Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 75, pp. 1–16. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.75 NORMALIZED SOLUTIONS OF FRACTIONAL KIRCHHOFF EQUATIONS IN THE DEFOCUSING CASE ZHENYU GUO, TIANQING ZHANG Abstract. In this article, we focus on the normalized solutions to the fractional Kirchhoff equa- tions with subcritical nonlinearities in the defocusing case. By applying distinct suppositions to the coefficients of nonlinearities, namely q < p, we prove the existence and nonexistence of normalized solutions. Also we obtain new results on the characterization of ground states of the fractional Kirchhoff equations. 1. Introduction The aim of article is to study the fractional Kirchhoff equation( a+ b ∫ R3 |(−∆)s/2u|2dx ) (−∆)su = λu+ µ|u|q−2u+ |u|p−2u in R3 (1.1) with a prescribed mass ∫ R3 u2dx = c2. (1.2) In this article, except for additional statements, we assume that a, b, c > 0, 0 < s < 1, 2 < q < p ≤ 2∗s = 6 3−2s (2∗s is the Sobolev critical exponent), N = 3, λ ∈ R is a Lagrange multiplier and µ < 0 is a parameter. For equation (1.1), we name the focusing case when µ > 0, and defocusing case when µ < 0. In equation (1.1), (−∆)s with s ∈ (0, 1)), namely the fractional Laplacian, is generally specified as (−∆)sv(x) = CsP.V. ∫ R3 v(x)− v(y) |x− y|3+2s dy = Cs lim ϵ→0+ ∫ R3\Bϵ(x) v(x)− v(y) |x− y|3+2s dy = −1 2 Cs ∫ R3 v(x+ y) + v(x− y)− 2v(x) |y|3+2s dy (1.3) for v ∈ S(R3), where S(R3) denotes the Schwartz space of rapidly decaying C∞ function, Bϵ(x) represents an open ball of radius ϵ centered at x, P.V. is the principle value, which is defined by the latter expression in (1.3), Cs = (∫ R3 1− cos(ξ1) |ξ|3+2s dξ )−1 . For u ∈ S(R3), the fractional Laplacian (−∆)s can be regulated by the Fourier transform (−∆)su = F−1(|ξ|2sFu), F denotes the usual Fourier transform. 2020 Mathematics Subject Classification. 35A01, 35R11. Key words and phrases. Fractional Kirchhoff equations; defocusing case; subritical nonlinear terms; normalized solutions. ©2025. This work is licensed under a CC BY 4.0 license. Submitted October 2, 2024. Published July 16, 2025. 1 2 Z. GUO, T. ZHANG EJDE-2025/75 When setting a = 1, s = 1 and b = 0 (that is to say, (1.1) turns into the Schrödinger equation), in 1997, Jeanjean [13] studied the existence of the normalized solutions to the Schrödinger equation in the case of the corresponding energy functional F (u) = 1 2 ∫ RN |∇u(x)|2dx− m∑ i=1 ai σi + 2 ∫ RN |u(x)|σi+2dx is unbounded from below on the L2-constraint set S(c) = {u ∈ H1(RN ), ∥u∥L2(RN ) = c} initially. Lately, such type of problems have attracted extensive attention in the field of partial differential equations. For instance, Soave [24] gave the existence and some properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities −∆u = λu+ µ|u|q−2u+ |u|p−2u in RN , N ≥ 1, on the normalized manifold ∫ RN |u|2dx = a2. As for other results of the Schrödinger equation, we refer readers to [4, 11, 25, 27] and the references therein. When considering the case a = 1, s ̸= 1 and b = 0, i.e., for the fractional Schrödinger equations, see [18, 29] and the references therein for results about the normalized solutions to the fractional Schrödinger equations. When b > 0 and s = 1, equation (1.1) becomes the classic Kirchhoff model; this type of problems has also been researched by many authors [1, 14, 21]. In fact, such model has relation to the stationary solutions of equation utt − ( a+ b ∫ RN |∇u|2dx ) ∆u = f(x, u). (1.4) This equation comes from the traditional D’Alembert wave equation which was given by Kirchhoff [14] in 1876 in the process of studying the changes in the length of the string during vibrations, where f(x, u) denotes a general nonlinear term. Additionally, it deserves attention that in [1] equation (1.4) models some physical systems, where u explains a process which is related to the average of itself. A lot of papers about the Kirchhoff type equations emerged with the emergence of this ground breaking article [21]. For example, considering a Kirchhoff model, together with a critical Trudinger-Moser nonlinearity f(x, u), a class of fractional Kirchhoff-type equation with Trudinger-Moser nonlinearity was discussed by Xiang, Rădulescu and Zhang [20]. Using appro- priate assumptions on the potential function V and some energy estimates techniques, Chen and Huang [6] obtained the existence results of normalized solutions for a fractional Kirchhoff-type equation ( a+ b ∫ RN |(−∆)s/2u|2dx ) (−∆)su+ V (x)u = c|u|p−2u+ µu in RN with doubly critical exponents (when considering the case N = 4s, the critical Sobolev exponent 2∗s = 2N N−2s and the fractional Gagliardo-Nirenberg-Sobolev critical exponent 2∗GNS = 2N+8s N are