Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 59, pp. 1–20. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.59 MINIMIZERS FOR FRACTIONAL SCHRÖDINGER EQUATIONS WITH INHOMOGENEOUS PERTURBATION LEI ZHANG, LINTAO LIU, HAIBO CHEN Abstract. In this article, we study a constrained minimization problem arising in fractional Schrodinger equations with inhomogeneous term m(x) ̸≡ 1. We obtain the existence and limit behavior of constraint minimizers. The argument relies on energy estimates, blow-up analysis, comparison principle and iteration methods. 1. Introduction Consider the following constraint minimizers of L2-subcritical fractional variational problem I(M) := inf u∈Hs(RN ), ∥u∥2 2=M E(u), (1.1) the energy functional E(u) is defined by E(u) := ∫ RN |(−∆)s/2u|2dx− 2 p+ 1 ∫ RN m(x)|u|p+1dx, (1.2) where N ≥ 2, s ∈ ( 12 , 1), p ∈ (1, 1 + 4s N ), M > 0 and the inhomogeneous term m(x) ̸≡ 1 satisfies the assumptions (A1) m(x) ∈ L∞ loc(RN ) ∩ C1,α(0 < α < 1), 0 < m(x) ≤ m(0) = maxx∈RN m(x) = 1, and 0 < infx∈RN m(x) = lim|x|→∞m(x) = m∞ < 1; (A2) 0 ∈ RN is the unique global maximum point of m(x), and 1−m(x) = |x|r+2(1 + o(1)) as |x| → 0, where r > 0. It is well known that the fractional Laplacian (−∆)s(s ∈ (0, 1)) can be defined by (−∆)sv(x) = CN,sP.V. ∫ RN v(x)− v(y) |x− y|N+2s dy = CN,s lim ε→0 ∫ RN\Bε(x) v(x)− v(y) |x− y|N+2s dy for v ∈ S(RN ), where P.V. denotes a Cauchy principal value, S(RN ) is the Schwartz space of rapidly decaying C∞ function, Bε(x) denotes an open ball of radius ε centered at x and the nor- malization constant CN,s = ( ∫ RN 1−cos(ζ1) |ζ|N+2s )−1 , see [5, 31, 37] and the references therein for more details. There are applications of operator (−∆)s in some areas such as fractional quantum me- chanics, physics and chemistry, obstacle problems, optimization and finance, conformal geometry and minimal surfaces, see [1, 4, 19, 20, 28, 32] and the references therein for more details. When s = 1, this problem is related to the orbital stability waves in nonlinear Schrödinger equations, which was proposed by Lions in [21]. After the pioneer work of Lions, much attention has been devoted to the study of the Schrödinger equation. Recently, many scholars have studied and extended the well-known Bose-Einstein condensates and time-independent Gross-Pitaevskii equation, the reader is referred to [6, 7, 13, 14, 15, 16, 17, 25, 26]. These works mainly studied the situation when m(x) ≡ 1. 2020 Mathematics Subject Classification. 35B44, 35J20, 35J60. Key words and phrases. Inhomogeneous fractional equation; energy estimate; blow-up analysis; mass concentration. ©2025. This work is licensed under a CC BY 4.0 license. Submitted March 28, 2025. Published June 6, 2025. 1 2 L. ZHANG, L. LIU, H. CHEN EJDE-2025/59 When s ∈ (0, 1), problem (1.1) originated from the fractional nonlinear Schrödinger equation (−∆)su+ V (x)u = f(x, u), x ∈ RN , where N ≥ 2, V : RN → R is an external function and f(x, u) is a nonlinearity. In recent years, the study on equations with the fractional Laplacian has been attracted much interest [2, 9, 10, 11, 23, 33, 38]. Cheng [2] considered the following fractional Schrödinger equation (−∆)su+ V (x)u = |u|p−1u, x ∈ RN , where V (x) is an unbounded potential and 1 < p < 1+ 4s N , they proved the existence of ground state by Lagrange multiplier method. Moreover, in [9], Dipierro, Palatucci and Valdinoci obtained the existence and symmetry results for solutions with V (x) ≡ 1. Felmer, Quaas and Tan [11] studied the same equation with a more general nonlinearity f(x, u) instead of |u|p−1u, they obtained the existence of positive solutions and analysed the regularity and symmetry properties of these solutions. Du, Tian, Wang and Zhang [10] studied the stationary (i.e., time-independent) fractional Schrödinger equation (−∆)su+ V (x)u = µu+ af(u), x ∈ RN , where N ≥ 2, V : RN → R is a trapping potential, µ ∈ R and a > 0 are parameters, and f is a subcritical nonlinearity. They proved that the optimal embedding constant for the fractional Gagliardo-Nirenberg-Sobolev inequality can be expressed by exact form, and established the ex- istence, nonexistence and mass concentration of L2-normalized solutions for the above equation. In addition, under a certain type of trapping potentials, by using some delicate energy estimates, the authors presented a detailed analysis of the concentration behavior of L2-normalized solutions in the mass critical case. We note that, when V (x) = 0 there is no result on existence and mass concentration behavior for inhomogeneous mass subcritical fractional problems. We are going to study the existence results of minimizers as M → ∞. Before describing more details, let us introduce the following fractional Gagliardo-Nirenberg-Sobolev inequality, see [10]. Lemma 1.1. Let p ∈ (1, 1 + 4s N ). Then∫ RN |u|p+1dx ≤ Copt (∫ RN |(−∆)s/2u|2dx) )N(p−1) 4s (∫ RN |u|2dx) ) 2s(p+1)−N(p−1) 4s (1.3) for u ∈ Hs(RN )\{0}. This equality is attained by a function Q(x) with the following properties: (i) Q(x) is radial, positive and strictly decreasing in |x|; (ii) Q(x) ∈ H2s+1(RN ) ∩ C∞(RN ) and satisfies C1 1 + |x|N+2s ≤ Q(x) ≤ C2 1 + |x|N+2s x ∈ RN ; (1.4) (iii) Q(x) is the unique solution of the fraction Schrödinger equation (−∆)su+ 2s(p+ 1)−N(p− 1) N(p− 1) u− 4s N(p− 1) up = 0; (1.5) (iv) Copt = p+ 1 2∥Q∥p−1 2 . (1.6) According to Lemma 1.1, by a simple calculation, we can observe that∫ RN |(−∆)s/2Q|2dx = ∫ RN |Q|2dx = 2 p+ 1 ∫ RN |Q|p+1dx. (1.7) We now state our main results. Theorem 1.2. If m(x) satisfies (A1), then there exists at least one minimizer of I(M) for any M ∈ (0,∞). EJDE-2025/59 MINIMIZERS FOR FRACTIONAL SCHRÖDINGER EQUATIONS 3 To prove Theorem 1.2, we first prove the strict subadditivity inequality of I(M), then applying the fractional Gagliardo-Nirenberg-Sobolev inequality (1.3) to obtain the uniform boundedness of the minimizing sequences. Moreover, by using the concentration-compactness principle [21, 22], the compactness of minimizing sequences can be obtained. Motivated by studies [13, 15, 25, 27, 36], we are concerned with the limit behavior of minimizers as M → ∞, we have the following result. Theorem 1.3. Suppose m(x) satisfies (A1) and (A2). Let vk be a nonnegative minimizer of I(Mk). Then, for any sequence {Mk} with Mk → ∞ as k → ∞, there exists a subsequence of vk, still denoted by vk, such that vk has a unique maximum point z̄k and satisfies lim k→∞ ε 2s p−1 k vk(εkx+ z̄k) = Q(x) u ∈ Hs(RN ), (1.8) where limk→∞ z̄k = 0, εk := (Mk a∗ )− p−1 4s−N(p−1) , a∗ = ∥Q(x)∥22 and Q(x) is the unique radially symmetric positive solution of (1.5). Motivated by Maeda [27], we rewrite the constraint variational problem (1.1) into the equivalent form IM := inf v∈Hs(RN ), ∥v∥2 2=1 EM (v), (1.9) where EM (v) := ∫ RN |(−∆)s/2v|2dx− 2M p−1 2 p+ 1 ∫ RN m(x)|v|p+1dx 1 < p < 1 + 4s N , (1.10) which implies that vM := M−1/2uM is a nonnegative minimizer of IM and IM = M−1I(M) if and only if uM is a nonnegative minimizer of I(M). In other words, the proof of Theorem 1.3 can be equivalent to analyzing the limit behavior of minimizers for (1.9) as M → ∞. Up to some necessary scaling of the minimizers, one can obtain the boundedness of minimizers as M → ∞. Another difficult in studying the limit behavior of vM is to locate the peak of vM as M → ∞. Inspired by the works in [24, 25, 27], we introduce the following new constraint variational problem ĨM := inf v∈Hs(RN ),∥v∥2 2=1 ẼM (v), (1.11) where ẼM (v) is defined by ẼM (v) := ∫ RN |(−∆)s/2v|2dx− 2M p−1 2 p+ 1 ∫ RN |v|p+1dx 1 < p < 1 + 4s N . (1.12) By establishing that IM−ĨM → 0 asM → ∞, one deduces that 2M p−1 2 p+1 ∫ RN (1−m(x))|vM |p+1dx→ 0 as M → ∞, which is a good way to locate the peak of minimizers. The rest of this article is organized as follows. Section 2 is devoted to proving Theorem 1.2 on the existence of minimizers for (1.1). While in remaining section, we give the proof of Theorem 1.3. Throughout this paper, we use the following notation: • The space Hs(RN ) is equipped with the norm ∥u∥2 = ∫ RN (|(−∆)s/2u|2 + |u|2)dx; • The norm in Lp(RN ) is denoted by ∥ · ∥p, where p ∈ [1,∞]; • C,C1, C2, . . . , denote different positive constants; • ”→” denoted strongly convergence, ”⇀” denoted weakly convergence. 2. Existence of minimizers for I(M) This section is concerned with the proof of Theorem 1.2 on the existence of minimizers for (1.1). We first establish the subadditivity inequality of I(M), and then prove Theorem 1.2 by applying the concentration-compactness principle. Now, we give the following Lemma about subadditivity. Lemma 2.1. Assume m(x) satisfies (A1), then for any M ∈ (0,∞), it holds that I(M) < 0. (2.1) 4 L. ZHANG, L. LIU, H. CHEN EJDE-2025/59 Moreover, we have the following strict subadditivity inequality I(M) < I(α) + I(M − α) ∀ α ∈ (0,M). (2.2) Proof. As for (2.1), set uς(x) := ςN/2u(ςx), where ς > 0 and u ∈ Hs(RN ) satisfies ∥u∥2 = M . One can deduce that, for any M ∈ (0,∞), I(M) ≤ E(uς) = ∫ RN |(−∆)s/2uς |2dx− 2 p+ 1 ∫ RN m(x)|uς |p+1dx = 2 ( ς2s 2 ∫ RN |(−∆)s/2u|2dx− ς N(p−1) 2 p+ 1 ∫ RN m( x ς )|u|p+1dx ) , it then follows that E(uς) < 0 for ς > 0 sufficiently small, because N(p−1) 2 < 2s and m(x) satisfies (A1), thus, (2.1) holds. As for (2.2), for any M ∈ (0,∞) and θ ∈ ( 1, Mα ] , we have I(θα) = inf u∈Hs(RN ),∥u∥2 2=θα E(u) = inf v∈Hs(RN ),∥v∥2 2=α E(θ1/2v) = inf v∈Hs(RN ),∥v∥2 2=α { θ ∫ RN |(−∆)s/2v|2dx− 2θ p+1 2 p+ 1 ∫ RN m( x ς )|v|p+1dx } = inf v∈Hs(RN ),∥v∥2 2=α { θ [ ∫ RN |(−∆)s/2v|2dx− 2 p+ 1 ∫ RN m( x ς )|v|p+1 ] + 2(θ − θ p+1 2 ) p+ 1 ∫ RN m( x ς )|v|p+1dx } < θI(α), where the last inequality holds because θ > 1 and p > 1. This implies that for any M ∈ (0,∞), I(θα) < θI(α) ∀α ∈ (0,M), θ ∈ (1, M α ]. (2.3) Furthermore, it follows from (2.3) that I(M) = M − α M I ( M M − α (M − α) ) + α M I (M α · α ) < I(M − α) + I(α) ∀α ∈ (0,M). Hence, the proof of Lemma 2.1 is complete. □ Next, we introduce the concentration-compactness principle to fractional Sobolev spacesHs(RN ), see for example [12]. Lemma 2.2. Let N ≥ 2, suppose {un}n≥1 ⊂ Hs(RN ) and satisfies∫ RN |un|2dx = ρ > 0, sup n≥1 ∥un∥Hs(RN ) <∞. Then there exists a subsequence {unk }k≥1 for which one of the following properties holds. (i) Compactness: there exists a sequence {yk}k≥1 in RN , such that, for any ε > 0, there exists 0 < r <∞ with ∫ |x−yk|≤r |unk |2dx ≥ ρ− ε. (ii) Vanishing: for all r <∞, it follows that lim k→∞ sup y∈RN ∫ |x−y|≤r |unk |2dx = 0. (iii) Dichotomy: there exist a constant β ∈ (0, ρ) and two bounded sequences {vk}k≥1, {wk}k≥1 ⊂ Hs(RN ) such that supp vk ∩ suppwk = ∅, |vk|+ |wk| ≤ |unk |, ∥vk∥22 → β, ∥wk∥22 → (ρ− β) as