ(c) and involving rigorous ar- guments, we will prove separately that Ep(·) possesses the mountain pass geometry on S(c), see lemma 2.3 below for details. In addition, there exists uc ∈ S(c) such that Ep(uc) = γ(c), and uc is a solution of (1.1) with some λ ∈ R−. Inspired by this fact and the proof of our first theorem, we try to study some characteristics of γ(c) by bringing in some new estimates of the observations and energies. Furthermore, as a side effect, we show that a critical point on the level γ(c) is known to be unique if uc ∈ S(c) is a critical point of Ep(·) by indeed a scaling of Q(x). So, we have the following theorem. Theorem 1.4. Suppose conditions (A1) or (A2) hold, and that t̄p be the unique maximum point of fp(t) at (0,+∞). Then γ(c) = fp(t̄p), which can be achieved by ūc = cλ̄ N p p ‖Q‖Lp Q(λ̄px), where λ̄p = ( t̄p cp ) 1 ps . Moreover, ūc is also a solution of (1.1) for some λ ∈ R−. Remark 1.5. Still let fp(·) be given by (1.11) and note that it has a unique maximum point in (0,+∞) once (A1) or (A2) is assumed. In Theorem 1.4 , ūc is the unique solution of (1.14). The significance is as follows: if E′p(ū)|S(c) = 0 and Ep(ū) = γ(c). (1.15) i.e., ū ∈ S(c) is a critical point of Ep(·) on S(c) and its energy equal to γ(c). Then, up to translations, ū = ūc. 2. Preliminaries We first give some useful notation and basic results for fractional Sobolev spaces. Let 0 < s < 1 < p < ∞ be real numbers. The fractional Sobolev space W s,p(RN ) is defined by W s,p(RN ) = { u ∈ Lp(RN ) : ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy } , 6 L. WANG, H. CHEN, L. YANG EJDE-2022/61 equipped with the norm ‖u‖W s,p(RN ) = ( ‖u‖p Lp(RN ) + ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy )1/p . Now, we introduce the fractional Gagliardo-Nirenberg-Sobolev inequality; for more details, see [4, 17, 19]. Lemma 2.1 ([4]). If u ∈W s,p(RN ) and p < q < p∗s − 2, then∫ RN |u|qdx ≤ pqs N(q − p)‖Q‖q−pLp (∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy )N(q−p) p2s (∫ RN |u|pdx ) q p− N(q−p) p2s . Moreover, Q is an optimizer of the fractional Gagliardo-Nirenberg inequality. The Pohozaev identity plays an important role in our discussion. We give it in the following lemma. Lemma 2.2 ([17, Lemma 2]). Let u ∈W s,p(RN ), N ≥ 2, satisfy the equation (−∆)spu = f(u). then (N − ps) p ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy = N ∫ RN F (u)dx, where F (s) = ∫ s 0 f(t)dt. To prove the theorem 1.4, we first introduce the following lemma, which indicates that if hypothesis (A1) or (A2) hold, Ep(·) has mountain path geometry. Lemma 2.3. Assume that (A1) or (A2) holds. Then there exists K(c) ∈ (0, 1) such that γ(c) := inf h∈Γ(c) max t∈[0,1] Ep ( h(t) ) > max{Ep(h(0)), Ep(h(1))}. Proof. On the one hand, for any u ∈ S(c) and ∫ RN ∫ RN |u(x)−u(y)|p |x−y|N+ps dx dy ≤ 2a b , we have Ep(u) ≤ a p ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy + b 2p (∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy )2 ≤ 2a p ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy. (2.1) On the other hand, if ∫ RN ∫ RN |u(x)−u(y)|p |x−y|N+ps dx dy is small enough, from (A2), we have Ep(u) ≥ a p ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy + b 2p (∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy )2 − psc ( q−N(q−p) ps ) N(q − p)‖Q‖(q−p)Lp (∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy )N(q−p) p2s ≥ a 2p ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy. EJDE-2022/61 FRACTIONAL p-KIRCHHOFF EQUATION 7 This and (2.1) imply that Ep(u)→ 0 as ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy → 0 and for K(c) small enough; moreover, if K(c) ≤ 2a b , we have sup u∈AK(c) Ep ( u ) ≤ 2a p ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy ≤ 2a p Kp(c) = a 2p 4Kp(c) ≤ inf u∈∂A4K(c) Ep ( u ) . (2.2) where ∂A4K(c) = { u ∈ S(c) : (∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy )1/p = 4K(c) } . Furthermore, for all u ∈ A4K(c), (2.1) indicates that Ep(u) ≥ 0. Next, we prove Γ(c) 6= ∅. Set uλ(x) = cλN/p ‖Q‖Lp Q(λx), (2.3) where λ > 0 will be determined later. Then uλ ∈ S(c) and it follows from (1.8) that∫ RN ∫ RN |uλ(x)− uλ(y)|p |x− y|N+ps dx dy = cpλps, ∫ RN |uλ|qdx = (pqs)cqλ N(q−p) p N(q − p)‖Q‖q−pLp . Consequently, Ep(uλ) = a p (cpλps) + b 2p (cpλps)2 − psc [ q−N(q−p) ps ] N(q − p)‖Q‖q−pLp (cpλps) N(q−p) p2s . (2.4) Thus, ∫ RN ∫ RN |uλ1(x)− uλ1(y)|p |x− y|N+ps dx dy < K(c), if λ1 < (K(c) cp ) 1 ps , which implies that uλ1 ∈ AK(c). As a consequence of (A1) or (A2) and (2.4), it is easy to check that Ep(u)→ −∞ as λ→∞. Hence, we choose λ2 > 0 large enough, such that Ep(uλ2 ) < 0. Taking g(t) = u((1−t)λ1+tλ2), we have g(0) = uλ1 ∈ AK(c), g(0) = uλ2 , Ep(uλ2 ) < 0. These means that g(t) ∈ Γ(c) 6= ∅. For any g(t) ∈ Γ(c), we know that g(0) ∈ AK(c) and Ep(g(1)) < 0. Since g(t) is continuous, then there exists a t̄ ∈ (0, 1), such that g(t̄) ∈ ∂A4K(c) 8 L. WANG, H. CHEN, L. YANG EJDE-2022/61 According to (2.2), we have max t∈[0,1] Ep ( g(t) ) ≥ Ep ( g(t̄) ) > max{Ep(g(0)), Ep(g(1))}. Moreover, γ(c) := inf h∈Γ(c) max t∈[0,1] Ep ( h(t) ) > max{Ep(h(0)), Ep(h(1))}. The proof is complete. � 3. Proof of main results In this section, we prove Theorems 1.2 and 1.4 by using some energy estimates and the Gagliardo-Nirenberg inequality (1.9). We first note that by simply rescal- ing, we can easily prove that I(c) ≤ 0 for all c > 0 and 0 < q < p∗s. (3.1) Furthermore, using (1.9), we observe that for any u ∈ S(c), Ep(u) ≥ a p ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy + b 2p (∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy )2 − psc [ q−N(q−p) ps ] N(q − p)‖Q‖(q−p)Lp (∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy )N(q−p) p2s = fp(t), (3.2) where fp(·) is given by (1.11) and let t = ∫ RN ∫ RN |u(x)−u(y)|p |x−y|N+ps dx dy. Proof of Theorem 1.2. (i) Because 0 < q < p+ p2s N , we can readily check that fp(t) (t ∈ (0,∞)) is minimized at a unique point, denoted by tp. Thus, from (3.2) we obtain I(c) = inf u∈S(c) Ep(u) ≥ fp(tp). (3.3) On the other hand, choosing λ = ( tp cp ) 1 ps , i.e., cpλps = tp, it can be seen from (2.4) that I(c) ≤ Ep(uλ) = fp(tp). From this and (3.3), we infer that I(c) = fp(tp) = inf t∈R+ fp(t), (3.4) and