equal, and 2N N−2s = 2N+8s N = 4). In addition, in 2024, the existence of the normalized solutions to the fractional Kirchhoff equation with subcritical nonlinearity( a+ b ∫ RN |(−∆)s/2u|2dx ) (−∆)su = λu+ µ|u|q−2u+ |u|p−2u in RN was studied in [8] in R3 with s ∈ (3/4, 1) and µ < 0. If the constraint condition (1.2) is considered on this basis, some universal methods do not take effect. As a result, we need to establish extra claims to solve the technical obstacles. As is known (such as in [13]), (1.2) has definite physical motivations. Consequently, it has sparked a wave of research on the normalized solutions. More specifically, the practical application background of operator (−∆)s includes the following aspects such as fractional quantum mechanics [15], physics and chemistry [19], conformal geometry and minimal surfaces [5], obstacle problems [23]. Caffarelli EJDE-2025/75 NORMALIZED SOLUTIONS OF FRACTIONAL KIRCHHOFF EQUATIONS 3 and Silvestre [2] adopted the extension method which converted this nonlocal problem (due to the nonlocal feature of the operator (−∆)s on RN (N ≥ 1)) to a local one in higher dimensions. What makes this amusing, of course, is that this method can go for nonlinear equations with a fractional Laplacian, we refer readers to [7, 9] and the references therein. About equation (1.1), there are usually two classes of treatments. On one hand, we can think of it as a fixed frequency problem, in other words, we look for solutions u ∈ Hs(R3) by hunting for critical points of the action functional M : Hs(R3) → R: M(u) := a 2 ∫ R3 |(−∆)s/2u|2dx+ b 4 (∫ R3 |(−∆)s/2u|2dx )2 − λ 2 ∫ R3 |u|2dx− µ q ∫ R3 |u|qdx− 1 p ∫ R3 |u|pdx, (1.5) where λ ∈ R is a fixed frequency, readers can see [12, 17] for more results. On the other hand, we can also search for solutions to (1.1) with a prescribed L2 norm. In this case, we see λ ∈ R as part of unknown quantity. As everyone knows, equation (1.1) has roots in the standing wave type solution ψ(x, t) = e−iλtu(x), λ ∈ R to the time-dependent nonlinear fractional equation defined by: i ∂ψ ∂t = ( a+ b ∫ R3 |(−∆)s/2ψ|2dx ) (−∆)sψ − f(|ψ|)ψ, in R3, (1.6) where s ∈ (0, 1), i represents the imaginary unit, and ψ = ψ(x, t) : R3× [0,+∞) → C. It is easy to see that ψ solves (1.6) if and only if the standing wave u(x) satisfies (1.1) with f(u) = µuq−2+up−2. After computations, we can see that solutions ψ ∈ C([0, T );Hs(R3)) to (1.6) has conservation of mass along time, therefore this method is extremely significative from the physical perspective. Now mention some publications that consider normalized solutions to (1.1). Li, Luo and Yang [16] proved the existence and properties of solutions to (1.1) with s = 1 under normalized constraint∫ R3 |u|2dx = c2 when a, b > 0 and µ > 0, namely the focusing case. As far as we know, the defocusing case of problem (1.1) with the condition (1.2) was mainly studied by Soave [24] with b = 0. The situation b > 0, µ < 0 and s = 1 was studied in [3]. The defocusing case of fractional Kirchhoff equation was part of Ding’s result [8]. We further extended his results (see Theorem 2.8 and 2.10). In this paper, we take the case of b > 0, µ < 0 and s ∈ (0, 1) into consideration. Before presenting the main results of our paper, let us first recall that the fractional Sobolev space Hs(R3) can be defined as follows: Hs(R3) = { u ∈ L2(R3) : ∫ R3 |(−∆) s 2u|2dx < +∞ } with the norm ∥u∥2 = ∫ R3 (|(−∆)s/2u|2 + |u|2)dx, where ∫ R3 |(−∆) s 2u|2dx = ∫∫ R6 |u(x)− u(y)|2 |x− y|3+2s dxdy. Also we define Hs r (R3) = { u ∈ Hs(R3) : u(x) = u(|x|), x ∈ R3 } . In this paper, we denote by | · |p the usual norm in the Lp(R3) space, Sr c = Sc ∩Hs r and u∗ the symmetric decreasing rearrangement of the modulus of u ∈ Hs(R3). The functional Eµ : Sc → R is regulated as Eµ(u) = a 2 ∫ R3 |(−∆)s/2u|2dx+ b 4 (∫ R3 |(−∆)s/2u|2dx )2 − µ q ∫ R3 |u|qdx− 1 p ∫ R3 |u|pdx, where Sc is the constraint space Sc = { u ∈ Hs(R3) : ∫ R3 |u|2dx = c2 } . 4 Z. GUO, T. ZHANG EJDE-2025/75 It is easy to check that Eµ ∈ C1(R3,R). Thus, the weak solutions of (1.1) under the constraint (1.2) can be obtained as critical points of the functional Eµ. We can easily prove that, if u ∈ Hs(R3) is a weak solution of (1.1), then we have the Pohožaev identity Pµ(u) := a|(−∆)s/2u|22 + b|(−∆)s/2u|42 − µδs,q|u|qq − δs,p|u|pp = 0, where δs,p = 3(p−2) 2sp , δs,q = 3(q−2) 2sq . Therefore, the critical points of Eµ is certainly contained in the Pohožaev set Pc,µ = { u ∈ Sc : Pµ(u) = 0 } (see lemma 2.3 for a proof). By simple calculations, we can show that δs,p ∈ (0, 1) (when 2 < p < 2∗s) and qδs,q ≤ 4 < pδs,p, if 2 < q ≤ 2 + 8s 3 < p < 6 3− 2s , where 2 + 8s 3 is the mass critical exponent for the Kirchhoff constrained minimization