k → ∞, EJDE-2025/59 MINIMIZERS FOR FRACTIONAL SCHRÖDINGER EQUATIONS 5 ∥unk − vk − wk∥q → 0 for q ∈ [2, 2∗s), lim inf k→∞ {⟨(−∆)sunk , unk ⟩ − ⟨(−∆)svk, vk⟩ − ⟨(−∆)swk, wk⟩} ≥ 0. Proof of the Theorem 1.2. We first claim that −∞ < I(M) < 0. For any given M ∈ (0,∞), assume ∥u∥22 = M . From Lemma 2.1, there holds that I(M) < 0. On the other hand, applying (A1) and Gagliardo-Nirenberg-Sobolev inequality (1.3) to E(u), then yields that E(u) = ∫ RN |(−∆)s/2u|2dx− 2 p+ 1 ∫ RN m(x)|u|p+1dx ≥ ∫ RN |(−∆)s/2u|2dx− ∥Q∥1−p 2 M 2s(p+1)−N(p−1) 4s (∫ RN |(−∆)s/2u|2dx )N(p−1) 4s ≥ −4s−N(p− 1) 4s (CN(p− 1) 4s ) 4s 4s−N(p−1) , (2.4) where C = ∥Q∥1−p 2 M 2s(p+1)−N(p−1) 4s . This implies that E(u) is bounded from below for any M ∈ (0,∞). Let {un} ⊂ Hs(RN ) be a minimizing sequence satisfies ∥u∥22 = M and limn→∞E(un) = I(M), then (2.4) shows that {un} is bounded uniformly in Hs(RN ). Moreover, since −∞ < I(M) < 0, we obtain E(un) ≤ I(M) 2 for n sufficiently large. Therefore, we can deduce from (1.2) that ∫ RN m(x)|un|p+1dx ≥ (p+ 1)I(M) 4 . (2.5) Applying Lemma 2.2 with ρ = 1, one can conclude that there exists a subsequence {unk } of {un} such that {unk } satisfies the compactness or the dichotomy or the vanishing. We first prove that the vanishing does not occur. If not, according to vanishing Lemma, we know that unk → 0 in Lp+1(RN ). This is a contradiction with (2.5). Next we prove that the dichotomy does not occur. If not, then Lemma 2.2(iii) shows that there exist two sequences {vk}, {wk} such that lim infk→∞ (E(unk )− E(vk)− E(wk)) ≥ 0, which implies lim sup k→∞ (E(vk) + E(wk)) ≤ I(M). (2.6) By direct calculation, it follows that E(u) = 1 ξ2 E(ξu) + 2(ξp−1 − 1) P + 1 ∫ RN m(x)|u|p+1dx. (2.7) Set ξk = 1 ∥vk∥2 , then ∥ξkvk∥22 = 1, (2.7) shows that E(vk) ≥ I(M) ξ2k + 2(ξp−1 k − 1) P + 1 ∫ RN m(x)|vk|p+1dx. (2.8) Similarly, we have E(wk) ≥ I(M) γ2k + 2(γp−1 k − 1) P + 1 ∫ RN m(x)|wk|p+1dx, (2.9) where γk = 1/∥wk∥2. By the fact that ξk → β−1/2 and γk → (1 − β)−1/2 as k → ∞, we deduce from (2.5),(2.8),(2.9) and Lemma 2.2(iii) that lim inf k→∞ (E(vk) + E(wk)) ≥ I(M) + 2(ζ − 1) p+ 1 lim inf k→∞ ∫ RN m(x)|unk |p+1dx ≥ I(M)− (ζ − 1) 2 I(M) > I(M), (2.10) where ζ := min{β−1/2, (1 − β)−1/2} > 1, due to β ∈ (0, 1) and I(M) < 0. Combining (2.6) and (2.10), we can conclude that the dichotomy does not occur. Now Lemma 2.2(i) shows that there exists a subsequence of {unk } (still denoted by {unk }) and some {yk} ⊂ (RN ), such that ûnk := unk (·+ yk) satisfies ûnk ⇀ u0 in Hs(RN ) for some u0 ∈ Hs(RN ), ûnk → u0 in Lq(RN ) for q ∈ [2, 2∗s). (2.11) 6 L. ZHANG, L. LIU, H. CHEN EJDE-2025/59 We can see that lim k→∞ ∫ RN m(x)|ûnk |p+1dx = ∫ RN m(x)|u0|p+1dx. By the weakly lower semicontinuity, we deduce that E(u0) ≤ lim k→∞ E(ûnk ), (2.12) we observe that I(M) ≤ limk→∞E(ûnk ) from the definition of I(M). If I(M) = limk→∞E(ûnk ), using (2.12) we have I(M) ≤ E(u0) ≤ lim k→∞ E(ûnk ) = I(M), (2.13) it follows that u0 is a minimizer of I(M) for any M ∈ (0,∞). On the other hand, I(M) < limk→∞E(ûnk ), we claim that {yk} is bounded in RN . Otherwise, assume yk → ∞, then we have I(M) = lim k→∞ E(unk ) = lim k→∞ {∫ RN |(−∆)s/2unk |2dx− 2 p+ 1 ∫ RN m(x)|unk |p+1dx } = lim k→∞ {∫ RN |(−∆)s/2ûnk |2dx− 2 p+ 1 ∫ RN m(x+ yk)|ûnk |p+1dx } ≥ lim k→∞ E(ûnk ) + 2 p+ 1 ∫ RN (m(x)−m(x+ yk))|ûnk |p+1dx ≥ lim k→∞ E(ûnk ), where the last inequality holds because of infx∈RN m(x) = lim|x|→∞m(x). This is a contradiction, thus the claim holds. Passing to a subsequence if necessary, we have limk→∞ yk → y0 for some y0 ∈ RN . It follows from (2.11) that unk → u0 in Lq(RN ) with q ∈ [2, 2∗s). This yields that lim k→∞ ∫ RN m(x)|unk |p+1dx = ∫ RN m(x)|u0(x− y0)|p+1dx. Similar to (2.12) and (2.13), we can deduce that I(M) ≤ E(u0(· − y0)) ≤ lim k→∞ E(unk ), which implies that u0(· − y0) is a minimizer of I(M). Then the proof is complete. □ 3. Mass concentration In this section, we prove Theorem 1.3 on the concentration behavior of minimizers for I(Mk) with Mk → ∞ as k → ∞. We first establish the following Theorem. Theorem 3.1. Suppose m(x) satisfies (A1), (A2). Let uk be a nonnegative minimizer of IMk with Mk → ∞ as k → ∞. Then passing to a subsequence if necessary, uk has a unique maximum point z̄k as k is large enough, and z̄k satisfies limk→∞ z̄k = 0. Moreover, there also holds that lim k→∞ ε̂ N/2 k uk(ε̂kx+ z̄k) = (a∗)− 2s 4s−N(p−1)Q ( (a∗)− p−1 4s−N(p−1)x ) in Hs(RN ), (3.1) where ε̂k :=M − p−1 4s−N(p−1) k , a∗ = ∥Q∥22 and Q(x) is the unique radially symmetric positive solution of (1.5). 3.1. Energy estimates of IM . This section is aimed to establishing the refined energy estimates of IM by employing the analysis of ĨM defined in (1.11). As for the estimate of ĨM , we have the following Lemma. Lemma 3.2. Suppose ũM is a nonnegative minimizer of ĨM . Then ĨM = −λ (M a∗ ) 2s(p−1) 4s−N(p−1) , (3.2) and up to translations, ũM satisfies ũM = 1√ a∗ α̃ N/2 M Q(αMx), (3.3) EJDE-2025/59 MINIMIZERS FOR FRACTIONAL SCHRÖDINGER EQUATIONS 7 where a∗ = ∥Q∥22, Q(x) is the unique radially symmetric positive solution of (1.5), α̃M := ( M a∗ ) p−1 4s−N(p−1) ( N(p− 1) 4s ) 2 4s−N(p−1) and λ is defined by λ := 4s−N(p− 1) 4s (N(p− 1) 4s ) N(p−1) 4s−N(p−1) . (3.4) Proof. Suppose ũM is a nonnegative minimizer of ĨM , and suppose ũ1 is a nonnegative minimizer of Ĩ1. First we claim that ĨM = α2s M