uλ with λ = ( tp cp ) 1 ps , i.e., uλ = uc = c ‖Q‖Lp ( tp cp ) N p2s Q (( tp cp ) 1 psx ) is a minimizer of (1.5). All that remains is to prove that uc (up to translations) is the unique minimizer of (1.5). In fact, if u0 ∈ S(c) is a minimizer, then it can be shown from (3.2) that I(c) = Ep(u0) ≥ fp(t0), with t0 := ∫ RN ∫ RN |u0(x)− u0(y)|p |x− y|N+ps dx dy, where the “=” in the second inequality holds if and only if u0 is an optimizer of (1.9). Which together with (3.4) further means that t0 = tp and fp(t0) = EJDE-2022/61 FRACTIONAL p-KIRCHHOFF EQUATION 9 Ep(u0). Therefore, u0 is an optimizer of (1.9) and it follows from (1.10) that up to translations, u0 must be the form of u0(x) = αQ(βx). Using∫ RN |u0|pdx = cp, ∫ RN ∫ RN |u0(x)− u0(y)|p |x− y|N+ps dx dy = tp, and combining with this (1.8), we have α = c ‖Q‖Lp ( tp cp ) N p2s and β = ( tp cp ) 1 ps , as a result, u0 = uc. (ii) Since q < p+ p2s N , i.e., N(q−p) p2s = 1, from (2.4), we can conclude that fp(t) = [a p − psc(q−p) N(q − p)‖Q‖q−pLp ] t+ b 2p t2. (3.5) If c ≤ [aN(q−p) p2s ] 1 q−p ‖Q‖Lp , we can easily derive from (3.2) that Ep(u) ≥ fp (∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+ps dx dy ) > 0 for all u ∈ S(c). In consideration of (3.1), this shows that (1.5) has no minimizer. In the next step, we move to the case of c > [ aN(q−p) p2s ] 1 q−p ‖Q‖Lp . We know from (3.5) that fp(t)(t ∈ (0,+∞)) attains its minimum at the unique point tp = p2sc(q−p) − aN(q − p)‖Q‖q−pLp bN(q − p)‖Q‖q−pLp . Following a similar argument as in part (i), we can demonstrate that, up to trans- lations that uc = cλ N/p p ‖Q‖Lp Q(λpx), where λp = [p2sc(q−p) − aN(q − p)‖Q‖q−pLp bN(q − p)cp‖Q‖q−pLp ] 1 ps . Therefore, uc is the unique minimizer of (1.5). (iii) For the case p+ p2s N < q < min{p+ 2p2s N , p∗s}, i.e., 1 < N(q−p) p2s < 2, let α = 2p2s−N(q − p) p2s , β = 1− α = N(q − p)− p2s p2s . It is obvious from Young’s inequality that for any t > 0, one has a p t+ b 2p t2 = α ( a pα t ) + β ( b 2pβ t2 ) ≥ ( a pα )α( b 2pβ )β tα+2β = [ aps 2p2s−N(q − p) ] 2p2s−N(q−p) p2s [ bps 2N(q − p)− 2p2s ]N(q−p)−p2s p2s t N(q−p) p2s . where the “=” in the second inequality holds if and only if a pα t = b 2pβ t2, i.e., t = t0 := 2βa αb = 2[N(q − p)− p2s]a [2p2s−N(q − p)]b . 10 L. WANG, H. CHEN, L. YANG EJDE-2022/61 We set c∗ = {N(q − p)‖Q‖q−pLp ps [ aps 2p2s−N(q − p) ] 2p2s−N(q−p) p2s } ps pqs−N(q−p) × {[ bps 2N(q − p)− 2p2s ]N(q−p)−p2s p2s } ps pqs−N(q−p) . (3.6) In view of (3.2) and (3.6), we consequently have Ep(u) ≥ (c∗) q−N(q−p) ps − cq− N(q−p) ps N(q−p)‖Q‖q−p Lp ps ( t0 )N(q−p) p2s = fp(t0) for all u ∈ S(c). (3.7) If c ≥ c∗, on the one hand, we deduce from (3.7) that I(c) ≥ fp(t0). On the other hand, let uλ(x) be as in (2.3) and set λ = ( t0cp ) 1 ps , then I(c) ≤ Ep(uλ) = fp(t0). Which shows that uλ is a minimizer of (1.5) and that for any c ≥ c∗, I(c) = fp(t0) = (c∗) q−N(q−p) ps − cq− N(q−p) ps N(q−p)‖Q‖q−p Lp ps {2 [ N(q − p)− p2s ] a[ 2p2s−N(q − p) ] b }N(q−p) p2s . Uniqueness of the minimizers can be proved by the same proofs in part (i). If c < c∗, we then deduce from (3.7) that Ep(u) > 