Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 57, pp. 1–23. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu LOCALIZED NODAL SOLUTIONS FOR SEMICLASSICAL NONLINEAR KIRCHHOFF EQUATIONS LIXIA WANG Abstract. In this article, we consider the existence of localized sign-changing solutions for the semiclassical Kirchhoff equation −(ε2a + εb ∫ R3 |∇u|2dx)∆u + V (x)u = |u|p−2u, x ∈ R3, u ∈ H1(R3) where 4 < p < 2∗ = 6, ε > 0 is a small parameter, V (x) is a positive function that has a local minimum point P . When ε → 0, by using a minimax char- acterization of higher dimensional symmetric linking structure via the sym- metric mountain pass theorem, we obtain an infinite sequence of localized sign-changing solutions clustered at the point P . 1. Introduction and main results In this article, we study the semiclassical states of nonlinear Kirchhoff equation − ( ε2a+ εb ∫ R3 |∇u|2dx ) ∆u+ V (x)u = |u|p−2u, x ∈ R3, u ∈ H1(R3), (1.1) where p ∈ (4, 2∗), 2∗ = 6, ε > 0 is a small parameter and V : R3 → R is a continuous function satisfying the following conditions: (A1) V ∈ C1(R3,R) and there exist n0 > m0 > 0 such that m0 ≤ V (x) ≤ n0 for any x ∈ R3. (A2) There is a bounded domain Λ ⊂ R3 with smooth boundary ∂Λ such that ~n(x) · ∇V (x) > 0 ∀x ∈ ∂Λ, (1.2) where ~n(x) denotes the outward normal to ∂Λ at x and · denotes the inner product in R3. Note that if V has an isolated local minimum set, the condition (A2) is satisfied. That is, V has a local trapping potential well. Under (A2), the set of critical points of V is A = {x ∈ Λ|∇V (x) = 0} 6= ∅, (1.3) and A is a compact subset of Λ. In the following, we will assume 0 ∈ A. 2020 Mathematics Subject Classification. 35J20, 35J60. Key words and phrases. Kirchhoff equations; nodal solutions; penalization method. ©2022. This work is licensed under a CC BY 4.0 license. Submitted April 18, 2022. Published August 2, 2022. 1 2 L. WANG EJDE-2022/57 Equation (1.1) or a more general version of − ( a+ b ∫ R3 |∇u|2dx ) ∆u+ V (x)u = f(x, u), x ∈ RN . (1.4) This equation has been studied recently under different conditions on f(x, u) and V (x), where N = 1, 2, 3 and a, b are two positive constants. It is well known that problem (1.4) is a nonlocal problem since the presence of the term b ∫ R3 |∇u|2dx. This fact indicates that (1.4) is not a pointwise identity. It causes some math- ematical difficulties, and in the mean time, makes the study of such a problem particularly interesting. For a pure power f(x, u) := |u|p−2u (3 < p ≤ 6), Li and Ye [13] studied the existence of a positive ground state solution by using a mono- tonicity trick and a new version of global compactness lemma. The authors used the constrained minimization on a new manifold which is related to the Pohozaev’s identity to get a positive ground state solution to (1.4). We note that if V (x) = 0 and RN is replaced by a bounded domain Ω ⊂ RN in (1.4), then we have the Kirchhoff Dirichlet problem − ( a+ b ∫ Ω |∇u|2dx ) ∆u = f(x, u), x ∈ Ω, u = 0, x ∈ ∂Ω, which is arises when studying wave solutions of the equation ρ ∂2u ∂t2 − (P0 h + E 2L ∫ L 0 |∂u ∂x |2dx )∂2u ∂x2 = 0. It is related to the stationary analogue of the Kirchhoff equation utt − ( a+ b ∫ R3 |∇u|2dx ) ∆u = g(x, t), (1.5) which is proposed by Kirchhoff [12] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. In [5], the authors pointed out that Problem (1.5) models several physical and biological systems, where u describes a process which depends on the average of itself (for example, population density). Motivated by the works above, in this paper we study the existence of local- ized sign-changing solutions to the semiclassical nonlinear Kirchhoff equation (1.1). Before giving our main results, we give some notations. Let H1(R3) be the usual Sobolev space endowed with the standard scalar and norm (u, v) = ∫ R3 (∇u∇v + uv) dx; ‖u‖ = (∫ R3 (|∇u|2 + |u|2) dx )1/2 . D1,2(R3) is the completion of C∞0 (R3) with respect to the norm ‖u‖D := ‖u‖D1,2(R3) = (∫ R3 |∇u|2 dx )1/2 . The norm on Ls = Ls(R3) with 1 < s <∞ is given by |u|s = ( ∫ R3 |u|sdx )1/s . Assume the functional space is Hε = { u ∈ H1(R3) : ‖u‖ε = (∫ R3 (|∇u|2 + V (εx)u2) dx )1/2 <∞ } . EJDE-2022/57 LOCALIZED NODAL SOLUTIONS FOR KIRCHHOFF EQUATIONS 3 Since 0 < m0 ≤ V (x) ≤ n0, we have min(1,m0)‖u‖2 ≤ ∫ R3 (|∇u|2 + V (εx)u2) dx ≤ max{1, n0}‖u‖2, Hε(R3) ↪→ Lp(R3) (2 ≤ p ≤ 6), |u|p ≤ Cp‖u‖ ≤ C ′p‖u‖ε. Moreover we make the following assumptions. For any set Ω ⊂ R3, ε > 0 and δ > 0, we set Ωε = {x ∈ R3 : εx ∈ Ω}, Ωδ = {x ∈ R3 : dist(x,Ω) := inf z∈Ω |x− z| < δ}. (1.6) A function u ∈ H1(R3) is called sign-changing if u+ 6= 0 and u− 6= 0, where u± = max{±u, 0}. Our main result reads as follows. Theorem 1.1. Suppose that 4 < p < 6, (A1) and (A2) hold. Then for any positive integer N , there exists εN > 0 such that if 0 < ε < εN , (1.1) has at least N pairs of sign-changing solutions ±vj,ε, j = 1, 2, . . . , N, satisfying that, for any δ > 0, there exist c = c(δ,N) > 0 and C = C(δ,N) > 0 such that |vj,ε(x)| ≤ C exp ( − cdist(x,Aδ) ε ) , 1 ≤ j ≤ N. In recent years, the existence and multiplicity solutions for (SKε) have been studied by many researchers under different assumptions on the potential and non- linearity. Figueiredo [8] constructed a family of positive solutions which concen- trates around the local minima of V as ε → 0, the nonlinearities is subcritical. Motivated by [8], He [11] extended the result of Figueiredo to the case where the nonlinearity is of critical growth, i.e. due to [20]. In [21] the authors consider the stability of ground states to a nonlinear focusing Schrödinger equation in presence of a Kirchhoff term. Especially, if a = 1, b = 0 and R3 replaced by RN , (1.1) is reduced to a singular perturbed Schrödinger equation i.e., − ε2∆u+ V (x)u = |u|p−2u, x ∈ RN , 2 < p < 2∗, N ≥ 1. (1.7) Floer and Weinstein [9] constructed a single peak solution which concentrates around any given non-degenerate critical point of the potential V . Oh [15] showed the existence of muli-peak solutions which concentrate around any finite subsets of the non-degenerate critical points of V . The methods in [9, 15] are mainly used a Lyapunov-Schmidt reduction. The concentration behavior of the positive solutions also has been considered by variational methods. When ε > 0 small enough, by using the Mountain-Pass The- orem, Rabinowitz [16] proved that (1.7) possesses a positive ground state solution under the condition (A3) V∞ = lim inf |x|→∞ V (x) > V0 = infx∈RN V (x) > 0. Other results on the concentration behavior for the family of positive ground solu- tion see [6]. By using the same arguments as in [6, 16], He and Zou [10] considered the existence, concentration and multiplicity of solutions for (1.1) with general nonlinearity f(u), and the potential V (x) satisfy the condition (A4) 0 < V0 := inf V (x) < lim inf |x|→∞ V (x) = V∞, where V∞ ≤ +∞ 4 L. WANG EJDE-2022/57 and f(u) ∈ C1(R+,R+) is a subcritical function satisfying the Ambrosetti-Rabinowtiz condition, which is concentrate on the minima of V (x) as ε→ 0. Now we give an outline of the proof, we set v(x) = u(εx). Then (1.1) is changed to −(a+ b ∫ R3 |∇v|2dx)∆v + V (εx)v = |v|p−2v, x ∈ R3, v ∈ H1(R3), (1.8) and the corresponding energy functional is Iε(v) = 1 2 ∫ R3 (a|∇v|2 + V (εx)v2) dx+ b 4 (∫ R3 |∇v|2dx )2 − 1 p ∫ R3 vpdx. (1.9) It is well known that, by using Rabinowitz [16], we can prove that Iε satisfies the (PS)c condition if c is smaller than the mountain pass value of the limiting functional I(v) = 1 2 ∫ R3 (a|∇v|2 + V0v 2) dx+ b 4 (∫ R3 |∇v|2dx )2 − 1 p ∫ R3 vpdx. where V0 = lim inf |x|→∞ V (x). However, we will construct the solutions in Theorem 1.1 have larger critical values. The variational problem does not satisfy the compact condition anymore. Instead we use some ideas for nonlinear Schrödinger equations (see [3]) in particular in the recent work of [4] in which for nonlinear Schrödinger equations have an infinite sequence of localized nodal solutions were constructed near a local minimum of the potential function V (x). This involves using the Byeon-Wang’s penalization method [3], we define Γε : Hε → R by Γε(v) = Iε(v) +Qε(v), where Qε(v) = (∫ R3 χεv 2dx− 1 )β + , χε(x) = { 0, if x ∈ Λε, ε−6ξ(dist(x,Λε)), if