Ĩ1, ũM = α N/2 M ũ1(αMx) with αM =M p−1 4s−N(p−1) . (3.5) Indeed, set ṽ1 := α −N 2 M ũM (α−1 M x). Simple calculations show that ∥ṽ1∥22 = 1 and ĨM = ẼM (ũM ) = ∫ RN |(−∆)s/2ũM |2dx− 2M p−1 2 p+ 1 ∫ RN |ũM |p+1dx = α2s M ∫ RN |(−∆)s/2v1|2dx− 2M p−1 2 p+ 1 α N(p−1) 2 M ∫ RN |ṽ1|p+1dx =M 2s(p−1) 4s−N(p−1) (∫ RN |(−∆)s/2v1|2dx− 2M p−1 2 p+ 1 ∫ RN |ṽ1|p+1dx ) ≥M 2s(p−1) 4s−N(p−1) Ĩ1 (3.6) Similarly, setting ṽM := α N/2 M ũ1(αMx), we have ĨM ≤ ẼM (ũM ) = α2s M Ĩ1. (3.7) From (3.6) and (3.7), we can deduce that (3.5) holds. Next we prove that Ĩ1 = −λ ( 1 a∗ ) 2s(p−1) 4s−N(p−1) (3.8) and ũ1 satisfies ũ1 = (N(p− 1) 4s ) N 4s−N(p−1)√ a∗ − 4s 4s−N(p−1)Q ((N(p− 1) 4s ) 2 4s−N(p−1)√ a∗ − 2(p−1) 4s−N(p−1)x ) . (3.9) Take a test function ṽε = εN/2ṽ0(εx), where 0 < ṽ0 ∈ Hs(RN ) satisfies ∥ṽ0∥22 = 1 and ε > 0 is a positive constant, we have Ĩ1 ≤ Ẽ1(ṽε) = ε2s ∫ RN |(−∆)s/2ṽ0|2dx− 2 p+ 1 ε N(p−1) 2 ∫ RN |ṽ0|p+1dx < 0, when ε is small enough, due to 2s > N(p−1) 2 . Note that the minimizer ũ1 of Ĩ1 satisfies (−∆)sũ1 = µ̃1ũ1 + ũp1 in RN , (3.10) where µ̃1 is a suitable Lagrange multiplier. It follows from (1.12) and (3.10) that µ̃1 = −Ĩ1 + p− 1 p+ 1 ∫ RN |ũ1|p+1dx > 0. (3.11) Similar to the proof of [35, Proposition 4.1], we know ũ1 > 0. In view of (1.6) and (3.10), it follows that ũ1(x) = µ̃ 1 p−1 1 ( 4s 2s(p+ 1)−N(p− 1) ) 1 p−1 Q ( µ̃ 1 2s 1 ( N(p− 1) 2s(p+ 1)−N(p− 1) ) 1 2s x ) . (3.12) Furthermore, since ∥ũ1∥22 = 1, one can deduce that µ̃1 satisfies µ̃1 = (N(p− 1) 4s ) 4s 4s−N(p−1) 2s(p+ 1)−N(p− 1) N(p− 1) √ a∗ − 4s(p−1) 4s−N(p−1) , 8 L. ZHANG, L. LIU, H. CHEN EJDE-2025/59 from which and (3.12), we can deduce that (3.9) holds. Then (1.7), (1.12) and (3.9) show that (3.8) holds. Thus, combining (3.5) and (3.8), we can see that (3.2), (3.3) hold. □ Based on Lemma 3.2, we can obtain the following energy estimates of IM . Lemma 3.3. Suppose m(x) satisfies (A1). Then we have lim n→∞ IM = −λ (M a∗ ) 2s(p−1) 4s−N(p−1) , (3.13) where a∗ = ∥Q∥22 and Q(x) is the unique radially symmetric positive solution of (1.5), λ is given by (3.4). Proof. Let uM be a nonnegative minimizer of IM . According to the definition of IM and ĨM , we deduce from Lemma 3.2 that IM = ẼM (uM ) + 2M p−1 2 p+ 1 ∫ RN (1−m(x))|uM |p+1dx ≥ ĨM = −λ (M a∗ ) 2s(p−1) 4s−N(p−1) as M → ∞. (3.14) On the other hand, set a cut-off function φ ∈ C∞ 0 (RN ), such that φ(x) = 1 for |x| ≤ 1, φ(x) = 0 for |x| ≥ 2, 0 ≤ φ(x) ≤ 1 and |∇φ| ≤ 2. Define uτ (x) := Aτ ∥Q∥22 τN/2φ(x)Q(τx), (3.15) where τ > 0, Aτ is chosen so that ∥uτ∥22 = 1. Applying (1.4), we have 1 ≤ A2 τ ≤ 1 + Cτ−N−4s as τ → ∞. (3.16) In fact, since ∥uτ∥22 = 1, it follows that 1 = A2 τ ∥Q∥22 ∫ RN φ2( x τ )Q2(x)dx ≤ A2 τ ∥Q∥22 ∥Q∥22 = A2 τ , which implies the left inequality of (3.16). On the other hand, 1 = A2 τ ∥Q∥22 ∫ RN φ2( x τ )Q2(x)dx ≥ A2 τ ∥Q∥22 ∫ Bτ (0) Q2(x)dx, thus, we have A2 τ ≤ ∥Q∥22∫ Bτ (0) Q2(x)dx = ∫ Bτ (0) Q2(x)dx+ ∫ RN\Bτ (0) Q2(x)dx∫ Bτ (0) Q2(x)dx ≤ 1 + ∫ RN\Bτ (0) | C2 1+|x|N+2s |2(x)dx∫ Bτ (0) Q2(x)dx ≤ 1 + Cτ−N−4s. Furthermore, A2 τ → 1 as τ → ∞. Combining (1.7) and (3.16), we have IM ≤ EM (uτ ) = ∫ RN |(−∆)s/2uτ |2dx− 2M p−1 2 p+ 1 ∫ RN |m(x)uτ |p+1dx = ∫ RN |(−∆)s/2uτ |2dx− 2M p−1 2 p+ 1 ∫ RN |uτ |p+1dx+ 2M p−1 2 p+ 1 ∫ RN (1−m(x))|uτ |p+1dx = A2 ττ 2s ∥Q∥22 ∫ RN |(−∆)s/2Q(x)|2dx− 2M p−1 2 p+ 1 Ap+1 τ τ N(p−1) 2 ∥Q∥p+1 2 ∫ RN Q(x)p+1dx EJDE-2025/59 MINIMIZERS FOR FRACTIONAL SCHRÖDINGER EQUATIONS 9 + 2M p−1 2 τ−r−2 p+ 1 Ap+1 τ τ N(p−1) 2 ∥Q∥p+1 2 ∫ RN Q(x)p+1dx ≤ (1 + Cτ−N−4s)τ2s − (1− τ−r−2) (M a∗ ) p−1 2 τ N(p−1) 2 as τ → ∞, where r > 0 is defined in (A2). Take τ = (N(p−1) 4s ) 2 4s−N(p−1) ( M a∗ ) p−1 4s−N(p−1) , then yields that IM ≤ (N(p− 1) 4s ) 4s 4s−N(p−1) (M a∗ ) 2s(p−1) 4s−N(p−1) − (N(p− 1) 4s ) N(p−1) 4s−N(p−1) (M a∗ ) 2s(p−1) 4s−N(p−1) + o(1) = −λ (M a∗ ) 2s(p−1) 4s−N(p−1) as M → ∞, (3.17) where λ is given by (3.4). Therefore, (3.13) follows from (3.14) and (3.17) directly, and Lemma 3.3 is then proved. □ 3.2. Blow-up analysis. The main purpose of this subsection is to establish blow-up analysis of IM with M → ∞. Motivated by Guo [18, Lemma 2.2] and Maeda [27, Lemma 4.2], combining a new trial function and the polynomial decay of Q, we first obtain the following Lemma. Lemma 3.4. Let uM be a nonnegative minimizer of IM , then we have 0 ≤ IM − ĨM → 0 as M → ∞. Proof. Set ûM = AM√ a∗ α̃ N/2 M φ(α̃−t M )Q(α̃Mx), t > 0, where α̃M and φ are given in Lemma 3.2, a∗ = ∥Q∥22, Q(x) is the unique radially symmetric positive solution of (1.5) and AM > 0 is chosen so that ∥ûM∥22 = 1. As in Lemma 3.2, we have 1 ≤ A2 M ≤ 1 + C̃α̃ −(t+1)(N+4s) M . (3.18) Using the nonlocal Leibniz rule, we have∫ RN |(−∆)s/2ûM |2dx− ∫ RN |(−∆)s/2ũM |2dx = A2 M α̃ 2s M a∗ ∫ RN |(−∆)s/2φ(α̃−t−1 M x)Q(α̃Mx)|2dx− ∫ RN |(−∆)s/2ũM |2dx = A2 M α̃ 2s M a∗ ∫ RN |Q(x)(−∆)s/2φ(α̃−t−1 M x) + φ(α̃−t−1 M x)(−∆)s/2Q(x) −B(φ(α̃−t−1 M x), Q(x))|2dx− ∫ RN |(−∆)s/2ũM |2dx := T1 + T2 + T3 + T4 + T5 + T6, (3.19) where T1 := A2 M α̃ 2s M a∗ ∫ RN φ2(α̃−t−1 M x)|(−∆)s/2Q(x)|2dx− ∫ RN |(−∆)s/2ũM |2dx, T2 := A2 M α̃ 2s M a∗ ∫ RN Q2(x)|(−∆)s/2φ(α̃−t−1 M x)|2dx, T3 := A2 M α̃ 2s M a∗ ∫ RN ∫ RN B2(φ(α̃−t−1 M x), Q(x)), T4 := 2 A2 M α̃ 2s M a∗ ∫ RN Q(x)φ(α̃−t−1 M x)(−∆)s/2Q(x)(−∆)s/2φ(α̃−t−1 M x)dx, T5 := −2 A2 M α̃ 2s M a∗ ∫ RN Q(x)(−∆)s/2φ(α̃−t−1 M x)B(φ(α̃−t−1 M x), Q(x))dx, T6 := −2 A2 M α̃ 2s M a∗ ∫ RN φ(α̃−t−1 M x)(−∆)s/2Q(x)B(φ(α̃−t−1 M