0 for all u ∈ S(c). Thus, problem (1.5) cannot be achieved for (3.1). (iv) On the one hand if q = p+ 2p2s N and c > [bN(q − p)‖Q‖q−pLp 2p2s ] ps pqs−N(q−p) , from (2.3) and (2.4), it follows that I(c) ≤ lim λ→+∞ Ep(uλ) = −∞, therefore, problem (1.5) cannot be achieved. On the other hand, if q = p+ 2p2s N and c ≤ [bN(q − p)‖Q‖q−pLp 2p2s ] ps pqs−N(q−p) , from (3.2) we have Ep(u) > 0 for all u ∈ S(c). This and (3.1) obviously indicate that problem (1.5) cannot be attained. Finally, if q > p+ 2p2s N , it follows (2.3) and (2.4) that I(c) ≤ lim λ→+∞ Ep(uλ) = −∞, and thus problem (1.5) cannot be attained. The proof is complete. � Proof of Theorem 1.4. First, for any q ≥ p + 2p2s N , we can prove the existence of K(c) > 0 by Lemma 2.3 and can choose K(c) small enough so that Ep(·) satisfies mountain pass geometry on S(c) if (A1) or (A2) is assumed. Therefore, in the following, we always hypothesize that K(c) < t̄p, where t̄p stands for the unique maximum point of fp(t) in (0,+∞). EJDE-2022/61 FRACTIONAL p-KIRCHHOFF EQUATION 11 For any r ∈ [0, 1] and h(r) ∈ Γ(c), we can derive from (3.2) that Ep(h(r)) ≥ fp (∫ RN ∫ RN | ( h(r) ) (x)− ( h(r) ) (y)|p |x− y|N+ps dx dy ) , (3.8) where “=” holds if and only if h(r) ∈ S(c) is an optimizer of (1.9), i.e., up to translations, (h(r))(x) = cβN/p ‖Q‖Lp Q(βx) for some β > 0. (3.9) Since h(0) ∈ AK(c) with K(c) < t̄p, and note that fp(t) > 0 ∀t ∈ (0, t̄p], thus we have ∫ RN ∫ RN | ( h(0) ) (x)− ( h(0) ) (y)|p |x− y|N+ps dx dy < t̄p < ∫ RN ∫ RN | ( h(1) ) (x)− ( h(1) ) (y)|p |x− y|N+ps dx dy. (3.10) As a result of (3.8) and (3.10), it holds that max r∈[0,1] Ep ( h(r) ) ≥ fp(t̄p) = max t∈R+ fp(t). (3.11) Thus, γ(c) ≥ fp(t̄p). (3.12) Instead, let uλ(x) be the test function given by (2.3), characterized by λ = λ̄p = ( t̄p cp ) 1 ps . Set g(r) := r N p2suλ(r 1 psx), One can then check that Ep(g(r)) = fp(t̄pr). Choosing 0 < t̃p < t̄p small enough such that g ( t̃p t̄p ) ∈ AK(c), and select t̂p > t̄p such that fp(t̂p) < 0. Let h(r) = g ( (1− r) t̃p t̄p + r t̂p t̄p ) , ∀r ∈ (0, 1). Then h(0) = g ( t̃p t̄p ) ∈ AK(c) and Ep ( h(1) ) = Ep ( g ( t̂p t̄p )) = fp(t̂p) < 0. This shows that h ∈ Γ(c), and γ(c) ≤ max t∈[0,1] E ( h(t) ) = Ep(uλ̄p ) = fp(t̄p). Combing this with (3.12), we deduce that γ(c) = fp(t̄p) and uλ̄p = ūc(x) = c ‖Q‖Lp ( t̄p cp ) N p2s Q (( t̄p cp ) 1 psx ) is a solution of problem (1.14). 12 L. WANG, H. CHEN, L. YANG EJDE-2022/61 Next we demonstrate that ūc satisfies equation (1.1) for some µ ∈ R−. In fact, in light of f ′p(t̄p) = 0 and λ̄p = ( t̄p cp ) 1 ps , we obtain c[q− N(q−p) ps ] p‖Q‖q−pLp (t̄p) N(q−p)−p2s p2s = a p + b p t̄p = a p + b p (∫ RN ∫ RN |ūc(x)− ūc(y)|p |x− y|N+ps dx dy ) . (3.13) Moreover, since Q(x) is a solution of (1.7) and λ̄p = ( t̄p cp ) 1 ps , it follows that ūc satisfies c [ q−N(q−p) ps ] ‖Q‖q−pLp (t̄p) N(q−p)−p2s p2s (−∆)spūc − |ūc|q−2ūc = − [pqs−N(q − p)](cλ̄ N p p )q−p N(q − p)‖Q‖q−pLp |ūc|p−2ūc. From this and (3.13) we conclud that ūcis a solution of (1.1) with µ = − [ pqs−N(q − p) ]( cλ̄ N p p )q−p N(q − p)‖Q‖q−pLp . We finally prove that ūc is a unique solution of