x /∈ Λε. The function Qε will act as a penalization to force the concentration phenomena to occur inside the set of A. The function Γε has an advantage that it has a higher threshold for (PS)c condition to hold. Indeed, for any positive integer L, there exists εL > 0 such that Γε satisfies the (PS)c condition for every c < L if 0 < ε < εL. By using a minimax theorem for sign-changing solutions (see [14]) and the genus (see [17]), we obtain that, for any positive integer N , there exists εN > 0 such that Γε has at least N pairs of sign-changing critical points vj,ε (1 ≤ j ≤ N) if 0 < ε < εN . To verify the critical point vj,ε of Γε is a solution of the original problem (1.8), we need a finer asymptotic analysis and the local Pohozaev identity. Moreover, we show that the concentration points of these solutions lie in A as ε→ 0. Remark 1.2. As it is pointed in [4, 18] considered the critical frequency case, that is, V satisfies (A5) lim inf |x|→∞ V (x) > infx∈RN V (x) = 0; EJDE-2022/57 LOCALIZED NODAL SOLUTIONS FOR KIRCHHOFF EQUATIONS 5 (A6) There exists a closed subset Z with a nonempty interior such that V (x) = 0 for x ∈ Z. By using minimax theorem, they obtained that for any integer N , there exists εN > 0 such that for 0 < ε < εN , (1.7) has at least N solutions. Under the as- sumption of critical frequency, one can use higher dimensional symmetric structures to construct minimax values below the mountain pass value of the limiting func- tional I. However, in our case with positive potentials, the energies of the sequence of localized nodal solutions tend to infinity. To the best of our knowledge, there is no result on the existence and concentra- tion of sign-changing solutions for Kirchhoff type equation under (A1) and (A2). In the present paper, we will adopt the ideas of Chen and Wang [4] to study the existence of sign-changing solutions for (1.1). But their method cannot be used directly because of the nonlocal term and more careful analysis is needed. Throughout this paper, the letters C,C ′ will be used to denote various positive constants which may vary from line to line and are not essential to the problem. E′ is a dual space for a Banach space E. The closure and the boundary of set G are denoted by Ḡ and ∂G respectively. For F ∈ C1(E,R), we denote the Fréchet derivative of F at u by F ′(u), and the Gateaux derivative of F by 〈F ′(u), v〉 for all u, v ∈ E. We denote ⇀ for weak convergence, and → for strong convergence. Also if we take a subsequence of a sequence {un}, we shall denote it again {un}. This article is organized as follows. In Section 2, we introduce the penalized function Γε, show that Γε satisfy (PS)c condition for c < L and ε small enough. In Section 3, when ε is small, we show the existence of multiple sign-changing solutions of the problem through an abstract critical point theorem. In Section 4, we give the proof of Theorem 1.1. We prove the solutions obtained in Section 3 are in fact solutions of the original problem for ε small. 2. Variational setting and compactness condition Set ξ ∈ C∞(R) be a cut-off function such that 0 ≤ ξ(t) ≤ 1 and ξ′(t) ≥ 0 for any t ∈ R. ξ(t) > 0 if t > 0, ξ(t) = 1 if t ≥ 1 and ξ(t) = 0 if t ≤ 0. Define χε(x) = { 0, if x ∈ Λε, ε−6ξ(dist(x,Λε)), if x /∈ Λε. Obviously, for ε small, χε is a C1 function and χε(x) = { 0, if x ∈ Λε, ε−6, if x /∈ (Λε) 1. For u ∈ H1(R3), we define the penalization function Qε(v) = (∫ R3 χεv 2dx− 1 )β + (2.1) which β satisfies 2 < 2β < p and (t)+ = max{t, 0}. For v ∈ H1(R3), we define Γε(v) = Iε(v) +Qε(v), (2.2) 6 L. WANG EJDE-2022/57 where Iε is defined by (1.9). For u, v ∈ H(R3), 〈Γ′ε(v), u〉 = ∫ R3 (a∇v∇u+ V (εx)vu) dx+ b ∫ R3 |∇v|2dx ∫ R3 ∇v∇u dx + 2β (∫ R3 χεv 2dx− 1 )β−1 + ∫ R3 χεvu dx− ∫ R3 |v|p−2vu dx. (2.3) The critical point v of Γε is a solution of − ( a+ b ∫ R3 |∇v|2dx ) ∆v+V (εx)v+ 2β (∫ R3 χεv 2dx− 1 )β−1 + χεv = |v|p−2v, (2.4) for any v ∈ H1(R3). If v is a critical point of Γε with Qε(v) = 0, then v is a solution of (1.8). Lemma 2.1. For any L > 0, there exists εL > 0 such that, for any ε ∈ (0, εL) and c < L, then Γε satisfies (PS)c condition. Proof. Let {un} ⊂ H1(R3) satisfy the conditions Γε(un)→ c, Γ′ε(un)→ 0 in (H1(R3))′. Now we can show that {un} contains a convergent subsequence in H1(R3). Note that o(‖un‖) + L ≥ o(‖un‖) + c = Γε(un)− 1 p 〈Γ′ε(un), un〉 = 1 2 ∫ R3 (a|∇un|2 + V (εx)u2 n) dx+ b 4 (∫ R3 |∇un|2dx )2 − 1 p ∫ R3 |un|pdx + (∫ R3 χεu 2 ndx− 1 )β + − 1 p ∫ R3 ( a|∇un|2 + V (ε)u2 n ) dx− b p (∫ R3 |∇un|2dx )2 + 1 p ∫ R3 |un|pdx − 2β p (∫ R3 χεu 2 n dx− 1 )β−1 + ∫ R3 χεu 2 n dx = (1 2 − 1 p ) ∫ R3 (a|∇un|2 + V (εx)u2 n) dx+ b (1 4 − 1 p )( ∫ R3 |∇un|2dx )2 + (∫ R3 χεu 2 ndx− 1 )β + − 2β p (∫ R3 χεu 2 ndx− 1 )β−1 + ∫ R3 χεu 2 n dx From this inequality and 2 < 2β < p, there exists ηL > 0 independent of ε such that ‖un‖ ≤ ηL and Qε(un) ≤ ηL. Suppose that un ⇀ u in H1(R3) as n→∞ and λn := 2β (∫ R3 χεu 2 ndx− 1 )β−1 + → λ, n→∞. It is easy to prove that u solves − ( a+ b ∫ R3 |∇u|2dx ) ∆u+ V (εx)u+ λχεu = |u|p−2u. (2.5) Hence, for v ∈ H1(R3), a ∫ R3 ∇(un − u)∇v dx+ b ∫ R3 |∇u|2dx ∫ R3 ∇(un − u)∇v dx EJDE-2022/57 LOCALIZED NODAL SOLUTIONS FOR KIRCHHOFF EQUATIONS 7 + b ∫ R3 (|∇un|2 − |∇u|2) dx ∫ R3 ∇un∇v dx+ ∫ R3 V (εx)(un − u)v dx + λ ∫ R3 χε(un − u)v dx+ (λn − λ) ∫ R3 χεunv dx− ∫ R3 (|un|p−2un − |u|p−2u)v dx = 〈Γ′ε(un), v〉 = o(‖v‖), as n→∞. Since Λ is a bounded set, there exists r0 > 0 satisfy Λ ⊂ B(0, r0). Let φε be a C∞ cut-off function such that 0 ≤ φε ≤ 1 and |∇φε| ≤ 4 in R3, φε(x) = 1 if |x| ≥ ε−1r0 + 2 and φε(x) = 0 if |x| ≤ ε−1r0 + 1. We choose v = φ2 ε(un−u) in (2.5) we obtain that( a+ b ∫ R3 |∇u|2dx )∫ R3 |∇(φε(un − u))|2dx+ ∫ R3 V (εx)φ2 ε(un − u)2dx + b ∫ R3 (|∇un|2 − |∇u|2) dx ∫ R3 ∇un ( 2φε∇φε(un − u) + φ2 ε∇(un − u) ) dx + λ ∫ R3 χεφ 2 ε(un − u)2dx+ (λn − λ) ∫ R3 χεφ 2 ε(un − u)2dx − (p− 1) ∫ R3 (θun + (1− θ)u)p−2φ2 ε(un − u)2dx − ( a+ b ∫ R3 |∇u|2dx )∫ R3 (un − u)2|∇φε|2dx = o(1) as n→∞. (2.6) where 0 < θ < 1 comes from the mean value theorem. By λn → λ as n → ∞, |∇φε|2 has compact support, and un ⇀ u in H1(R3) as n ⇀∞, then we have (λn − λ) ∫ R3 χεφ 2 ε(un − u)un dx = o(1), ∫ R3 (un − u)2|∇φε|2 = o(1). as n→∞. Then by V ≥ m0 in R3 and (2.6), we obtain min { a+ b ∫ R3 |∇u|2dx,m0 } ‖φε(un − u)‖2 ≤ ( a+ b ∫ R3 |∇u|2dx )∫ R3 |∇φε(un − u)|2dx+ ∫ R3 V (εx)φ2 ε(un − u)2dx + b ∫ R3 (|∇un|2 − |∇u|2) dx ∫ R3 ∇un{2φε∇φε(un − u) + φ2 ε∇(un − u)}dx ≤ (p− 1) ∫ R3 |θun + (1− θ)u|p−2φ2 ε(un − u)2dx+ o(1) ≤ (p− 1) (∫ R3 |θun + (1− θ)u|pdx )(p−2)/p(∫ R3 φpε(un − u)pdx )2/p + o(1) ≤ C(p− 1) {(∫ |x|≥ε−1r0+1 |un|pdx )(p−2)/p + (∫ |x|≥ε−1r0+1 |u|pdx )(p−2)/p} ‖φε(un − u)‖2 + o(1) as n→∞, (2.7) where C > 0 is a constant independent of n and ε. By Fatou’s Lemma, we have∫ R3 (∇unφ2 ε∇un −∇unφ2 ε∇u) dx = ∫ R3 (|∇un|2φ2 ε − |∇un||∇u|φ2 ε) dx ≥ 0, 8 L. WANG EJDE-2022/57 then ∫ R3 2∇unφε∇φε(un − u)un dx = o(1). By Qε(un) ≤ ηL and (Aε) 1 ⊂ B(0, ε−1r0 + 1), we have∫ |x|≥ε−1r0+1 u2 n dx ≤ (1 + η̂L)1/βε6. It follows that ∫ |x|≥ε−1r0+1 u2dx ≤ (1 + η̂L)1/βε6. Assume p < q < 6. By the inequality |u|p ≤ |u|t2|u|1−tq ≤ C ′|u|t2‖u‖1−t, where the positive C ′ is independent of n and ε, and 1 p = t 2 + 1−t q , by above two inequalities and ‖un‖ ≤ η̂L, we infer that there is a constant CL > 0 independent of ε and n such that ∫ |x|≥ε−1r0+1 upn dx ≤ CLε3pt, ∫ |x|≥ε−1r0+1 updx ≤ CLε3pt. Let εL > 0 satisfying that, for 0 < ε < εL, C(p− 1)(2C p−2 p L ε3(p−2)t) < 1 2 min{a+ b ∫ R3 |∇u|2dx,m0}. Then by (2.6) we obtain lim n→∞ ‖φε(un − u)‖ = 0. (2.8) Choosing v = (1 − φε)2(un − u) in (2.5), we obtain that (2.6) still holds if we replace φε with 1 − φε. Indeed, 1 − φε has a compact support and un → u in Lqloc(R3) for any 2 ≤ q < 2∗,( a+ b ∫ R3 |∇u|2dx )∫ R3 |∇((1− φε)2(un − u))|2dx − ( a+ b ∫ R3 |∇u|2dx )∫ R3 |∇(1− φε)|2(un − u)2dx + b ∫ R3 (|∇un|2 − |∇u|2) dx + ∫ R3 {∇un∇(1− φε)2(un − u) +∇un(1− φε)2∇(un − u)}dx + ∫ R3 V (εx)(1− φε)2(un − u)2dx+ λ ∫ R3 χε(1− φε)2(un − u)2dx + (λn − λ) ∫ R3 χε(1− φε)2(un − u)2dx − (p− 1) ∫ R3 (θun + (1− θ)u)p−2(1− φε)2(un − u)2dx = o(1), as n→∞. This implies lim n→∞ ‖(1− φε)(un − u)‖ = 0 (2.9) and lim n→∞ ‖un − u‖ = lim n→∞ ‖(1− φε)(un − u) + φε(un − u)‖ ≤ lim n→∞ ‖(1− φε)(un − u)‖+ lim n→∞ ‖φε(un − u)‖ = 0. EJDE-2022/57 LOCALIZED NODAL SOLUTIONS FOR KIRCHHOFF EQUATIONS 9 The proof is