problem, namely, 2+ 8s N is the threshold exponent for many dynamic problems, see [28] for more information. In this article, we will be concerned with ground state solutions, wihci are defined as follows. Definition 1.1. We say that ũ is a ground state of (1.1) on Sc if it is a solution to (1.1) having minimal energy among all the solutions which belong to Sc: dEµ|Sc (ũ) = 0 and Eµ(ũ) = inf{Eµ(u) : dEµ|Sc (u) = 0, and u ∈ Sc}. And the set of ground states will be denoted by Zc,µ. To go over the obstacles about the convergence of the Palais Smale (hereinafter referred to as PS) sequence of Eµ, we build the sequence {un}n∈N satisfying Pµ(un) → 0, when n→ ∞. Thanks to the normalized condition (1.2), we define the dilations (ω ∗ u)(x) = e3ω/2u(eωx), a.e. in R3 which retain the L2 norm, more precisely,∫ R3 (ω ∗ u)2dx = ∫ R3 u2dx, and it is a continuous map from R×Hs(R3) into Hs(R3). Furthermore, we introduce the following fiber map Jµ u (ω) := Eµ(ω ∗ u) = ae2sω 2 |(−∆)s/2u|22 + be4sω 4 |(−∆)s/2u|42 − µ eqδs,qsω q |u|qq − epδs,psω p |u|pp, where δs,q = 3(q−2) 2sq and δs,p = 3(p−2) 2sp . By using the functional Jµ u , we cast a function into the Pohožaev set. Soave [24] and Li, Luo and Yang [16] have also applied such idea. The main results of this paper are organized as follows: • If µ < 0, 2 < q < p = 2 + 8s 3 = p̄, we prove that (1.1) under the condition (1.2) does not have solution. • If 2 < q ≤ 2 + 8s 3 < p < 6 3−2s are given constants and µ < 0 satisifies an additional assumption, we prove that there exists λ < 0 such that (1.1) under the condition (1.2) has a solution. The solution is radially symmetric, and is a ground state on Sc. • If 2 < q ≤ 2 + 8s 3 < p < 6 3−2s are given constants and µ < 0 satisfies an additional assumption, we also give a characterization to the set of ground states. For overcoming some technical difficulties, the main proof of our results involves the techniques used by Soave [24], Li, Luo and Yang [16]. This paper is organized as follows. In section 2 we give the notation and assumptions and we enunciate our main results. In section 3 we prove some results concerning the subcritical case (namely, the main results in this paper). EJDE-2025/75 NORMALIZED SOLUTIONS OF FRACTIONAL KIRCHHOFF EQUATIONS 5 2. Preliminaries and main results Lemma 2.1 ([7]). Let s ∈ (0, 1) and p ∈ [1,+∞) be such that sp < N . Then, there exists a positive constant Ss = Ss(N, p, s) such that, for any measurable and compactly supported function u : RN → R, we have Ss|u|22∗s ≤ ∫∫ R2N |u(x)− u(y)|2 |x− y|N+2s dxdy, (2.1) where 2∗s = 2N N−2s is the so-called fractional critical exponent. Moreover, (2.1) becomes equality if and only if ũ = K(µ̄2 + |x − x0|2)− N−2s 2 with K ∈ R\{0}, µ̄ > 0, x0 ∈ RN fixed constants, Ss is the best Sobolev embedding constant. It is known that if p ∈ (2, 2∗s), then there exists an optimal constant C(s, p) such that |u|p ≤ C(s, p)|(−∆)s/2u|δs,p2 |u|1−δs,p 2 , (2.2) holds for all u ∈ Hs(RN ). (2.2) is called of the fractional Gagliardo-Nirenberg inequality. Lemma 2.2 ([18]). Let u ∈ Hs(RN ), N ≥ 2 satisfy the equation (−∆)su = g(u), then N − 2s 2 ∫ RN |(−∆)s/2u|2dx = N ∫ RN G(u)dx, where G(u) = ∫ u 0 g(t)dt. Lemma 2.3. Let p, q ∈ (2, 2N N−2s ] and λ, µ ∈ R. If u ∈ Hs(RN ) is a weak solution of the equation in the N -dimensional space corresponding to equation (1.1), then it satisfies the Pohožaev identity Pµ(u) := a|(−∆)s/2u|22 + b|(−∆)s/2u|42 − µδs,q|u|qq − δs,p|u|pp = 0, (2.3) where δs,q = N(q−2) 2sq and δs,p = N(p−2) 2sp . Proof. Set A = a+ b ∫ RN |(−∆)s/2u|2dx. According to the lemma 2.2, we have (−∆)su = 1 A (λu+ µ|u|q−2u+ |u|p−2u). (2.4) Let f(u) = 1 A (λu+ µ|u|q−2u+ |u|p−2u), then F (u) = 1 A (λ2 |u| 2 + µ q |u| q + 1 p |u| p). According to the assumption that u is a weak solution to the equation in the N -dimensional space corresponding to equation (1.1) (in other words, multiplying the above equation by u and integrating), one has that ∫ RN A(−∆)su · udx = (λ|u|22 + µ|u|qq + |u|pp). (2.5) The above equality implies that A|(−∆)s/2u|22 = (λ|u|22 + µ|u|qq + |u|pp). (2.6) Multiplying by N N−2s , it holds that N N − 2s A|(−∆)s/2u|22 = N N − 2s λ|u|22 + N N − 2s µ|u|qq + N N − 2s |u|pp. (2.7) By lemma 2.2, we have that |(−∆)s/2u|22 = 2N A(N − 2s) (λ 2 |u|22 + µ q |u|qq + 1 p |u|pp ) . (2.8) Combining the above two equations, we obtain that( 1− N N − 2s ) A|(−∆)s/2u|22 = µ ( 2N q(N − 2s) − N N − 2s ) |u|qq + ( 2N p(N − 2s) − N N − 2s ) |u|pp. From the above equality, one deduces that A|(−∆)s/2u|22 = µδs,q|u|qq + δs,p|u|pp. 6 Z. GUO, T. ZHANG EJDE-2025/75 Thus, the fractional Pohožaev identity holds, namely, a|(−∆)s/2u|22 + b|(−∆)s/2u|42 = µδs,q|u|qq + δs,p|u|pp. □ For convenience, we decompose the set Pc,µ into three disjoint sets as follows: Pc,µ = P+ c,µ ∪ P0 c,µ ∪ P− c,µ, where P+ c,µ = {u ∈ Pc,µ, 2a|(−∆)s/2u|22 + 4b|(−∆)s/2u|42 − µqδ2s,q|u|qq − pδ2s,p|u|pp > 0} = { u ∈ Pc,µ, (J µ u ) ′′ (0) > 0 } , P− c,µ = {u ∈ Pc,µ, 2a|(−∆)s/2u|22 + 4b|(−∆)s/2u|42 − µqδ2s,q|u|qq − pδ2s,p|u|pp < 0} = { u ∈ Pc,µ, (J µ u ) ′′ (0) < 0 } , P0 c,µ = {u ∈ Pc,µ, 2a|(−∆)s/2u|22 + 4b|(−∆)s/2u|42 − µqδ2s,q|u|qq − pδ2s,p|u|pp = 0} = { u ∈ Pc,µ, (J µ u ) ′′ (0) = 0 } , where (Jµ u ) ′′ (0) = ( 2a|(−∆)s/2u|22 + 4b|(−∆)s/2u|42 − µqδ2s,q|u|qq − pδ2s,p|u|pp ) s2. Next we state a lemma that is a type of minimax principle. But first, we state a related definition. Definition 2.4. Let X be a topological space and B be a closed subset of X. We say that a class F of compact subsets of X is a homotopy-stable family with extended boundary B if for any set A in F and any η ∈ C([0, 1]×X;X) satisfying η(t, x) = x for all (t, x) ∈ ({0} ×X) ∪ ([0, 1]×B), we have that η({1} ×A) is in F . Lemma 2.5 ([10, Theorem 5.2]). Let Φ be a C1 functional on a complete connected C1-Finsler manifold X and consider a homotopy-stable family F with an extended closed boundary B. Set m = m(Φ, F ) = infA∈F maxx∈A Φ(x) and let F be a closed subset of X satisfying (1) A ∩ F\B ̸= ∅ for each A ∈ F . (2) supΦ(B) ≤ m ≤ inf Φ(F ). Then, for any sequence of sets {An}n in F such that limn supAn Φ = m, there exists a sequence {xn}n in X\B such that lim n→∞ Φ(xn) = m, lim n→∞ ∥dΦ(xn)∥ = 0, lim n→∞ dist(xn, F ) = 0, lim n→∞ dist(xn, An) = 0. Lemma 2.6 ([18]). Let N ≥ 2, then Hs r (RN ) is compactly embedding into Lp(RN ) for p ∈ (2, 2∗s). Lemma 2.7 ([18]). Let s ∈ (0, 1). For any u ∈ Hs(RN ), the following inequality holds∫∫ R2N (u∗(x)− u∗(y))2 |x− y|N+2s dxdy ≤ ∫∫ R2N (u(x)− u(y))2 |x− y|N+2s dxdy. Next, we state the main results of this paper. Theorem 2.8 (Subcritical case). Let N = 3 and 2 < q < p = 2 + 8s 3 = p̄. If b ≥ 4 p̄C(s, p̄) p̄cp̄−4, then there is no solution to problem (1.1)-(1.2) for any µ < 0. Now, we define the constant C0 := ( a δs,pC(s, p)pcp(1−δs,p) ) 1 pδs,p−2 . Theorem 2.9 (Subcritical case). Let N = 3 and 2 < q ≤ 2 + 8s 3 < p < 2∗s = 6 3−2s be given constants. If µ < 0 satisfies (1− 1 δs,p )(a+ bC2 0 )C 2−qδs,q 0 + µ (δs,q δs,p − 1 ) C(s, q)qcq(1−δs,q) := ϵ0 < 0, (2.9) then Eµ|Sc has a critical point ũ at a positive level m(c, µ) = infu∈Pc,µ Eµ(u) > 0 satisfying: ũ is radially symmetric, it solves (1.1) for some λ̃ < 0 and it is a ground state of (1.1) on Sc, where EJDE-2025/75 NORMALIZED SOLUTIONS OF FRACTIONAL KIRCHHOFF EQUATIONS 7 C(s, q) is an optimal constant such that the fractional Gagliardo-Nirenberg inequality holds (see inequality (2.2)). The following theorem characterizes the ground states. Theorem 2.10 (Subcritical case). Under the assumptions of Theorem 2.9, if u ∈ Zc,µ (see Definition 1.1 for the definition of Zc,µ), then eiθ|u| ∈ Zc,µ for each θ ∈ R. Moreover, if u is a ground state, then the associated Lagrange multiplier λ is negative. 3. Subcritical case In this part we prove Theorems 2.8, 2.9 and 2.10. First we list some lemmas. Since some of these lemmas appear in the references, we omit their proof here. Lemma 3.1 (Jeanjean [13]). For u ∈ Sc and s ∈ R, the map ϕ 7→ s ∗ ϕ from TuSc to Ts∗uSc is a linear isomorphism with inverse ψ 7→ (−s) ∗ ψ, where TuSc = {ϕ ∈ Sc : ∫ RN uϕdx = 0}. Next, we will discuss the convergence of a class of special PS sequences satisfying appropriate additional assumptions. The idea used in the proof was first introduced by Jeanjean [13]. Then Soave [24] applied this idea to study the normalized solutions to the nonlinear Schrödinger equation with mixed nonlinearities. Lemma 3.2 (Compactness of PS sequences). Let 2 < q ≤ 2 + 8s 3 < p < 6 3−2s be given constants. We suppose that {un}n∈N ⊂ Sc is a PS sequence for Eµ|Sc at level c ̸= 0 and it holds (i) Pµ(un) → 0 as n→ ∞. (ii) µ < 0 and (2.9) holds. Then, going to a subsequence, un → u strongly in Hs(R3), and u ∈ Sc is a radial solution to (1.1) for some λ < 0. Proof. In this lemma, we argue directly. Since Pµ(un) → 0 as n→ ∞, we have that a|(−∆)s/2un|22 + b|(−∆)s/2un|42 − µδs,q|un|qq − δs,p|un|pp = o(1), (3.1) as n→ ∞. Therefore, c+ 1 ≥ Eµ(un) = a 2 ∫ R3 |(−∆)s/2un|2dx+ b 4 (∫ R3 |(−∆)s/2un|2dx )2 − µ q ∫ R3 |un|qdx− 1 p ∫ R3 |un|pdx = a (1 2 − 1 pδs,p )∫ R3 |(−∆)s/2un|2dx+ b (1 4 − 1 pδs,p )(∫ R3 |(−∆)s/2un|2dx )2 − µ q ( 1− qδs,q pδs,p )∫ R3 |un|qdx+ o(1) ≥ a (1 2 − 1 pδs,p )∫ R3 |(−∆)s/2un|2dx+ b (1 4 − 1 pδs,p )(∫ R3 |(−∆)s/2un|2dx )2 + o(1), as n → ∞, where we used the fact that Eµ(un) → c, the equivalent deformation of (3.1) and −µ q ( 1 − qδs,q pδs,p ) > 0 (since µ < 0, 0 < qδs,q < pδs,p). From the above inequality and the fact that |un|22 = c2, we deduce that {un} is a bounded sequence in Hs(R3). Besides, Hilbert space Hs(R3) is a reflexive Banach space. In the reflexive Banach space Hs(R3), bounded sequence {un} has weakly convergent subsequence {un} (for the sake of brevity, the subsequence of {un} is still represented by {un}). According to lemma 2.6, Hs r (R3) ↪→ Lp(R3) compactly for p ∈ (2, 2∗s), there exists u ∈ Hs r (R3) such that un ⇀ u in Hs r (R3), un → u in Lp(R3), un(x) → u(x) a.e. in R3, (3.2) as n→ ∞. 