γ(c) before translation. Assume that ū is a solution of γ(c) and that it satisfies (1.15), then there exists µ ∈ R such that E′(u) = µ|ū|p−2ū, so we have a ∫ RN ∫ RN |ū(x)− ū(y)|p |x− y|N+ps dx dy + b (∫ RN ∫ RN |ū(x)− ū(y)|p |x− y|N+ps dx dy )2 − ∫ RN |ū|qdx = µ ∫ RN |ū|pdx. (3.14) It then follows from the Pohozaev identity that a(N − ps) p ∫ RN ∫ RN |ū(x)− ū(y)|p |x− y|N+ps dx dy + b(N − ps) p × (∫ RN ∫ RN |ū(x)− ū(y)|p |x− y|N+ps dx dy )2 − N q ∫ RN |ū|qdx = µN p ∫ RN |ū|pdx. (3.15) From (3.14) and (3.15), we deduce that a ∫ RN ∫ RN |ū(x)− ū(y)|p |x− y|N+ps dx dy + b (∫ RN ∫ RN |ū(x)− ū(y)|p |x− y|N+ps dx dy )2 − N(q − p) pqs ∫ RN |ū|qdx = 0. (3.16) EJDE-2022/61 FRACTIONAL p-KIRCHHOFF EQUATION 13 Letting ĥ(r) := r N p2s ū(r 1 psx), we obtain g(r) := Ep(ĥ(r)) = ar p ∫ RN ∫ RN |ū(x)− ū(y)|p |x− y|N+ps dx dy + br2 2p (∫ RN ∫ RN |ū(x)− ū(y)|p |x− y|N+ps dx dy )2 − r N(q−p) p2s q ∫ RN |ū|qdx. Equality (3.16) shows that g(r) (r ∈ (0,+∞)) reaches its maximum at the unique point r = 1, and lim r→+∞ g(r) = −∞. Choosing 0 < s̃ < 1 < ŝ such that ĥ(s̃) ∈ AK(c) and g(ŝ) < 0, we have h0(r) := ĥ((1− r)s̃+ rŝ) ∈ Γ(c), max r∈[0,1] Ep(h0(r)) = Ep(ū). Through arguments such as (3.8) and (3.11), one sees that fp(t̄p) = γ(c) = Ep(ū) = max r∈[0,1] Ep(h0(r)) ≥ max t∈R+ fp(t) = fp(t̄p). From (3.9), this implies that ū must be the form cβN/p ‖Q‖Lp Q(βx) for some β > 0. Translating this into the equality fp(t̄p) = Ep(ū), then we can obtain that ū = ūc and β = λ̄p. The proof is complete. � Acknowledgements. H. Chen was supported by the National Science Foundation of China 12071486. L. Wang was supported by the Research Funds for the Central South University (No. 2021zzts0043). References [1] V. Ambrosio, T. Isernia, V. Radulescu; Concentration of positive solutions for a class of fractional p-Kirchhoff type equation, Proc. Royal Soc. Edinburgh A: Math., 151 (2021), no. 2, 601–651. [2] J. Bellazzini, L. Jeanjean, T. Luo; Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations, Proc. Lond. Math. Soc. 107 (2013), no. 3, 303–339. [3] G. Che, T. Wu; Multiple positive solutions for a class of Kirchhoff type equations with in- definite nonlinearities. Adv. Nonlinear Anal. 11 (2022), no. 1, 598–619. [4] N. Dao, J. Dı́az, Q. Nguyen; Generalized Gagliardo-Nirenberg inequalities using Lorentz spaces, BMO, Hölder spaces and fractional Sobolev spaces, Nonlinear Analysis. 173 (2018), 146–153. [5] F. Faraci, K. Silva; On the Brezis-Nirenberg problem for a Kirchhoff type equation in high dimension, Calc. Var. Partial Differential Equations, 60 (2021), no. 1, 22. [6] R. Frank, E. Lenzmann, L. Silvestre; Uniqueness of radial solutions for the fractional Lapla- cian. Commun. Pure Appl. Math. 69 (2016), no. 9, 1671–1726. [7] Z. Feng, Y. Su; Lions-type theorem of the fractional Laplacian and applications. Dyn. Partial Differ. Equ. 202 (2021), no. 6, 211–230. [8] Z. Feng, C.Tan, L. Wei; Uniqueness and asymptotic behavior of positive solution of quasilin- ear elliptic equations with Hardy potential, Nonlinear Anal. 202 (2021), Paper No. 112152, 24 pp. [9] H. Hajaiej, X. Yu, Z. Zhai; Fractional Gagliardo-Nirenberg and Hardy inequalities under Lorentz norms, J. Math. Anal. Appl. 369 (2012), no. 2, 569–577. [10] X. He, W. Zou; Existence and concentration behavior of positive solutions for a fractional p-Kirchhoff equation in R3, J. Differential Equations, 2 (2012), 1813–1834. 