x), Q(x))dx, 10 L. ZHANG, L. LIU, H. CHEN EJDE-2025/59 and B(φ(α̃−t−1 M x), Q(x)) = CsP.V. ∫ RN (φ(α̃−t−1 M x)− φ(α̃−t−1 M y))(Q(x)−Q(y)) |x− y|N+s dy. To estimate T1 − T6, two inequalities are established below. By direct calculation, we have |(−∆)s/2φ(α̃−t−1 M x)| = CsP.V. ∣∣ ∫ RN φ(α̃−t−1 M x)− φ(α̃−t−1 M y) |x− y|N+s dy ∣∣ ≤ Cα̃−t−1 M ∫ |x−y|≤α̃t+1 M |∇φ(α̃−t−1 M ξ)| |x− y|N+s dy + C ∫ |x−y|≥α̃t+1 M 1 |x− y|N+s dy ≤ Cα̃ −s(t+1) M , (3.20) where ξ = y + θ(x− y) with θ ∈ (0, 1). By the Hölder inequality and (3.20), we obtain B(φ(α̃−t−1 M x), Q(x)) = CsP.V. ∫ RN (φ(α̃−t−1 M x)− φ(α̃−t−1 M y))(Q(x)−Q(y)) |x− y|N+s dy ≤ C (∫ RN |φ(α̃−t−1 M x)− φ(α̃−t−1 M y)|2 |x− y|N+s dy )1/2(∫ RN |Q(x)−Q(y)|2 |x− y|N+s dy )1/2 ≤ 2C (∫ RN |φ(α̃−t−1 M x)− φ(α̃−t−1 M y)| |x− y|N+s dy )1/2(∫ RN |Q(x)−Q(y)|2 |x− y|N+s dy )1/2 ≤ Cα̃ − s 2 (t+1) M (∫ RN |Q(x)−Q(y)|2 |x− y|N+s dy )1/2 (3.21) In view of (1.7), (3.3) and (3.18), it follows that T1 = (A2 M − 1)α̃2s M a∗ ∫ RN φ2(α̃−t−1 M x)|(−∆)s/2Q(x)|2dx + α̃2s M a∗ ∫ RN (φ2(α̃−t−1 M x)− 1)|(−∆)s/2Q(x)|2dx + α̃2s M a∗ ∫ RN |(−∆)s/2Q(x)|2dx− ∫ RN |(−∆)s/2ũM |2dx ≤ (A2 M − 1)α̃2s M a∗ ∫ RN φ2(α̃−t−1 M x)|(−∆)s/2Q(x)|2dx ≤ Cα̃ 2s−(t+1)(N+4s) M as M → ∞. (3.22) Using (1.7), (3.18) and (3.20), we have |T2| ≤ C A2 M α̃ 2s M a∗ α̃ −2s(t+1) M ∫ RN |Q|2dx ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃−2st M as M → ∞. (3.23) Applying (1.7), (3.18),(3.21) and [29, Proposition 3.4], we obtain |T3| ≤ C A2 M α̃ 2s M a∗ α̃ −s(t+1) M ∫ RN ∫ RN |Q(x)−Q(y)|2 |x− y|N+s dy dx ≤ C A2 M α̃ s(1−t) M a∗ ∫ RN |ξ|s|Q̌(ξ)|2dξ ≤ C 1 + C̃α̃ −(t+1)(N+4s) M a∗ α̃ s(1−t) M ∫ RN (1 + |ξ|2s)|Q̌(ξ)|2dξ ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃ s(1−t) M (3.24) EJDE-2025/59 MINIMIZERS FOR FRACTIONAL SCHRÖDINGER EQUATIONS 11 as M → ∞, where Q̌ denote the Fourier transform of Q. By the Hölder inequality, (1.7), (3.18) and (3.20), we obtain that |T4| ≤ C A2 M α̃ 2s M a∗ (∫ RN Q2(x)|(−∆)s/2φ(α̃−t−1 M x)|2dx )1/2 × (∫ RN φ2(α̃−t−1 M x)|(−∆)s/2Q|2dx )1/2 ≤ C A2 M α̃ 2s M a∗ ( C3α̃ −2s(t+1) M ∫ RN Q2(x)dx )1/2(∫ RN |(−∆)s/2Q|2dx )1/2 ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃ s(1−t) M as M → ∞. (3.25) From the Hölder inequality, (1.7), (3.18), (3.20) and (3.21), it follows that |T5| ≤ C A2 M α̃ 2s M a∗ (∫ RN Q2(x)|(−∆)s/2φ(α̃−t−1 M x)|2dx )1/2 × (∫ RN B2(φ(α̃−t−1 M x), Q(x))dx )1/2 ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃2s M α̃ −s(t+1) M α̃ − s(t+1) 2 M ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃ s( 1 2− 3 2 t) M asM → ∞. (3.26) Similarly, |T6| ≤ C A2 M α̃ 2s M a∗ (∫ RN φ2(α̃−t−1 M x)|(−∆)s/2Q|2dx )1/2(∫ RN B2(φ(α̃−t−1 M x), Q(x))dx )1/2 ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃2s M α̃ − s(t+1) 2 M ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃ s( 3 2− 1 2 t) M asM → ∞. (3.27) By the fact that t > 3, we then deduce from (3.19), (3.22)-(3.27) that∫ RN |(−∆)s/2ûM |2dx− ∫ RN |(−∆)s/2ũM |2dx→ 0 asM → ∞. (3.28) By (1.4) and (3.18), we obtain that M p−1 2 Ap+1 M α̃ N(p−1) 2 M√ a∗ p+1 ∫ RN m(x)(1− φp+1(α̃−t−1 M x))Qp+1(x)dx ≤M p−1 2 Ap+1 M α̃ N(p−1) 2 M√ a∗ p+1 ∫ |α̃−t−1 M x≥1| Qp+1(x)dx ≤ C(1 + C̃α̃ −(t+1)(N+4s) M ) p+1 2 (a∗)−1 (M a∗ ) p−1 2 Ap+1 M α̃ N(p−1) 2 M α̃ (t+1)[N−(N+2s)(p+1)] M ≤ Cα̃ 2s+(t+1)[N−(N+2s)(p+1)] M , then from (3.3) and (M2), we conclude that M p−1 2 ∫ RN m(x)|ûM |p+1dx = (M a∗ ) p−1 2 Ap+1 M α̃ N(p−1) 2 M a∗ ∫ RN m(α̃−1 M x)φp+1(α̃−t−1 M x)Qp+1(x)dx = 4s N(p− 1) Ap+1 M α̃2s M a∗ ∫ RN (m(α̃−1 M x)− 1)φp+1(α̃−t−1 M x)Qp+1(x)dx + 4s N(p− 1) Ap+1 M α̃2s M a∗ ∫ RN (φp+1(α̃−t−1 M x)− 1)Qp+1(x) dx 12 L. ZHANG, L. LIU, H. CHEN EJDE-2025/59 + 4s N(p− 1) Ap+1 M α̃2s M a∗ ∫ RN Qp+1(x)dx ≥ −C4A p+1 M α̃2s M α̃ −(r+2) M − C5A p+1 M α̃2s M α̃ (t+1)[N−(N+2s)(p+1)] M + 4s N(p− 1) Ap+1 M α̃2s M a∗ (M a∗ ) p−1 2 N(p− 1) 4s a∗ p−1 2 ∫ RN |ũM |p+1dx = −C4α̃ 2s−(r+2) M − C5α̃ 2s+(t+1)[N−(N+2s)(p+1)] M +M p−1 2 ∫ RN |ũM |p+1dx as M → ∞. which implies that M p−1 2 ∫ RN |ũM |p+1dx−M p−1 2 ∫ RN m(x)|ûM |p+1dx→ 0 as M → ∞. (3.29) Moreover 0 ≤ IM − ĨM ≤ EM (ûM )− ẼM (ũM ) = ∫ RN |(−∆)s/2ûM |2dx− ∫ RN |(−∆)s/2ũM |2dx + 2M p−1 2 p+ 1 ∫ RN |ũM |p+1dx− 2M p−1 2 p+ 1 ∫ RN m(x)|ûM |p+1dx, for large k. (3.30) it follows from (3.28)-(3.30) that 0 ≤ IM − ĨM → 0 asM → ∞. Thus, we completes the proof. □ Remark 3.5. Obviously, Lemma 3.4 shows that IM (uM )− ĨM (uM ) = 2M p−1 2 p+ 1 ∫ RN (1−m(x))|uM |p+1dx→ 0. Let uk be a nonnegative minimizer of IMk with Mk → ∞ as k → ∞. we define ε̂k :=M − p−1 4s−N(p−1) k with Mk → ∞ as k → ∞; (3.31) ŵk := ε̂ N/2 k uk(ε̂kx) with Mk → ∞ as k → ∞. (3.32) By simple analysis, we know that there exist some positive constants C6, C7, C ′ 6 and C ′ 7, such that C6 ≤ ∫ RN |(−∆)s/2ŵk|2dx ≤ C7 and C ′ 6 ≤ ∫ RN |(−∆)s/2ŵk|p+1dx ≤ C ′ 7. (3.33) Indeed, some calculations yields that∫ RN |(−∆)s/2ûk|2dx = ε̂−2s k ∫ RN |(−∆)s/2ŵk|2dx,∫ RN |ûk|p+1dx = ε̂ −N(p−1) 2 k ∫ RN |ŵk|2dx. (3.34) Applying (1.9), (3.31) and (3.34), we conclude that IMk = ε̂−2s k [ ∫ RN |(−∆)s/2ŵk|2dx+ ε̂2sk 2M p−1 2 k p+ 1 ∫ RN (1−m(x))|ũk|p+1dx− 2 p+ 1 ∫ RN |ŵk|p+1dx ] . This implies from Lemma 3.3 and Remark 3.5 that (a∗) 2s(p−1) 4s−N(p−1) (∫ RN |(−∆)s/2ŵk|2dx− 2 p+ 1 ∫ RN |ŵk|p+1dx ) → −λ < 0. (3.35) EJDE-2025/59 MINIMIZERS FOR FRACTIONAL SCHRÖDINGER EQUATIONS 13 If ∫ RN |(−∆)s/2ŵk|2dx → ∞ as k → ∞. Setting γ2k := ∫ RN |(−∆)s/2ŵk|2dx and ŵk(x) := γ N/2 k uk(γkx), one can