8 Z. GUO, T. ZHANG EJDE-2025/75 Since {un} is a bounded PS sequence of Eµ|Sc , by applying the Lagrange multipliers rule, we conclude that there exists λn ∈ R such that a ∫ R3 (−∆)s/2un(−∆)s/2ϕdx+ b|(−∆)s/2un|22 ∫ R3 (−∆)s/2un(−∆)s/2ϕdx − µ ∫ R3 |un|q−2unϕdx− ∫ R3 |un|p−2unϕdx− λn ∫ R3 unϕdx = o(1)∥ϕ∥Hs , (3.3) for all ϕ ∈ Hs(R3). Letting ϕ = un, we have that a|(−∆)s/2un|22 + b|(−∆)s/2un|42 − µ|un|qq − |un|pp − λn|un|22 = o(1)∥un∥Hs . (3.4) Therefore, λn = 1 c2 ( a|(−∆)s/2un|22 + b|(−∆)s/2un|42 − µ|un|qq − |un|pp ) + o(1)∥un∥Hs . (3.5) As {un} is a bounded sequence in Hs(R3)∩Lp(R3)∩Lq(R3), from the above equation we obtain that {λn} is a bounded sequence. Thus, going if necessary to a subsequence, there exists λ ∈ R such that λn → λ (3.6) as n→ ∞. In the remaining of this proof, we prove that λ < 0. From Pµ(un) → 0 as n→ ∞, we obtain that a|(−∆)s/2un|22 + b|(−∆)s/2un|42 = µδs,q|un|qq + δs,p|un|pp + o(1) ≤ δs,p|un|pp + o(1). (3.7) Employing the fractional Gagliardo-Nirenberg inequality (2.2), we derive that a|(−∆)s/2un|22 ≤ a|(−∆)s/2un|22 + b|(−∆)s/2un|42 ≤ δs,p|un|pp + o(1) ≤ δs,pC(s, p) p|(−∆)s/2un| pδs,p 2 |un| p(1−δs,p) 2 + o(1). (3.8) It is easy to obtain that u ̸≡ 0: assuming by contradiction that u ≡ 0, then we obtain that limn→∞ |un|qq = limn→∞ |un|pp = 0. Using that Pµ(un) → 0 we deduce that Eµ(un) → 0, while this contradicts the assumption that Eµ(un) → c ̸= 0. Thus we have u ̸≡ 0. Since un ∈ Sc and the weak lower semi-continuity of the norm, it follows that |u|2 ≤ c, then we have that C0 = ( a δs,pC(s, p)pcp(1−δs,p) ) 1 pδs,p−2 ≤ B, (3.9) where B = limn→∞ |(−∆)s/2un|2. That is to say, for n large enough, we obtain that |(−∆)s/2un|2 ≥ C0. (3.10) Inserting (3.1) into (3.5), we have that λn = 1 c2 [( 1− 1 δs,p )( a+ b|(−∆)s/2un|22 ) |(−∆)s/2un|22 + µ (δs,q δs,p − 1 ) |un|qq ] + o(1). (3.11) According to the fractional Gagliardo-Nirenberg inequality (2.2), we obtain that |un|qq ≤ C(s, q)q|(−∆)s/2un| qδs,q 2 |un| q(1−δs,q) 2 . (3.12) By un ∈ Sc, we have that |un|qq ≤ C(s, q)q|(−∆)s/2un| qδs,q 2 cq(1−δs,q). (3.13) Further, by 1 − 1 δs,p < 0 (since 0 < δs,p < 1), combining (3.9), (3.11) with (3.13), we can deduce that λn ≤ 1 c2 |(−∆)s/2un| qδs,q 2 [( 1− 1 δs,p ) (a+bC2 0 )C 2−qδs,q 0 +µ (δs,q δs,p −1 ) C(s, q)qcq(1−δs,q) ] +o(1). (3.14) EJDE-2025/75 NORMALIZED SOLUTIONS OF FRACTIONAL KIRCHHOFF EQUATIONS 9 By (3.10) and (2.9), we derive that λn ≤ 1 c2 C qδs,q 0 [( 1− 1 δs,p ) (a+ bC2 0 )C 2−qδs,q 0 + µ (δs,q δs,p − 1 ) C(s, q)qcq(1−δs,q) ] + o(1). (3.15) Therefore, considering condition (2.9), we obtain that λn ≤ 1 c2 C qδs,q 0 ϵ0 + o(1), (3.16) for n adequately large. Taking the limit of the above formula as n→ ∞, we have that λ ≤ 1 c2 C qδs,q 0 ϵ0 < 0. (3.17) Thus λ < 0, and the claim is proved. Finally, we prove that un → u strongly in Hs(R3). By taking the limit of (3.3), one has a ∫ R3 (−∆)s/2u(−∆)s/2ϕdx+ bB2 ∫ R3 (−∆)s/2u(−∆)s/2ϕdx− µ ∫ R3 |u|q−2uϕdx − ∫ R3 |u|p−2uϕdx− λ ∫ R3 uϕdx = 0, (3.18) for all ϕ ∈ Hs(R3), that is, u satisfies (a+B2b)(−∆)su = λu+ µ|u|q−2u+ |u|p−2u, namely, u solves (1.1) for some λ < 0. Testing (3.18), (3.3) with ϕ = un − u, we can see that (a+B2b) ∫ R3 |(−∆)s/2(un − u)|2dx− λ ∫ R3 |un − u|2dx→ 0, as n→ ∞. Since λ < 0, we deduce that {un} converges strongly to u in Hs(R3). □ Lemma 3.3. Let µ < 0, and 2 < q ≤ 2 + 8s 3 < p < 2∗s be given constants. Then P0 c,µ = ∅ and Pc,µ is a smooth manifold of codimension 2 in Hs(R3). Proof. Suppose by contradiction that this is not the case, namely we set P0 c,µ ̸= ∅, then from the definition of P0 c,µ, we can derive that there exists u ∈ Sc such that Pµ(u) = 0 and (Jµ u ) ′′ (0) = 0. Therefore, a|(−∆)s/2u|22 + b|(−∆)s/2u|42 = µδs,q|u|qq + δs,p|u|pp, (3.19) 2a|(−∆)s/2u|22 + 4b|(−∆)s/2u|42 = µqδ2s,q|u|qq + pδ2s,p|u|pp. (3.20) Combining (3.19) with (3.20), one has (pδs,p − 2)a|(−∆)s/2u|22 + (pδs,p − 4)b|(−∆)s/2u|42 = µδs,q(pδs,p − qδs,q)|u|qq ≤ 0, (3.21) where pδs,p > 4 ≥ qδs,q by 2 < q ≤ 2 + 8s 3 < p < 2∗s, δs,q > 0 by q > 2. Then, it follows that |(−∆)s/2u|2 = 0. (3.22) Further, from (3.19), (3.21) and the fractional Gagliardo-Nirenberg inequality (2.2), we obtain that |u|q = 0 and |u|p = 0. Thus, we deduce that u ≡ 0, which is in contradiction with u ∈ Sc. So we obtain P0 c,µ = ∅. To proof that Pc,µ is a smooth manifold of codimension 2 in Hs(R3) is very similar to the one of [24, lemma 5.2], therefore we omit it here. □ Since P0 c,µ = ∅ by lemma 3.3, we observe that Pc,µ is a natural constraint in the following sense. Lemma 3.4. Let µ < 0 and 2 < q ≤ 2 + 8s 3 < p < 2∗s be given constants. If u ∈ Pc,µ is a critical point for Eµ|Pc,µ , then u is a critical point for Eµ|Sc . 