14 L. WANG, H. CHEN, L. YANG EJDE-2022/61 [11] X. Huang, Y. Zhang; Existence and uniqueness of minimizers for L2-constrained problems related to fractional Kirchhoff equation. Math. Methods Appl. Sci. 43 (2020), no. 15, 8763– 8775. [12] L. Jeanjean, T. Luo; Sharp nonexistence results of prescribed L2-norm solutions for some class of Schrödinger-Poisson and quasi-linear equations, Z. Angew. Math. Phys. 64 (2013) 937–954. [13] G. Kirchhoff; Mechanik, Teubner, Leipzig, 1883. [14] Y. Li, W. Ni; Radial symmetry of positive solutions of nonlinear elliptic equations in R. Comm. Partial Differential Equations. 18 (1993), no. 5, 1043–1054. [15] X. L. Lin, S. Zheng; Multiplicity and asymptotic behavior of solutions to fractional (p,q)- Kirchhoff type problems with critical Sobolev-Hardy exponent, Electron. J. Differential Equa- tions, 2021 (2021), no. 66, 1–20. [16] Z. Liu, M. Squassina, J. Zhang; Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension, NoDEA Nonlinear Differential Equations Appl. 24 (2017), no. 4, Paper No. 50, 32. [17] H. Luo, Z. Zhang; Normalized solutions to the fractional Schrödinger equations with combined nonlinearities, Calc. Var. Partial Differential Equations, 59 (2020), no. 4, 1–35. [18] J. Sirren, X. Tang; Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math., 49 (2000), no. 3, 897–923. [19] N. Soave; Normalized ground states for the NLS equation with combined nonlinearities, J. Differential Equations, 269 (2020), no. 9, 6941-6987. [20] Y. Su, H. Chen; Fractional Kirchhoff-type equation with Hardy-Littlewood-Sobolev critical exponent, Computers and Mathematics with Applications, 78(2019), no. 6, 2063–2082. [21] Y. Su, Z. Feng; Ground state solutions for the fractional problems with dipole-type potential and critical exponent, Commun. Pure Appl. Anal. 21(2022), no. 6, 1953–1968. [22] M. Weinstein; Nonlinear Schrödinger equations and sharp interpolations estimates, Comm. Math. Phys., 87 (1983), 567–576. [23] H. Ye; The existence of normalized solutions for L2-critical constrained problems related to fractional p-Kirchhoff equations, Z. Angew. Math. Phys. 66 (2015), 1483–1497. [24] X. Zeng, Y Zhang; Existence and uniqueness of normalized solutions for the Kirchhoff equa- tion. Appl. Math. Lett. 74 (2017), 52–59. Lixiong Wang School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China Email address: wanglixiong2018@163.com Haibo Chen (Corresponding author) School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China Email address: math chb@163.com Liu Yang College of Mathematics and Statistics, Hengyang Normal University, Hengyang, 421008 Hunan, China Email address: yangliuyanzi@163.com 1. Introduction and statement of main results 2. Preliminaries 3. Proof of main results Acknowledgements References