deduce that ∥(−∆)s/2uk∥22 = ∥uk∥22 = 1. Applying the Gagliardo-Nirenberg- Sobolev inequality (1.3), it yields ∥uk∥p+1 p+1 ≤ Copt. Moreover, we can deduce that∫ RN |ŵk|p+1dx∫ RN |(−∆)s/2ŵk|2dx = ∫ RN |ûk|p+1dx∫ RN |(−∆)s/2ûk|2dx γ N(p−1)−4s 2 k ≤ Coptγ N(p−1)−4s 2 k → 0 (3.36) as k → ∞. However, from (3.35), we have∫ RN |ŵk|p+1dx∫ RN |(−∆)s/2ŵk|2dx → 1, as k → ∞, which contradicts (3.36), thus ∫ RN |(−∆)s/2ŵk|2dx ≤ C7. This fact combined with Gagliardo- Nirenberg-Sobolev inequality (1.3), we can conclude ∫ RN |ŵk|p+1dx ≤ C ′ 7. On the other hand, (3.35) shows that ∫ RN |ŵk|p+1dx ≥ C ′ 6. Then (1.3) further implies that ∫ RN |(−∆)s/2ŵk|2dx ≥ C6. Lemma 3.6. Let uk be a nonnegative minimizer of IMk as k → ∞. Then there exist a sequence {yk}, R0 > 0 and η > 0, such that the function wk(x) := ŵk(x+ yk) = ε̂ N/2 k uk(ε̂kx+ ε̂kyk), (3.37) satisfies lim inf k→∞ ∫ BR0 (0) wp+1 k dx ≥ η > 0, (3.38) where ε̂k is given in (3.31). Moreover, {ε̂kyk} is uniformly bounded as k → ∞, and for any sequence {yk}, there exists a subsequence, still denoted by {yk}, such that zk := ε̂kyk k−→ y0, for some point y0 ∈ RN and y0 is a global maximum point of m(x), that is y0 = 0. Proof. First, we claim that there exist a sequence {yk} ∈ RN and R0, η > 0 such that lim inf k→∞ ∫ BR0 (yk) ŵp+1 k dx ≥ η > 0. (3.39) Suppose by contradiction that for any R > 0, there exists a subsequence {ŵk}, such that lim k→∞ sup y∈RN ∫ BR0 (y) ŵp+1 k dx = 0. By the fact that 1 < p < 1 + 4s N , we deduce from [12, Lemma 2.8] that ŵk → 0 as k → ∞ in Lp+1(RN ), which contradicts (3.33). Then (3.37) and (3.39) show that (3.38) holds. Next, we show that {ε̂kyk} is bounded. By contradiction, suppose that {ε̂kyk} is unbounded. Then for any sequence {yk}, there exists a subsequence {yk} such that limk→∞ |ε̂kx+ ε̂kyk| → ∞ and limk→∞m(ε̂kx + ε̂kyk) = m∞ < 1, where the constants m∞ ≥ 0 and R0 > 0 are given in (A1) and (3.38), respectively. By Remark 3.5 and (3.37), we have 2M p−1 2 k p+ 1 ε̂ −N(p−1) 2 k ∫ RN (1−m(ε̂kx+ ε̂kyk))|ŵk|p+1dx→ 0 as k → ∞, from which and (3.38), we obtain that 0 = 2M p−1 2 k p+ 1 ε̂ −N(p−1) 2 k ∫ RN (1−m(ε̂kx+ ε̂kyk))|ŵk|p+1dx ≥ 2 p+ 1 ε̂−2s k (1−m∞)η as k → ∞, which is a contradiction. Thus, {ε̂kyk} is uniformly bounded as k → ∞. Moreover, zk := ε̂kyk k−→ y0, for some point y0 ∈ RN . Finally, using (3.38) and Fatou’s Lemma, we know that lim inf k→∞ ∫ RN (1−m(ε̂kx+ ε̂kyk))|ŵk|p+1dx ≥ (1−m(y0)) ∫ RN lim inf k→∞ |ŵk|p+1dx ≥ (1−m(y0))η, which with Remark 3.5, we can conclude that y0 = 0. By the assumption (A2) that 0 is the unique maximum point of m(x). Then the proof is complete. □ 14 L. ZHANG, L. LIU, H. CHEN EJDE-2025/59 To obtain the limitting behavior of wk, we first concentrate on the L∞-estimate and decay estimate of wk. Lemma 3.7. Let wk be given by Lemma 3.6. Then there exists a constant C > 0, such that ∥wk∥∞ ≤ C forallk ∈ N. (3.40) Moreover, wk → 0 as |x| → ∞, uniformly for large k. (3.41) Proof. Since uk is a nonnegative minimizer of IMk , we can derive that uk satisfies the Euler- Lagrange equation (−∆)suk = µkuk +M p−1 2 k m(x)upk in RN , (3.42) where µk ∈ R is a suitable Lagrange multiplier and satisfies µk = ∫ RN |(−∆)s/2uk|2dx−M p−1 2 k ∫ RN m(x)|uk|p+1dx = IMk − p− 1 p+ 1 M p−1 2 k ∫ RN m(x)|uk|p+1dx. Then from (3.37), we have (−∆)swk = µkε̂ 2s k wk +m(ε̂kx+ ε̂kyk)w p k. (3.43) Combining (1.9), (3.37), (3.43), Lemma 3.3 and Lemma 3.4, it follows that ε̂2sk µk = ε̂2sk ( IMk + p− 1 p+ 1 M p−1 2 k ∫ RN (1−m(x))|uk|p+1dx− p− 1 p+ 1 M p−1 2 k ∫ RN |uk|p+1dx ) = ε̂2sk IMk − p− 1 p+ 1 ∫ RN |wk|p+1dx = −λ(a∗)− 2s(p−1) 4s−N(p−1) − p− 1 p+ 1 ∫ RN |wk|p+1dx < 0 as k → ∞. (3.44) By (3.43) and (3.44), we have (−∆)swk ≤ wp k. (3.45) By a similar argument to the one in [3, Lemma 2.4] or [8, Proposition A.1], (3.40) holds. Now we show wk → 0 as |x| → ∞ uniformly for large k. The problem (3.43) can be rewritten as (−∆)swk = hk(x) in RN , where hk(x) := µkε̂ 2s k wk + (m(ε̂kx+ ε̂kyk)− 1)wp k + wp k. According to the argument in [8, Proposition A.1], we know that wk(x) ∈ Lr(RN ) for 2 ≤ r <∞. Using (3.33) and (3.44), we can conclude that the uniform boundedness {µkε̂ 2s k } as k → ∞. And by Lemma 3.6 that {ε̂kyk} is bounded in RN . As for the inhomogeneous term, from Remark 3.5 and (3.32), one can deduce that∫ RN (m(ε̂kx+ ε̂kyk)− 1)wp+1 k dx = ε̂ N(p−1) 2 k ∫ RN (m(x)− 1)up+1 k dx→ 0 as k → ∞. (3.46) Combining the above facts with (3.40), we deduce that hk ∈ C∞(RN ) for large k. Thus, using (3.40) and [32, Proposition 2.9], we have ∥wk∥C1,α(RN ) ≤ C(∥wk∥∞ + ∥hk∥∞) ≤ C for α < 2s− 1, as k → ∞. Finally, the fact wk ∈ Lr(RN ) ∩C1,α(RN ) for 2 ≤ r <∞, which implies that lim|x|→∞ wk(x) = 0 uniformly for large k. Thus, (3.41) holds. □ Lemma 3.8. Let wk be given by Lemma 3.6. Then there exists a constant C > 0, such that ∥wk∥∞ ≤ C 1 + |x|N+2s uniformly for large k, where C > 0 is a constant independent of k. EJDE-2025/59 MINIMIZERS FOR FRACTIONAL SCHRÖDINGER EQUATIONS 15 Proof. By the boundedness of {µkε̂ 2s k }, we may assume that µkε̂ 2s k → −γ, (3.47) for some γ ∈ R+, as k → ∞. According [32, Lemma 4.3], there exists a function ψ, such that 0 ≤ ψ ≤ C 1 + |x|N+2s , (3.48) (−∆)sψ + γ 2 ψ = 0 in RN\BR1(0), (3.49) for some suitable R1 > 0. By (3.47), we deduce that there exists R2 > 0 sufficiently large such that for large k, (−∆)swk + γ 2 wk ≤ µkε̂ 2s