10 Z. GUO, T. ZHANG EJDE-2025/75 Proof. From lemma 3.3, we see that Pc,µ is a smooth manifold of codimension 2 in Hs(R3) and P0 c,µ = ∅. If u ∈ Pc,µ is a critical point for Eµ|Pc,µ , then by the Lagrange multipliers rule, one gets that there exist λ, ν ∈ R such that ⟨E′ µ(u), ϕ⟩ − λ ∫ R3 uϕdx− ν⟨P ′ µ(u), ϕ⟩ = 0 for any ϕ ∈ Hs(R3), that is, u solves[ (1− 2ν)a+ (1− 4ν)b ∫ R3 |(−∆)s/2u|2dx ] (−∆)su = λu+ µ(1− νqδs,q)|u|q−2u+ (1− νpδs,p)|u|p−2u in R3. (3.23) By lemma 2.2, we obtain that 3− 2s 2 (1− 2ν)a ∫ R3 |(−∆)s/2u|2dx+ 3− 2s 2 (1− 4ν)b (∫ R3 |(−∆)s/2u|2dx )2 − 3 2 λ ∫ R3 u2dx+ 3µ(νqδs,q − 1) q ∫ R3 uqdx+ 3(νpδs,p − 1) p ∫ R3 updx = 0 in R3. (3.24) Multiplying (3.23) by u and integrating, then combining it with (3.24), we obtain that (1− 2ν)a ∫ R3 |(−∆)s/2u|2dx+ (1− 4ν)b (∫ R3 |(−∆)s/2u|2dx )2 + µδs,q(νqδs,q − 1) ∫ R3 uqdx+ δs,p(νpδs,p − 1) ∫ R3 updx = 0. (3.25) By (3.25) and Pµ(u) = 0, we have that ν ( 2a|(−∆)s/2u|22 + 4b|(−∆)s/2u|42 − µqδ2s,q|u|qq − pδ2s,p|u|pp ) = 0, which implies that ν = 0 since u /∈ P0 c,µ: from lemma 3.3, we see that Pc,µ is a smooth manifold of codimension 2 in Hs(R3) and P0 c,µ = ∅, namely,( 2a|(−∆)s/2u|22 + 4b|(−∆)s/2u|42 − µqδ2s,q|u|qq − pδ2s,p|u|pp ) ̸= 0. □ Lemma 3.5. For every u ∈ Sc, there exists a unique tu ∈ R such that tu ∗u ∈ Pc,µ. Moreover, tu is the unique critical point of Jµ u and it is a strict maximum point at the positive level. Moreover, (i) Pc,µ = P− c,µ. (ii) Jµ u is strictly decreasing and concave on (tu,+∞) and tu < 0 implies that Pµ(u) < 0. (iii) The function u ∈ Sc 7→ tu is of class C1. (iv) If Pµ(u) < 0, then tu < 0. Proof. For every u ∈ Sc, according to the definition of Jµ u (τ), we obtain lim τ→−∞ Jµ u (τ) = 0+ and lim τ→+∞ Jµ u (τ) = −∞. Thus, Jµ u has at least one global maximum point tu at positive level. Besides, this is the one and only critical point of Jµ u . For each u ∈ Sc, we define h(t) = at2s 2 |(−∆)s/2u|22 + bt4s 4 |(−∆)s/2u|42 − tpδs,ps p |u|pp − µ tqδs,qs q |u|qq. Indeed, Jµ u (τ) = h(eτ ), then we obtain Jµ u ′(τ) = h′(eτ )eτ . Thus, it is sufficient to study the function h. The derivative of h can be written as h′(t) = t4s−1η(t), where η(t) = ast−2s|(−∆)s/2u|22 + bs|(−∆)s/2u|42 − δs,pst (pδs,p−4)s|u|pp − µsδs,qt (qδs,q−4)s|u|qq, and η satisfies lim t→0+ η(t) = +∞, lim t→+∞ η(t) = −∞, η′(t) < 0, for all t > 0. EJDE-2025/75 NORMALIZED SOLUTIONS OF FRACTIONAL KIRCHHOFF EQUATIONS 11 From the above analysis for the function η, we obtain it has a unique zero point t̃ in (0,+∞). By h′(t) = t4s−1η(t), it holds that the function h has a unique critical point t̃ and tu = ln t̃. By lim t→0+ h(t) = 0+, and lim t→+∞ h(t) = −∞, we deduce that h(t̃) > 0. From the above arguments and sPµ(tu ∗ u) = (Jµ u ) ′(tu), we deduce that for each u ∈ Sc, there exists a unique tu ∈ R such that Pµ(tu ∗ u) = 0, namely, tu ∗ u ∈ Pc,µ. Let u ∈ Pc,µ, then we obtain tu = 0 and as tu is a maximum point of Jµ u , we obtain that (Jµ u ) ′′ (0) ≤ 0. Since P0 c,µ = ∅, we conclude that (Jµ u ) ′′ (0) < 0. Thus, Pc,µ = P− c,µ. From the calculus above, we can also deduce that Jµ u is strictly decreasing and concave on (tu,+∞). Since (Jµ u ) ′(t) < 0 if and only if t > tu, we conclude that Pµ(u) = 1 s (J µ u ) ′(0) < 0 if and only if tu < 0. Item (iii) holds when applying the implicit function theorem to the function Φ(τ, u) = (Jµ u ) ′(τ). We use that Φ(tu, u) = (Jµ u ) ′(tu) = 0, that ∂τΦ(tu, u) = (Jµ u ) ′′ (tu) < 0, and the fact that it is not possible to pass with continuity from P+ c,µ to P− c,µ (since P0 c,µ = ∅), therefore we obtain u 7→ tu is C1. □ Lemma 3.6. It holds that m(c, µ) = inf u∈Pc,µ Eµ(u) > 0. Proof. Setting u ∈ Pc,µ, by fractional Gagliardo-Nirenberg inequality and µ < 0 we obtain that a|(−∆)s/2u|22 ≤ δs,p|u|pp ≤ δs,pC(s, p) p| −∆s/2u|pδs,p2 |u|p(1−δs,p) 2 . (3.26) From the definition of Pc,µ, we obtain u ∈ Sc, namely, |u|2 = c. Therefore, |(−∆)s/2u|2 ≥ ( a δs,pC(s, p)pcp(1−δs,p) ) 1 pδs,p−2 ⇒ inf Pc,µ |(−∆)s/2u|2 > 0. (3.27) For every u ∈ Pc,µ, we obtain that Eµ(u) = (1 2 − 1 pδs,p ) a|(−∆)s/2u|22 + (1 4 − 1 pδs,p ) b|(−∆)s/2u|42 − µ q ( 1− qδs,q pδs,p ) |u|qq ≥ (1 2 − 1 pδs,p ) a|(−∆)s/2u|22. Then pδs,p > 2 and (3.27) imply that m(c, µ) = inf Pc,µ Eµ(u) > 0. □ Lemma 3.7. For 2 < q ≤ 2 + 8s 3 < p < 2∗s and µ < 0, there exists k > 0 small enough such that 0 < sup Āk Eµ(u) < m(c, µ) and u ∈ Āk ⇒ Eµ(u), Pµ(u) > 0, where Āk = {u ∈ Sc : |(−∆)s/2u|22 ≤ k}. Proof. For u ∈ Āk with k small enough, thanks to the fractional Gagliardo-Nirenberg inequality and the fact that pδs,p > 4 (because of p > 2 + 8s 3 ), we have that Eµ(u) = a 2 |(−∆) s 2u|22 + b 4 |(−∆) s 2u|42 − µ q |u|qq − 1 p |u|pp ≥ a 2 |(−∆) s 2u|22 − 1 p C(s, p)pcp(1−δs,p)|(−∆) s 2u|pδs,p2 > 0 and that Pµ(u) = a|(−∆) s 2u|22 + b|(−∆) s 2u|42 − µδs,q|u|qq − δs,p|u|pp ≥ a|(−∆) s 2u|22 − C(s, p)pcp(1−δs,p)|(−∆) s 2u|pδs,p2 δs,p > 0. 12 Z. GUO, T. ZHANG EJDE-2025/75 Then, if u ∈ Āk, we have that supĀk Eµ(u) ≥ Eµ(u) > 0 and Pµ(u) > 0. Now, replacing k with a smaller quantity, recalling that m(c, µ) > 0 by lemma 3.6 and applying fractional Gagliardo- Nirenberg inequality, we conclude that Eµ(u) ≤ a 2 |(−∆) s 2u|22 + b 4 |(−∆) s 2u|42 + |µ| q C(s, q)qcq(1−δs,q)|(−∆) s 2u|qδs,q2 < m(c, µ). Thus supĀk Eµ(u) < m(c, µ). □ Now, we define Ec µ = {u ∈ Sc : Eµ(u) ≤ c} and the minimax class τ̃ := {γ = (α, β) ∈ C([0, 1],R× Sr c ) : γ(0) ∈ (0, Āk) and γ(1) ∈ (0, E0 µ)}, where Sr c = Sc ∩Hs r . We define the minimax level as follows: σ(c, µ) = inf γ∈τ̃ max (τ,u)∈γ([0,1]) Ẽµ(τ, u), where Ẽµ(τ, u) = Eµ(τ ∗ u) = Jµ u (τ) = ae2sτ 2 |(−∆)s/2u|22 + be4sτ 4 |(−∆)s/2u|42 − µ eqδs,qsτ q |u|qq − epδs,psτ p |u|pp. Proof of Theorem 2.8. Assume by contradiction that if there exists a solution u to (1.1)-(1.2), then by the Pohožaev identity Pµ(u) = 0, we obtain that a|(−∆)s/2u|22 + b|(−∆)s/2u|42 = µδs,q|u|qq + 4 p̄ |u|p̄p̄, where δs,p̄ = 4 p̄ when p = 2 + 8s 3 = p̄. Since b ≥ 4 p̄C(s, p̄) p̄cp̄−4, by the fractional Gagliardo- Nirenberg inequality, we have that a|(−∆)s/2u|22 + b|(−∆)s/2u|42 − 4 p̄ |u|p̄p̄ ≥ a|(−∆)s/2u|22 + b|(−∆)s/2u|42 − 4 p̄ C(s, p)p̄cp̄−4|(−∆)s/2u|42 ≥ 0, and hence we deduce that 0 > µδs,q|u|qq = a|(−∆)s/2u|22 + b|(−∆)s/2u|42 − 4 p̄ |u|p̄p̄ ≥ 0, which is a contradiction. □ Proof of Theorem 2.9. Through simple analysis we can see that it is suffice to prove that there exists a PS sequence such that conditions (i) and (ii) of lemma 3.2 hold. By lim τ→−∞ Jµ u (τ) = 0+ and lim τ→+∞ Jµ u (τ) = −∞, it follows that there exist τ1 and τ2 ∈ R satisfying Jµ u (τ) ≤ k for all τ < τ1 (k > 0), Jµ u (τ) ≤ 0 for all τ > τ2. Next, we define γu : [0, 1] → R× Sr c by γu(p) = ( 0, ((1− p)τ1 + pτ2) ∗ u ) , (3.28) which is a path in τ̃ , therefore σ(c, µ) is a real number. Now, we aim to prove the claim that for all γ ∈ τ̃ , there exists τγ ∈ (0, 1) such that α(τγ) ∗ β(τγ) ∈ P− c,µ. (3.29) Actually, γ(0) = (0, β(0)) ∈ (0, Āk) and sPµ(t ∗ u) = (Jµ u ) ′(t). According to lemma 3.7, we obtain Pµ(β(0)) = 1 s (J µ u ) ′(0) > 0, further, by lemma 3.5, we obtain t0∗β(0) = tβ(0) > 0. In addition, by Eµ(β(1)) = Ẽµ(γ(1)) ≤ 0, we deduce that tα(1)∗β(1) = tβ(1) < 0. Indeed, Jβ(1)(τ) > 0 for each τ ∈ (−∞, tβ(1)], and Jβ(1)(0) = Eµ(β(1)) ≤ 0, it is necessary that tβ(1) < 0. EJDE-2025/75 NORMALIZED SOLUTIONS OF FRACTIONAL KIRCHHOFF EQUATIONS 13 Furthermore, lemma 3.5 implies that the function u ∈ Sc 7→ tu ∈ R is continuous. Then there exists τγ ∈ (0, 1) such that tα(τγ)∗β(τγ) = 0. This amounts to α(τγ) ∗β(τγ) = tα(τγ)∗β(τγ) ∗ ( α(τγ) ∗ β(τγ) ) ∈ Pc,µ = P− c,µ. By (3.29), it follows that max γ([0,1]) Ẽµ ≥ Ẽµ(γ(τγ)) = Eµ(α(τγ) ∗ β(τγ)) ≥ inf P− c,µ∩Sr c Eµ. Then we obtain that σ(c, µ) ≥ inf P− c,µ∩Sr c Eµ. (3.30) Besides, taking u ∈ P− c,µ ∩ Sr c and γu the corresponding path defined in (3.28), we derive that Eµ(u) = Ẽµ(0, u) = max γu([0,1]) Ẽµ ≥ σ(c, µ), then we obtain that inf P− c,µ∩Sr c Eµ ≥ σ(c, µ). (3.31) Formulas (3.30) and (3.31) imply that inf P− c,µ∩Sr c Eµ = σ(c, µ). (3.32) Next, we prove a claim that: inf Pc,µ∩Sr c Eµ = inf Pc,µ Eµ. (3.33) This is equivalent to verifying that infPc,µ Eµ ≥ infPc,µ∩Sr c Eµ. Suppose by contradiction that there exists u ∈ Pc,µ \ Sr c with Eµ(u) < infPc,µ∩Sr c Eµ. Then we set v := |u|∗, the symmetric decreasing rearrangement of the modulus of u, which belongs to Sr c . By lemma 2.7, we obtain Eµ(v) ≤ Eµ(u) and Pµ(v) ≤ Pµ(u). From u ∈ Pc,µ \ Sr c , we have Pµ(u) = 0. If Pµ(v) = 0, we immediately derive a contradiction, hence we assume that Pµ(v) < 0. In this case, from lemma 3.5, we know that tv < 0. But then we obtain a contradiction in the following way Eµ(u) < Eµ(tv ∗ v) = ae2stv 2 |(−∆)s/2v|22 + be4stv 4 |(−∆)s/2v|42 − µ q eqδs,qstv |v|qq − 1 pδs,p [ ae2stv |(−∆)s/2v|22 + be4stv |(−∆)s/2v|42 − µδs,qe qδs,qstv |v|qq ] = a( 1 2 − 1 pδs,p )e2stv |(−∆)s/2v|22 + b( 1 4 − 1 pδs,p )e4stv |(−∆)s/2v|42 − µ q (1− qδs,q pδs,p )eqδs,qstv |v|qq ≤ a( 1 2 − 1 pδs,p )|(−∆)s/2u|22 + b( 1 4 − 1 pδs,p )|(−∆)s/2u|42 − µ q (1− qδs,q pδs,p )|u|qq = Eµ(u), where we used that tv ∗ v and u lie in Pc,µ. This proves that infPc,µ∩Sr c Eµ = infPc,µ Eµ. By the above claim, (3.33), and lemma 3.6, we obtain that m(c, µ) = σ(c, µ). (3.34) We