k wk +m(ε̂kx+ ε̂kyk)w p k + γ 2 wk ≤ 0 for |x| ≥ R2. (3.50) Let R3 = max{R1, R2}. For large k, set d := inf BR3(0) ψ > 0 and w̌k = (δ + 1)ψ − dwk, (3.51) where δ = sup ∥wk∥∞ <∞, we claim that w̌k ≥ 0 uniformly for large k. Indeed, if not, then there exists a sequence {xj} such that inf x∈RN w̌k(x) = lim j→∞ w̌k(xj) < 0. (3.52) Combining (3.41) and (3.48), we deduce that lim |x|→∞ w̌k(x) = 0 uniformly for large k. (3.53) Then, (3.52) and (3.53) show that {xj} is bounded. Moreover, up to a subsequence, let xj → x∗ for some x∗ ∈ RN as j → ∞. From (3.52), we have inf x∈RN w̌k(x) = w̌k(x∗) < 0, (3.54) which implies that (−∆)sw̌k(x∗) = −Cs 2 ∫ RN w̌k(x∗ + y) + w̌k(x∗ − y)− 2w̌k(x∗) |y|N+2s dy ≤ 0, (3.55) from which and (3.51), we obtain w̌k(x∗) = δψ + ψ − dwk ≥ δd + ψ − dδ > 0 in BR3(0). Thus, (3.54) shows that x∗ ∈ RN\BR3(0). (3.56) It follows from (3.48)-(3.51) that (−∆)sw̌k + γ 2 w̌k ≥ 0 ∈ RN\BR3(0). (3.57) Using (3.54)-(3.57), we deduce that 0 ≤ (−∆)sw̌k(x∗)+ γ 2 w̌k(x∗) < 0, which is a contradiction, so w̌k ≥ 0 uniformly for large k. Then (3.47) shows that wk ≤ (δ + 1)d−1ψ ≤ C 1 + |x|N+2s uniformly for large k, thus, we complete the proof of Lemma 3.8. □ The next Lemma shows the limitting behavior of wk. Lemma 3.9. Let wk be given by Lemma 3.6. Then passing to a subsequence if necessary, we have lim k→∞ wk(x) = (a∗)− 2s 4s−N(p−1)Q[(a∗)− p−1 4s−N(p−1)x+ x̃0] in Hs(RN ) for some x̃0 ∈ RN , where a∗ = ∥Q∥22 and Q(x) is the unique radially symmetric positive solution of (1.5). 16 L. ZHANG, L. LIU, H. CHEN EJDE-2025/59 Proof. From (3.33), we can know that wk is bounded uniformly in Hs(RN ). Using (3.46) and taking k → ∞, passing to a subsequence, then we have wk ⇀ w0 inH s(RN ) for some w0 ∈ Hs(RN ) satisfies (−∆)sw0 = −γw0 + wp 0 in RN , (3.58) where γ > 0 be given by Lemma 3.8. By (3.38), we know that w0 ̸≡ 0. Similar to the proof of [34, Proposition 4.4], we have that w0 ∈ C1,α(RN ) for some α ∈ (0, 1). Applying [29, Lemma 3.2], we have (−∆)sw̌0(x) = −Cs 2 ∫ RN w̌0(x+ y) + w̌0(x− y)− 2w̌0(x) |y|N+2s dy in RN . Assume that there exists x̄ ∈ RN , such that w0(x̄) = 0, this together with w0 ≥ 0 and w0 ̸≡ 0 leads to (−∆)sw̌0(x̄) = −Cs 2 ∫ RN w̌0(x̄+ y) + w̌0(x̄− y) |y|N+2s dy < 0. However, it is evident that (−∆)sw0(x̄) = −γw0(x̄) + wp 0(x̄) = 0, which is a contradiction. Thus, w0(x) > 0 for x ∈ RN . Since the equation (1.5), up to translations, admits a unique positive solution Q, it then follow from (3.58) that, there exists x̃0 ∈ RN such that w0(x) = γ 1 p−1Q(γ 1 2sx+ x̃0) for some x̃0 ∈ RN . (3.59) We now claim that∫ RN m(ε̂kx+ ε̂kyk)w p+1 k (x)dx→ ∫ RN m(ε̂kx+ ε̂kyk)w p+1 0 (x)dx. (3.60) Noting that ∣∣∣∣∫ RN m(ε̂kx+ ε̂kyk)w p+1 k (x)dx− ∫ RN m(ε̂kx+ ε̂kyk)w p+1 0 (x)dx ∣∣∣∣ ≤ ∫ RN |wp+1 k (x)− wp+1 0 (x)|dx = ∫ BR(0) |wp+1 k (x)− wp+1 0 (x)|dx+ ∫ RN\BR(0) |wp+1 k (x)− wp+1 0 (x)|dx := Ak +Bk where R > 0 is arbitrary. By the fact that |am − bm| ≤ L|a− b|m for a, b ≥ 0,m ≥ 1 and L ≥ 1, we have Ak ≤ L ∫ BR(0) |wp+1 k (x)− wp+1 0 (x)|dx→ 0 as k → ∞, (3.61) due to Sobolev embedding theorem and the uniform boundedness of wk in RN . On the other hand, from Lemma 3.8, (1.4) and (3.59), for any ε > 0, there exists a constant Rε > 0 independent of k, such that Bk ≤ ∫ RN\BR(0) |wp+1 k (x) + wp+1 0 (x)|dx ≤ ∫ RN\BR(0) C |x|(N+2s)(p+1) dx ≤ CRN−(N+2s)(p+1) ≤ Cε ∀R > Rε as k → ∞, (3.62) since the arbitrariness of ε, we conclude from (3.61) and (3.62) that (3.60) holds. Next, we prove that ∥w0∥22 = 1. By contradiction, we assume that ∥w0∥22 = l, where l ∈ (0, 1), due to ∥w0∥22 ≤ limk→∞ ∥wk∥22 = 1. Using the Brézis-Lieb Lemma, we have ∥wk∥22 = ∥w0∥22 + ∥wk − w0∥22 + o(1) as k → ∞, ∥wk∥p+1 p+1 = ∥w0∥p+1 p+1 + ∥wk − w0∥p+1 p+1 + o(1) as k → ∞, ∥(−∆)s/2wk∥22 = ∥(−∆)s/2w0∥22 + ∥(−∆)s/2(wk − w0)∥22 + o(1) as k → ∞. (3.63) EJDE-2025/59 MINIMIZERS FOR FRACTIONAL SCHRÖDINGER EQUATIONS 17 Set wl := w0 l , it follows from (3.8), (3.34)-(3.37), (3.60), (3.63) and Lemma 3.3 that Ĩ1 = lim k→∞ ε̂2sk (∫ RN |(−∆)s/2uk|2dx− 2M p−1 2 k p+ 1 ∫ RN m(x)|uk|p+1dx ) = lim k→∞ (∫ RN |(−∆)s/2wk|2dx− 2 p+ 1 ∫ RN m(ε̂kx+ ε̂kyk)|w0|p+1dx ) ≥ lim k→∞ (∫ RN |(−∆)s/2wk|2dx− 2 p+ 1 ∫ RN |w0|p+1dx ) = ∫ RN |(−∆)s/2w0|2dx+ lim k→∞ ∫ RN |(−∆)s/2(wk − w0)|2dx− 2 p+ 1 ∫ RN |w0|p+1dx ≥ ∫ RN |(−∆)s/2w0|2dx− 2 p+ 1 ∫ RN |w0|p+1dx = l ∫ RN |(−∆)s/2wl|2dx− l p+1 2 p+ 1 ∫ RN |wl|p+1dx = l (∫ RN |(−∆)s/2wl|2dx− 1 p+ 1 ∫ RN |wl|p+1dx ) ≥ lĨ1, which implies that l > 1, due to Ĩ1 < 0, this is a contradiction. Thus, ∥w0∥22 = 1. Moreover, we can obtain that γ = ∥Q∥ − 4s(p−1) 4s−N(p−1) 2 . Since ∥wk∥22 = ∥w0∥22 = 1, one can derive that wk → w0 in L2(RN ) as k → ∞. Then using the interpolation inequality and the Sobolev inequality, we have wk → w0 in Lq(RN ), q ∈ [2, 2∗s) as k → ∞. Further, substituting γ = ∥Q∥ − 4s(p−1) 4s−N(p−1) 2 into (3.59), we can deduce that Lemma 3.9 holds. □ 3.3. Proof of main resutls. In this section, we prove Theorem 1.3 on the concentration behavior of minimizers for IM with M → ∞. We first give the proof of Theorem 3.1. Proof of Theorem 3.1. From (3.41), one knows that uk admits at least one global maximum point. Let z̄k be any global maximum point of uk, according to Lemma 3.6 that zk := ε̂kyk → 0 as k → ∞. Hence, we deduce from (3.37) that wk(x) attains its global maximum point at xk = z̄k−zk ε̂k . One can verify that { z̄k−zk ε̂k } is bounded