also obtain that m(c, µ) = σ(c, µ) > sup (Āk∪E0 µ)∩Sr c Eµ = sup ((0,Āk)∪(0,E0 µ))∩(R×Sr c ) Ẽµ. (3.35) The rest of the proof is the same as in Soave [24], existence of a second critical point of mountain pass type for Eµ|Sc , thus we omit it here (namely, we use lemma 2.5 to obtain a PS sequence {un} for Eµ|Sr c at level σ(c, µ) > 0 and dist(un,Pc,µ) → 0, i.e., Pµ(un) → 0). To verify that u is a ground state, we show that u achieves infPc,µ Eµ = m(c, µ). From the above proof, we know that σ(c, µ) = infγ∈τ̃ max(τ,u)∈γ([0,1]) Ẽµ(τ, u) = Eµ(u) = infPc,µ∩Sr c Eµ, hence we have to show that infPc,µ Eµ = infPc,µ∩Sr c Eµ. By claim (3.33), this equality holds, hence u is a ground state. Therefore, Theorem 2.9 is proved. □ 14 Z. GUO, T. ZHANG EJDE-2025/75 Proof of Theorem 2.10. We start by describing the structure of Zc,µ of ground states. If u ∈ Zc,µ, then u ∈ Pc,µ and Eµ(u) = m(c, µ) = infPc,µ Eµ. We claim that u ∈ Zc,µ ⇒ |u| ∈ Zc,µ, |(−∆)s/2|u||2 = |(−∆)s/2u|2. (3.36) To prove the claim, we observe that Eµ(|u|) ≤ Eµ(u) and Pµ(|u|) ≤ Pµ(u) = 0. Then by lemma 3.5, there exists t|u| ≤ 0 with t|u| ∗ |u| ∈ Pc,µ and by definition of t|u|, one has m(c, µ) ≤ Eµ(t|u| ∗ |u|) = a (1 2 1 pδs,p ) e2st|u| |(−∆)s/2|u||22 + b ( 1 4 − 1 pδs,p ) e4st|u| |(−∆)s/2|u||42 − µ q ( 1− qδs,q pδs,p ) eqδs,qst|u| |u|qq ≤ a (1 2 − 1 pδs,p ) e2st|u| |(−∆)s/2u|22 + b (1 4 − 1 pδs,p ) e4st|u| |(−∆)s/2u|42 − µ q ( 1− qδs,q pδs,p ) eqδs,qst|u| |u|qq ≤ [ a (1 2 − 1 pδs,p ) |(−∆)s/2u|22 + b (1 4 − 1 pδs,p ) |(−∆)s/2u|42 − µ q ( 1− qδs,q pδs,p ) |u|qq ] ekst|u| = ekst|u|Eµ(u) = ekst|u|m(c, µ), where k = min { 2, qδs,q } , and we used the fact that u, t|u| ∗ |u| ∈ Pc,µ, and Eµ(u) = m(c, µ). By t|u| ≤ 0, we deduce that necessarily t|u| = 0, that is Pµ(|u|) = 0, and since also Pµ(u) = 0, it holds that |u| ∈ Pc,µ, |(−∆)s/2|u||2 = |(−∆)s/2u|2 and Eµ(|u|) = m(c, µ). This proves claim (3.36). After proving that |u| minimizes Eµ on Pc,µ, we obtain that |u| is a non- negative solution to (1.1) for some λ ∈ R, by lemma 3.4. By regularity and the strong maximum principle, it is a C2 positive solution. Using also that |(−∆)s/2|u||2 = |(−∆)s/2u|2, and u ∈ Zc,µ (then we obtain |u| ∈ Zc,µ, namely, Eµ(|u|) = m(c, µ)), we will prove that eiθ|u| ∈ Zc,µ for any θ ∈ R. By the definition of Eµ(u) and that the modulus of eiθ = 1 for any θ ∈ R, we obtain that Eµ(e iθ|u|) = a 2 |(−∆)s/2eiθ|u||22 + b 4 |(−∆)s/2eiθ|u||42 − µ q |eiθ|u||qq − 1 p |eiθ|u||pp = a 2 |(−∆)s/2|u||22 + b 4 |(−∆)s/2|u||42 − µ q |u|qq − 1 p |u|pp = Eµ(|u|) = m(c, µ). In the remaining of this proof, we prove that if u ∈ Zc,µ, then the associated Lagrange multiplier λ is negative. Recalling that u ∈ Pc,µ, then Pµ(u) = 0, and we have that a|(−∆)s/2u|22 + b|(−∆)s/2u|42 = µδs,q|u|qq + δs,p|u|pp ≤ δs,p|u|pp. Then, by the fractional Gagliardo-Nirenberg inequality, we derive that a|(−∆)s/2u|22 ≤ a|(−∆)s/2u|22 + b|(−∆)s/2u|42 ≤ δs,p|u|pp ≤ δs,pC(s, p) p|(−∆)s/2u|pδs,p2 |u|p(1−δs,p) 2 . Next, we prove the claim that u ̸≡ 0. Since otherwise Eµ(u) = 0 = m(c, µ), in contradiction with lemma 3.6. By the definition of Pc,µ, we obtain |u|2 = c, then we can deduce that |(−∆)s/2u|2 ≥ ( a δs,pC(s, p)pcp(1−δs,p) ) 1 pδs,p−2 . (3.37) EJDE-2025/75 NORMALIZED SOLUTIONS OF FRACTIONAL KIRCHHOFF EQUATIONS 15 Now, since u is a weak radial and positive solution to (a+ b ∫ R3 |(−∆)s/2u|2dx)(−∆)su = λu+ µ|u|q−2u+ |u|p−2u in R3. (3.38) By the Pohožaev identity, we infer that Pµ(u) = 0, i.e., δs,p|u|pp = a|(−∆)s/2u|22 + b|(−∆)s/2u|42 − µδs,q|u|qq. (3.39) Testing (3.38) with u and using (3.39), we obtain that λ|u|22 = a ( 1− 1 δs,p ) |(−∆)s/2u|22 + b ( 1− 1 δs,p ) |(−∆)s/2u|42 + µ (δs,q δs,p − 1 ) |u|qq, where 1− 1 δs,p < 0 since 0 < δs,p < 1, while µ ( δs,q δs,p −1 ) > 0 since µ < 0. Using again the fractional Gagliardo-Nirenberg inequality and estimate (3.37), we infer that λ|u|22 ≤ a ( 1− 1 δs,p ) |(−∆)s/2u|22 + b ( 1− 1 δs,p ) |(−∆)s/2u|42 + µ (δs,q δs,p − 1 ) C(s, q)q|(−∆)s/2u|qδs,q2 |u|q(1−δs,q) 2 ≤ |(−∆)s/2u|qδs,q2 [ a ( 1− 1 δs,p ) |(−∆)s/2u|2−qδs,q 2 + b ( 1− 1 δs,p ) |(−∆)s/2u|4−qδs,q 2 + µ (δs,q δs,p − 1 ) C(s, q)qcq(1−δs,q) ] ≤ |(−∆)s/2u|qδs,q2 [ a ( 1− 1 δs,p )( a δs,pC(s, p)pcp(1−δs,p) ) 2−qδs,q pδs,p−2 + b ( 1− a δs,p )( 1 δs,pC(s, p)pcp(1−δs,p) ) 4−qδs,q pδs,p−2 + |µ| ( 1− δs,q δs,p ) C(s, q)qcq(1−δs,q) ] . It is not difficult to check that the right hand side is strictly negative when (2.9) holds, finally implying that λ < 0, as desired, hence the proof is complete. □ Acknowledgents. This work was supported by a special fund for basic scientific research ex- penses of universities in the Liaoning Province (No. LJ212410165018). References [1] C. O. Alves, F. J. S. A. Corrêa, T. F. 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