uniformly in RN . Otherwise, it follows from (3.41) that limk→∞ ∥wk∥∞ = 0 as |xk| → ∞, which contradicts to (3.38). This further implies that, passing to a subsequence if necessary lim k→∞ z̄k = lim k→∞ zk = 0. (3.64) By Lemma 3.9, we have wk(x) → (a∗)− 2s 4s−N(p−1)Q ( (a∗)− p−1 4s−N(p−1)x+ x̃0 ) (3.65) in Hs(RN ) for some x̃0 ∈ RN , as k → ∞, where wk is given by Lemma 3.6. We next prove that wk(x) → (a∗)− 2s 4s−N(p−1)Q ( (a∗)− p−1 4s−N(p−1)x+ x̃0 ) in C2 loc(RN ) as k → ∞. (3.66) Indeed, it follows from (3.37) that (−∆)swk = µkε̂ 2s k wk +m(ε̂kx+ ε̂kyk)w p k := g(wk) in RN , (3.67) where µk ∈ R is a suitable Lagrange multiplier. Similar to the proof of Lemma 3.6, we know that ∥wk∥∞ ≤ C uniformly in k, then (3.64) shows that g(wk) ∈ L∞(RN ). Applying [32, Proposition 2.9], we have ∥wk∥C1,α(RN ) ≤ C(∥wk∥∞ + ∥g(wk)∥∞) ≤ C for some α ∈ (0, 1) as k → ∞. (3.68) From this, (A1) and ∥wk∥∞ ≤ C, we assert that ∥m(ε̂kx+ ε̂kyk)w p k∥C0,α(RN ) 18 L. ZHANG, L. LIU, H. CHEN EJDE-2025/59 ≤ C + sup x ̸=y |m(ε̂kx+ ε̂kyk)−m(ε̂ky + ε̂kyk)||wp k(x)| |x− y|α + sup x̸=y |m(ε̂ky + ε̂kyk)||wp k(x)− wp k(y)| |x− y|α ≤ C + C8 sup x ̸=y |m(ε̂kx+ ε̂kyk)−m(ε̂ky + ε̂kyk)| |x− y|α + sup x ̸=y |m(ε̂ky + ε̂kyk)||wk(x)− wk(y)| |x− y|α |wp−1 k (x) + wp−1 k (y)| ≤ C + C8 sup x ̸=y |m(ε̂kx+ ε̂kyk)−m(ε̂ky + ε̂kyk)| |x− y|α + C9 sup x̸=y |wk(x)− wk(y)| |x− y|α , for any t > 0 fixed, we obtain |wk(x)− wk(y)| |x− y|α ≤ CT 1−α ≤ C if |x− y| ≤ T, |wk(x)− wk(y)| |x− y|α ≤ CT−α ≤ C if |x− y| > T, (3.69) and |m(ε̂kx+ ε̂kyk)−m(ε̂ky + ε̂kyk)| |x− y|α ≤ CT 1−α ≤ C if |x− y| ≤ T, |m(ε̂kx+ ε̂kyk)−m(ε̂ky + ε̂kyk)| |x− y|α ≤ CT−α ≤ C if |x− y| > T. (3.70) It follows from (3.69) and (3.70) that ∥m(ε̂kx+ ε̂kyk)w p k∥C0,α(RN ) ≤ C for large k. (3.71) The same procedure may be easily adapted to obtain ∥wk∥C0,α(RN ) ≤ C for large k. (3.72) Hence, combining (3.71) and (3.72), we conclude ∥g(wk)∥C0,α(RN ) ≤ C for large k. Applying [32, Proposition 2.8], we obtain ∥wk∥C1,ϱ1 (RN ) ≤ C(∥wk∥∞ + ∥g(wk)∥C0,α(RN )) ≤ C for large k, (3.73) where ϱ1 = α+ 2s− 1. Differentiating (3.67), we obtain (−∆)s(wk)xi = µkε̂ 2s k (wk)xi + ε̂km ′(ε̂kx+ ε̂kyk)w p k +m(ε̂kx+ ε̂kyk)pw p−1 k (wk)xi := G(x) for i = 1, 2, . . . , N . Obviously, (3.73) shows that ∥(wk)xi∥C0,ϱ1 (RN ) ≤ C and ∥m′(ε̂kx+ ε̂kyk)w p k∥C0,ϱ1 (RN ) ≤ C for large k. (3.74) Using (3.73), (3.74) and (A1), we conclude that ∥m(ε̂kx+ ε̂kyk)w p−1 k (wk)xi∥C0,α(RN ) ≤ C + sup x ̸=y ∣∣∣m(ε̂kx+ ε̂kyk)w p−1 k (x)∇wk(x)−m(ε̂ky + ε̂kyk)w p−1 k (y)∇wk(y) ∣∣∣ |x− y|α ≤ C + sup x ̸=y m(ε̂kx+ ε̂kyk)|wp−1 k (x)− wp−1 k (y)||∇wk(x)| |x− y|α + sup x̸=y m(ε̂kx+ ε̂kyk)|wp−1 k (y)||∇wk(x)−∇wk(y)| |x− y|α + sup x̸=y |wp−1 k (y)||∇wk(y)||m(ε̂kx+ ε̂kyk)−m(ε̂ky + ε̂kyk)| |x− y|α ≤ C for large k. (3.75) We also conclude that ∥(wk)xi ∥C0,α(RN ) ≤ C and ∥m′(ε̂kx+ ε̂kyk)w p k∥C0,α(RN ) ≤ C for large k, (3.76) EJDE-2025/59 MINIMIZERS FOR FRACTIONAL SCHRÖDINGER EQUATIONS 19 in the same way. By the fact that ∥wk∥∞ ≤ C uniformly in k, we then derive from (3.75) and (3.76) that ∥G(x)∥∞ ≤ C. Applying [32, Proposition 2.8] again, we deduce that for some ϱ2 ∈ (0, 1), ∥(wk)xi ∥C1,ϱ2 (RN ) ≤ C(∥(wk)xi ∥∞ + ∥G(x)∥∞) ≤ C for large k, (3.77) which implies that ∥wk∥C2(RN ) ≤ C. Thus, (3.66) holds. Noting that the origin is a local maximum point of wk for all k > 0, and it follows from (3.66) that it is also a global maximum point of Q. Moreover, it is evident that x = 0 is the unique global maximum point of Q, then we conclude that x̃0 = 0. Thus, we have wk → (a∗)− 2s 4s−N(p−1)Q ( (a∗)− p−1 4s−N(p−1)x ) in Hs(RN ) as k → ∞. (3.78) Finally, we shall prove the uniqueness of global maximum point of wk as k is large enough. From the definition of z̄k and (3.67), we have (−∆)swk(z̄k) ≥ C > 0. Using Remark 3.5, (3.47) and (3.67), one can deduce thatwx(z̄k) ≥ C > 0 for large k. It then follows from (3.41) that all local maximum points of wk must stay in a finite ball BR̄(0) in RN as k is large enough, where R̄ is independent of k. Letting R̄ > 0 small enough, such that Q′′(ι) < 0 for 0 < ι < R̄. Then [30, Lemma 4.2] implies that {wk} has no critical points other than the origin, that is, xk is the unique maximum point of uk as k is large enough. Hence, we complete the proof of Theorem 3.1. □ Proof of the Theorem 1.3. Now, based on the Theorem 3.1, let vk be a nonnegative minimizer of I(Mk) and εk := ( Mk a∗ )− p−1 4s−N(p−1) . Note from Theorem 3.1 that εk = (a∗) p−1 4s−N(p−1) ε̂k and vk =M 1/2 k uk, it follows that ε 2s p−1 k vk(εkx+ z̄k) = (a∗) 2s 4s−N(p−1) ε̂ 2s p−1 k M 1/2 k uk(εkx+ z̄k) = (a∗) 2s 4s−N(p−1) ε̂ N/2 k uk(εkx+ z̄k) = (a∗) 2s 4s−N(p−1) (a∗)− 2s 4s−N(p−1)Q ( (a∗)− p−1 4s−N(p−1) εk ε̂k x ) = Q(x) in Hs(RN ) as k → ∞, (3.79) which implies that (1.8) holds. 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Lei Zhang School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha 410083, China Email address: math zl317@163.com Lintao Liu Department of Mathematics, North University of China, Taiyuan 030051, China Email address: liulintao math@nuc.edu.cn Haibo Chen (corresponding author) School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha 410083, China Email address: math chb@csu.edu.cn 1. Introduction 2. Existence of minimizers for I(M) 3. Mass concentration 3.1. Energy estimates of IM 3.2. Blow-up analysis 3.3. Proof of main resutls References