complete. � 3. Existence of multiple sign-changing critical points of Γε We will use an abstract critical point theorem in [14] to obtain multiple sign- changing critical points for Γε. First we give some definitions and notation. Let X be a Banach space. For P ⊂ X, define −P = {−u : u ∈ P}. The genus (see [17]) of a closed symmetric subset B(i.e. − B = B) of X is denoted by γ(B). For J ∈ C1(X,R) and c ∈ R, denote Jc = {u ∈ X : J(u) ≤ c}, Kc = {u ∈ X : J(u) = c, J ′(u) = 0}. Definition 3.1 ([14]). Let J ∈ C1(X,R) be an even functional. Let P ⊂ X be a non-empty open set and W = P ∪ (−P ). P is called an admissible invariant set with respect to J at level c, if the following deformation property holds, there is τ0 > 0 and a symmetric open neighborhood M of Kc \W with γ(M) < ∞, such that for τ ∈ (0, τ0), there exists η ∈ C(X,X) satisfying (1) η(∂P ) ⊂ P, η(∂(−P )) ⊂ −P, η(P ) ⊂ P, η(−P ) ⊂ −P ; (2) η(−u) = −η(u), for all u ∈ X; (3) η|Jc−2τ = id. (4) η(Jc+τ \ (M ∪W )) ⊂ Jc−τ . Proposition 3.2. Assume J ∈ C1(X,R) is an even functional, P ⊂ X is a non- empty open set, M = P ∩ (−P ),W = P ∪ (−P ) and Σ = ∂P ∩ ∂(−P ). Let P be an admissible invariant set with respect to J for c ∈ [c∗, L] for some L > c∗, where c∗ = infu∈Σ J(u) and for any n ∈ N, there is a continuous map φn : Bn := {x ∈ Rn : |x| ≤ 1} → X satisfying (1) φn(0) ∈M,φn(−t) = −φn(t) for all t ∈ Bn; (2) φn(∂Bn) ∩M = ∅; (3) max{J(0), supu∈φn(∂Bn) J(u)} < c∗. For j ∈ N, we define cj = inf B∈Λj sup u∈B\W J(u), where Λj = { B : B = φ(Bn \ Y ) for some φ ∈ Gn, n ≥ j, and open Y ⊂ Bn such that −Y = Y and γ(Y ) ≤ n− j } and Gn = {φ : φ ∈ C(Bn, X), φ(−t) = −φ(t) for any t ∈ Bn, φ|∂Bn = φn|∂Bn}. Then for j ≥ 2, if L > cj, we have Kcj \W 6= ∅. (3.1) Furthermore, if j ≥ 2 and L > c := cj = · · · = cj+m ≥ c∗, we have γ(Kc \W ) ≥ m+ 1. (3.2) The above proposition was proved in [14, Theorem 2.5]. If we choose k = 1 and G = −id in [14], we can obtain (3.1). The result (3.2) is proved in [4] by a variant of the argument in the proof of [14, Theorem 2.5]. So we omit it here. 10 L. WANG EJDE-2022/57 Let P± := {u ∈ H1(R3) : u ≥ (≤)0}. For σ > 0, let Pσ+ := {u ∈ H1(R3) : distH1(u, P+) < σ}, Pσ− := {u ∈ H1(R3) : distH1(u, P−) < σ}, where distH1(u,B) := infv∈B ‖u − v‖ for u ∈ H1(R3) and B ⊂ H1(R3). It is easy to see that Pσ− = −Pσ+. To apply Proposition 3.2 to obtain multiple sign-changing critical points of Γε, we let X = H1(R3), P = Pσ+, W = Pσ− ∪ Pσ+, J = Γε (3.3) in Definition 3.1 and Proposition 3.2. It is easy to know that W is a symmetric and open subset of H1(R3) and sign-changing functions are contained in H1(R3) \W . Furthermore, since 0 is a strict local minimum point of Γε, the constant c∗ in Proposition 3.2 satisfies c∗ = inf ∂(Pσ−)∩∂(Pσ+) Γε > 0, when σ > 0 is small enough. Without loss of generality, we assume that 0 ∈ A. (3.4) For z ∈ R3 and r > 0, we define B(z, r) = {x ∈ R3 : |x − z| < r}. From (3.3), we obtain B(0, 1) ⊂ Λε (3.5) if ε > 0 small enough. Now we define a function J0(u) = 1 2 ∫ B(0,1) (a|∇u|2 + n0u 2) dx− 1 p ∫ B(0,1) |u|pdx+ b 4 (∫ B(0,1) |∇u|2dx )2 , u ∈ H1 0 (B(0, 1)). Assume En := span{e1, . . . , en}, where {en} ⊂ H1 0 (B(0, 1)) is an orthonormal basis. From p > 2, we can infer that there is an increasing sequence of positive numbers {Rn} satisfying J0(u) < 0, for all u ∈ En and ‖u‖ ≥ Rn. We also we define φn ∈ C(Bn, H 1 0 (B(0, 1))) as φn(t) = Rn n∑ i=1 tiei, t = (t1, . . . , tn) ∈ Bn. (3.6) One can easily prove that under (3.4), φn satisfied (1)–(3) in Proposition 3.2. For j ∈ N, we define four sets Λj = { B : B = φ(Bn \ Y ) for some φ ∈ Gn, n ≥ j, and open Y ⊂ Bn such that − Y = Y and γ(Y ) ≤ n− j } , Λ̃j = { B : B = φ(Bn \ Y ) for some φ ∈ G̃n, n ≥ j, and open Y ⊂ Bn such that − Y = Y and γ(Y ) ≤ n− j } , Gn = {φ : φ ∈ C(Bn, H 1(R3)), φ(−t) = −φ(t) for all t ∈ Bn, φ|∂Bn = φn|∂Bn}, G̃n = {φ : φ ∈ C(Bn, H 1 0 (B(0, 1))), φ(−t) = −φ(t) ∀t ∈ Bn, φ|∂Bn = φn|∂Bn}. EJDE-2022/57 LOCALIZED NODAL SOLUTIONS FOR KIRCHHOFF EQUATIONS 11 Then we can give the the minimax values cεj = inf B∈Λj sup u∈B\W Γε(u), c̃j = inf B∈Λ̃j sup u∈B\W J0(u) We obtain 0 < cε2 ≤ cε3 ≤ . . . , c̃2 ≤ c̃3 ≤ . . . . (3.7) Because χε = 0 in Λε, from V ≤ n0 and (3.5), for all u ∈ H1 0 (B(0, 1)), we can obtain that Γε(u) ≤ J0(u). And then by Λ̃j ⊂ Λj , for any j ≥ 2 and sufficiently small ε > 0, we have 0 < cεj ≤ c̃j . (3.8) Proposition 3.3. Assume that σ0 > 0 and L > 0. Then for any σ ∈ (0, σ0) and ε ∈ (0, εL), Pσ+ is an admissible invariant set with respect to Γε for c < L, where εL is from Lemma 2.1. We prove the above proposition in the appendix. Proposition 3.4. For any N ∈ N, there exists ε′N > 0 such that, for any ε ∈ (0, ε′N ), Γε has at least N pairs of sign-changing critical points {±vj,ε : 1 ≤ j ≤ N} satisfying Γε(vj,ε) = cεj+1 ≤ c̃N+1, 1 ≤ j ≤ N. The above proposition follows from 3.2, Proposition 3.3, using (3.3), (3.7), and (3.8). 4. Proof of Theorem 1.1 In this part, we first verify that the sign-changing critical points {vj,ε} obtained in Proposition 3.4 are solutions of (1.8), then we can prove the main theorem. Lemma 4.1. For any N ∈ N and 0 < ε < ε′N , there exist ρ = ρ(a,m0, p) > 0 and ηN > 0 such that ρ ≤ ‖vj,ε‖ ≤ ηN , Qε(vj,ε) ≤ ηN , 1 ≤ j ≤ N, where ηN is independent of ε. Proof. Since c̃N+1 ≥ cεj+1 = Γε(vj,ε)− 1 p 〈Γ′ε(vj,ε), vj,ε〉 = Iε(vj,ε) +Qε(vj,ε)− 1 p 〈Γ′ε(vj,ε), vj,ε〉 = (1 2 − 1 p ) ∫ R3 (a|∇vj,ε|2 + V (εx)v2 j,ε) dx+ b (1 4 − 1 p )( ∫ R3 |∇vj,ε|2dx )2 + (∫ R3 χεv 2 j,εdx− 1 )β + − 2β p (∫ R3 χεv 2 j,εdx− 1 )β−1 + ∫ R3 χεv 2 j,εdx and 2 < 2β < p, we can have that there exists ηN > 0 independent of ε such that ‖vj,ε‖ ≤ ηN and Qε(vj,ε) ≤ ηN . From 〈Γ′ε(vj,ε), vj,ε〉 = 0, we obtain min{a,m0}‖vj,ε‖2 ≤ ∫ R3 (a|∇vj,ε|2 + V (εx)v2 j,ε) dx+ b (∫ R3 |∇vj,ε|2dx )2 + 2β (∫ R3 χεv 2 j,εdx− 1 )β−1 + ∫ R3 χεv 2 j,εdx 12 L. WANG EJDE-2022/57 = ∫ R3 |vj,ε|pdx ≤ C‖vj,ε‖p, where C = C(p) is the constant in Sobolev inequality. Since vj,ε are sign-changing functions, vj,ε 6= 0 and p > 2, it follows that there exists ρ = ρ(a,m0, p) > 0 such that ‖vj,ε‖ ≥ ρ for 1 ≤ j ≤ N . The proof is complete. � Lemma 4.2. If δ > 0, then limε→0 ‖vj,ε‖L∞(R3\(Λε)δ) = 0 for 1 ≤ j ≤ N . Proof. From Qε(vj,ε) ≤ ηN and the definition of cut-off function χε, we have that, for any δ > 0, there exists a positive constant C = C(δ,N) such that∫ R3\(Λε)δ v2 j,εdx ≤ Cε6, 1 ≤ j ≤ N. (4.1) Because vj,ε solves (2.4), ‖vj,ε‖ ≤ ηN , Then by (4.1) and using the bootstrap argument, we have ‖vj,ε‖L∞(R3\(Λε)δ) ≤ Cε 3, 1 ≤ j ≤ N. The proof is complete. � Lemma 4.3. Assume ς > 0, {yε} ⊂ R3, and {vε} ⊂ H1(R3) ∩ L∞(R3) satisfy sup ε>0 ‖vε‖ < +∞, (4.2)∫ B(yε,1) v2 εdx ≥ ς, (4.3) sup{〈Γ′ε(vε), u〉 : u ∈ H1 0 (Λε), ‖u‖H1 0 (Λε) ≤ 1} → 0 as ε→ 0, (4.4) and for δ > 0, lim ε→0 ‖vε‖L∞(R3\(Λε)δ) = 0. (4.5) Then yε ∈ Λε and limε→0 dist(yε, ∂Λε) = +∞. Proof. It follows from (4.3) and (4.5) that yε ∈ (Λε) 1. Assume wε = vε(· + yε). Then by (4.3), we obtain ∫ B(0,1) w2 εdx ≥ ς. (4.6) If not, we suppose lim ε→0 dist(yε, ∂Λε) = l < +∞. (4.7) By changing variables, without loss of generality, we may assume that yε = 0 (4.8) and there exists zε = (aε, 0, . . . , 0) ∈ ∂Λε such that |αε| = dist(yε, ∂Λε)→ l as ε→ 0. (4.9) Up to a subsequence, we assume limε→0 αε = α. By yε ∈ (Λε) 1 and (4.7)–(4.9), we can infer that −1 ≤ α < +∞. Because ‖wε‖ = ‖vε‖ and (4.2), we can set that wε ⇀ w in H1(R3) as ε → 0. From (4.6), we can obtain w 6= 0. And from (4.5) and (4.9), if x1 ≥ α, we obtain w(x) = 0, (4.10) EJDE-2022/57 LOCALIZED NODAL SOLUTIONS FOR KIRCHHOFF EQUATIONS 13 where x = (x1, x2, x3). By χε = 0 in Λε and (4.8), we obtain a ∫ R3 ∇wε∇u dx+ ∫ R3 V (εx)wεu dx+ b ∫ R3 |∇wε|2dx ∫ R3 ∇wε∇u dx = ∫ R3 |wε|p−2wεu dx+ 〈Γ′ε(wε), u〉, ∀u ∈ H1 0 (Λε). (4.11) By (4.4), (4.10), and (4.11), we infer that w is a weak solution of − ( a+ b ∫ R3 |∇wε|2dx ) ∆w + V (0)w = |w|p−2w in {x ∈ R3 : x1 < α}, w|x1=α = 0. By [7, Theorem I.1], the only solution of this equation in H1(R3) is w = 0. This contradicts with w 6= 0. The proof is complete. � Lemma 4.4. Let vj,ε ⇀ ṽ0 in H1(R3) as ε→ 0. If lim infε→0 ‖vj,ε− ṽ0‖Lp(R3) > 0, then there exists mj ∈ N,mj nonzero functions ṽi in H1(R3), 1 ≤ i ≤ mj and mj sequences {yij,ε} ⊂ Λε, 1 ≤ i ≤ mj satisfy (i) limε→0 |yij,ε| = +∞, limε→0 dist(yij,ε, ∂Λε) = +∞, 1 ≤ i ≤ mj, and lim ε→0 |yij,ε − yi ′ j,ε| = +∞, if i 6= i′; (ii) ṽ0 is a solution of − (a+ bAj)∆v + V (0)v = |v|p−2v, v ∈ H1(R3), (4.12) where Aj := lim ε→0 ∫ R3 |∇vj,ε|2dx, ∫ R3 |∇ṽ0|2dx ≤ Aj . For every 1 ≤ i ≤ mj, ṽi is a nontrivial solution of − (a+ bAj)∆v + V (yij)v = |v|p−2v, v ∈ H1(R3), (4.13) where yij = limε→0 εy i j,ε ∈ Λ̄; (iii) For any 2 < q < 6, lim ε→0 ‖vj,ε − ṽ0 − mj∑ i=1 ṽi(· − yij,ε)‖Lq(R3) = 0. (4.14) Proof. Since ‖vj,ε‖ and Q(vj,ε) are bounded and vj,ε solves (2.4), we can prove that ṽ0 is a solution of (4.12). Indeed, since vj,ε ⇀ ṽ0 in H1(R3) as ε → 0, we assume that for some constant Aj ∈ R, lim ε→0 ∫ R3 |∇vj,ε|2dx = Aj . For any φ ∈ C∞0 (R3), we have that 〈Γ′ε(vj,ε), φ〉 → 0, i.e.,( a+ b ∫ R3 |∇vj,ε|2dx )∫ R3 ∇vj,ε∇φdx+ ∫ R3 V (εx)vj,εφdx− ∫ R3 |vj,ε|p−2vj,εφdx + 2β (∫ R3 χεv 2 j,ε dx− 1 )β−1 + ∫ R3 χεvj,εφdx = o(1) which implies that as ε→ 0, (a+ bAj) ∫ R3 ∇ṽ0∇φdx+ V (0) ∫ R3 ṽ0φdx− ∫ R3 |v0|p−2v0φdx = 0. 14 L. WANG EJDE-2022/57 Since C∞0 (R3) is dense in H1(R3), we have that ṽ0 solves −(a+ bAj)∆v + V (0)v = |v|p−2v, v ∈ H1(R3). Let v1 j,ε = vj,ε − ṽ0 and {y1 j,ε} ⊂ R3 be such that∫ B(y1j,ε,1) (v1 j,ε) 2dx = sup y∈R3 ∫ B(y,1) (v1 j,ε) 2dx := ς1ε . Since v1 j,ε ⇀ 0 as ε → 0, we have |y1 j,ε| → ∞ as ε → 0 if lim infε→0 ςε > 0. Since vj,ε solves (2.4) and ṽ0 solves (4.12), we have − a∆v1 j,ε − b ∫ R3 |∇vj,ε|2dx∆v1 j,ε − b (∫ R3 |∇vj,ε|2dx−Aj ) ∆ṽ0 + V (εx)v1 j,ε + (V (εx)− V (0))ṽ0 + ξεχεv 1 ε + ξεχεṽ0 = |vj,ε|p−2vj,ε − |ṽ0|p−2ṽ0, (4.15) where ξε = 2β (∫ R3 χεv 2 j,εdx− 1 )β−1 + . (4.16) From (2.3) and (4.15), for u ∈ H1(Λε), we have 〈Γ′ε(v1 j,ε), u〉 = ∫ R3 (|vj,ε|p−2vj,ε − |ṽ0|p−2ṽ0 − |v1 j,ε|p−2v1 j,ε)u dx − ∫ R3 (V (εx)− V (0))ṽ0u dx + b (∫ R3 (|∇v1 j,ε|2 − |∇vj,ε|2) )∫ R3 ∇v1 j,ε∇u dx − b (∫ R3 |∇vj,ε|2 −Aj )∫ R3 ∇ṽ0∇u dx. (4.17) By [19, Lemma 8.1] and sup {∫ R3 (V (εx)− V (0))ṽ0u dx : u ∈ H1(R3), ‖u‖ ≤ 1 } → 0 (4.18) as ε→ 0. and (4.17), we obtain sup { 〈Γ′ε(v1 j,ε), u〉 : u ∈ H1 0 (Λε), ‖u‖H1 0 (Λε) ≤ 1 } → 0 (4.19) as ε→ 0. Since ṽ0 ∈ H1 0 (R3) and ṽ0 solves (4.12), we have that lim|x|→∞ ṽ0(x) = 0. By Lemma 4.2, for any δ > 0, we have lim ε→0 ‖v1 j,ε‖L∞(R3\(Λε)δ) = 0. (4.20) By Lions Lemma [19] and lim infε→0 ‖vj,ε−ṽ0‖Lp(R3) > 0, we have lim infε→0 ς 1 ε > 0. Then by Lemma 4.3, (4.19) and (4.20), we obtain that y1 j,ε ∈ Λε, lim ε→0 dist(y1 j,ε, ∂Λε) = +∞. (4.21) Let w1 j,ε = v1 j,ε(·+ y1 j,ε). Then lim inf ε→0 ∫ B(0,1) (w1 j,ε) 2dx = lim inf ε→0 ς1ε > 0. (4.22) EJDE-2022/57 LOCALIZED NODAL SOLUTIONS FOR KIRCHHOFF EQUATIONS 15 Let w1 j,ε ⇀ ṽ1 as ε→ 0. By (2.3) and (4.19), we deduce that for any u ∈ H1 0 (y1 j,ε + Λε), as ε→ 0,( a+ b ∫ R3 |∇w1 j,ε|2dx )∫ R3 ∇w1 j,ε∇u dx+ ∫ R3 V (ε(x+ y1 j,ε))w 1 j,εu dx − ∫ R3 |w1 j,ε|p−2w1 j,εu dx→ 0, (4.23) where y1 j,ε + Λε = {x + y1 j,ε : x ∈ Λε}. By (4.21)and(4.23), we know that ṽ1 is a solution of (4.13) with i = 1. From Lemma 4.1, we obtain that there exists a positive constant ρ depending only on a,m0 and p such that ‖ṽ1‖ ≥ ρ. (4.24) Let v2 j,ε = v1 j,ε − ṽ1(· − y1 j,ε). Since ṽ1 ∈ H1(R3) is a solution of (4.13), we can deduce that lim|x|→∞ ṽ1(x) = 0. Then by (4.20) and (4.21), we obtain that, for all δ > 0, lim ε→0 ‖v2 j,ε‖L∞(R3\(Λε)δ) = 0. (4.25) Since vj,ε, ṽ0 and ṽ1 solves (2.4), (4.12), and (4.13) with i = 1 respectively, we have − a∆v2 j,ε − b ∫ R3 |∇vj,ε|2dx∆v2 j,ε − b (∫ R3 |∇vj,ε|2 −Aj ) ∆ṽ0 − b (∫ R3 |∇vj,ε|2 −Aj ) ∆ṽ1 + ξεχεv 2 j,ε + ξεχεṽ0 + ξεχεṽ1 + V (εx)v2 j,ε + (V (εx)− V (0))ṽ0 + (V (εx)− V (yij))ṽ1(· − y1 j,ε) = |vj,ε|p−2vj,ε − |ṽ0|p−2ṽ0 − |ṽ1|p−2ṽ1(· − y1 j,ε). (4.26) By (2.3), (4.26) and [19, Lemma 8.1 ], for any u ∈ H1 0 (Λε) with ‖u‖H1 0 (Λε) ≤ 1, we have 〈Γ′ε(v2 j,ε), u〉 = ∫ R3 ( |vj,ε|p−2vj,ε − |ṽ0|p−2ṽ0 − |ṽ1(· − y1 j,ε)|p−2ṽ1(· − y1 j,ε)− |v2 j,ε|p−2v2 j,ε ) u dx − ∫ R3 (V (εx)− V (0)) ṽ0u dx− ∫ R3 (V (εx)− V (y1 j ))ṽ1(· − y1 j,ε)u dx + b (∫ R3 (|∇v2 j,ε|2 − |∇vj,ε|2 )∫ R3 ∇v2 j,ε∇u dx − b (∫ R3 |∇vj,ε|2 −Aj )∫ R3 ∇ṽ0∇u dx − b (∫ R3 |∇vj,ε|2 −Aj )∫ R3 ∇ṽ1(· − y1 j,ε)∇u dx+ o(1) (4.27) as ε→ 0. Since limε→0 εy 1 j,ε = y1 j , we obtain sup {∫ R3 (V (ε(x+ y1 j,ε))− V (y1 j ))ṽ1u dx : u ∈ H1(R3), ‖u‖ ≤ 1 } → 0 as ε→ 0. It follows that sup {∫ R3 (V (εx))− V (y1 j ))ṽ1(· − y1 j,ε)u dx : u ∈ H1(R3), ‖u‖ ≤ 1 } → 0 (4.28) 16 L. WANG EJDE-2022/57 as ε→ 0. By (4.18), (4.27), (4.28), and ∫ R3 |∇vj,ε|2dx→ Aj , we have sup { 〈Γ′ε(v2 j,ε), u〉 : u ∈ H1 0 (Λε), ‖u‖H1 0 (Λε) ≤ 1 } → 0 as ε→ 0. (4.29) Let {y2 j,ε} ⊂ R3 be such that∫ B(y2j,ε,1) (v2 j,ε) 2dx = sup y∈R3 ∫ B(y,1) (v2 j,ε) 2dx := ς2ε . By (4.25),(4.29) and Lemma 4.3, we have that y2 j,ε ∈ Λε,dist(y2 j,ε, ∂Λε) → +∞ as ε→ 0, limε→0 |y2 j,ε| = +∞, limε→0 |y2 j,ε − y1 j,ε| = +∞ if lim infε→0 ς 2 ε > 0. Iterating the above argument we can know that the iteration procedure has to stop in finite number of steps, since ‖vj,ε‖ ≤ ηN , ‖ṽi‖ ≥ ρ for all 1 ≤ i ≤ mj , and ‖vij,ε‖2 = ‖vi−1 j,ε ‖ 2 − ‖ṽi−1‖2 + o(1) = ‖vj,ε‖2 − i−1∑ n=1 ‖ṽn‖2 + o(1), as ε→ 0. (4.30) Hence, we obtain mj ∈ N such that v mj+1 j,ε = v mj j,ε − ṽmj (· − y mj j,ε ) satisfies sup y∈R3 ∫ B(y,1) (v mj+1 j,ε )2dx = 0 as ε→ 0. (4.31) It follows from the Lions lemma and (4.31) that for any 2 < q < 2∗ = 6,∫ R3 |vmj+1 j,ε |qdx = 0 as ε→ 0. Hence, we obtain mj nonzero functions ṽi in H1(R3), 1 ≤ i ≤ mj and mj sequences {yij,ε} ⊂ Λε, 1 ≤ i ≤ mj such that the results (i), (ii), and (iii) hold. The proof is complete. � Next, for each ε > 0 and 1 ≤ j ≤ N , we assume that y0 j,ε = 0. Let εn > 0 be such that lim n→∞ εn = 0. Up to a subsequence, we assume that limn→∞ εny i j,εn exists for every i. We may write the set of these limiting points by {x∗1, . . . , x∗sj} = { lim n→∞ εny i j,εn : 0 ≤ i ≤ mj} ⊂ Λ̄, (4.32) for some 1 ≤ sj ≤ mj . Set θ∗ = { 1 100 min{|x∗s − x∗s′ | : 1 ≤ s < s′ ≤ sj}, if sj ≥ 2 +∞, if sj = 1. (4.33) Lemma 4.5. If 0 < δ < θ∗, then there exist two positive constants C and c independent of n such that, for every 0 ≤ i ≤ mj, when n is large enough, |∇vj,εn(x)|+ |vj,εn(x)| ≤ C exp(−cε−1 n ), for x ∈ ∂B(yij,εn , δε −1 n ). Proof. Define Ain = B(yij,εn , 3 2δε −1 n ) \ B(yij,εn , 1 2δε −1 n ). From 0 < δ < θ∗, we can deduce that for every 0 ≤ i, i′ ≤ mj , dist(yi ′ j,εn , A i n)→∞ as n→∞. (4.34) EJDE-2022/57 LOCALIZED NODAL SOLUTIONS FOR KIRCHHOFF EQUATIONS 17 From Lemma 4.4, (4.34), and lim R→∞ ∫ R3\B(yij,εn ,R) |ṽi(· − yij,εn)|pdx = 0, 0 ≤ i ≤ mj , (4.35) we obtain that lim R→∞ ∫ Ain |vj,εn |pdx = 0 for 0 ≤ i ≤ mj . (4.36) Then there exists n1 ∈ N such that for n ≥ n1, ‖vj,εn‖ p−2 L∞(Ain) < a/2. (4.37) For m ∈ N, let Rm = B(yij,εn , 3 2 δε−1 n −m) \B(yij,εn , 1 2 δε−1 n +m). Let ζm be a cut-off function satisfying that 0 ≤ ξm(t) ≤ 1 for all t ∈ R, ζm(t) = { 0, if t ≤ 1 2δε −1 n +m− 1 or t ≥ 3 2δε −1 n −m+ 1, 1, if 1 2δε −1 n +m ≤ t ≤ 3 2δε −1 n −m, and |ζ ′m(t)| ≤ 4 for all t. For x ∈ R3, let ψm(x) = ζm(|x− yij,εn |). Multiplying both sides of (2.4) by ψ2 mvj,εn and integrating on R3, by (4.37) we have that( a+ b ∫ R3 |∇vj,εn |2dx )∫ Rm−1 |∇vj,εn |2ψ2 m dx+ ∫ Rm−1 V (εx)v2 j,εnψ 2 m dx + ξn ∫ Rm−1 χεnv 2 j,εnψ 2 m dx− ∫ Rm−1 |vj,εn |pψ2 m dx ≥ min{a+ b Aj 2 , m0 2 } ∫ Rm (|∇vj,εn |2 + v2 j,εn) dx, (4.38) and( a+ b ∫ R3 |∇vj,εn |2dx )∫ Rm−1 |∇vj,εn |2ψ2 m dx+ ∫ Rm−1 V (εx)v2 j,εnψ 2 m dx + ξn ∫ Rm−1 χεnv 2 j,εnψ 2 m dx− ∫ Rm−1 |vj,εn |pψ2 m dx ≤ 8(a+ bAj) ∫ Rm−1\Rm (|∇vj,εn |2 + v2 j,εn) dx, (4.39) where ξn := 2β (∫ R3 χεnv 2 j,εndx− 1 )β−1 + , here we have used the fact, limn→∞ ∫ R3 |∇vj,εn |2dx = Aj , then there exists n2 ∈ N, such that ∫ R3 |∇vj,εn |2dx > Aj 2 when n > n2. By above inequalities, letting C = 8(a+ bAj) min{a+ bAj/2,m0/2} , we have∫ Rm (|∇vj,εn |2 + v2 j,εn) dx ≤ C ∫ Rm\Rm−1 (|∇vj,εn |2 + v2 j,εn) dx. (4.40) Let am = ∫ Rm (|∇vj,εn |2 +v2 j,εn ) dx, we obtain that am ≤ C(am−1−am) which gives am ≤ θam−1 with θ = C/(1 + C) < 1. Therefore am ≤ a0θ m. By Lemma 4.1, we 18 L. WANG EJDE-2022/57 obtain a0 ≤ η2 N . Hence, for sufficiently large n, am ≤ η2 Ne mlnθ. Denote [x] be the integer part of x. Choosing m = [δε−1/2]−1 and noting that [δε−1/2]−1 ≤ δε−1/4 when n is large enough, we obtain∫ Din (|∇vj,εn |2 + v2 j,εn) dx ≤ am ≤ η2 N exp(([δε−1/2]− 1)lnθ) ≤ η2 Nexp( 1 4 δε−1 n lnθ), (4.41) where Di n = B(yij,εn , δε −1 n + 1) \B(yij,εn , δε −1 n − 1). By the standard regularity of elliptic equations, we can obtain the result of this lemma. The proof is complete. � Lemma 4.6. For each 0 ≤ i ≤ mj, limε→0 dist(εyij,ε, A) = 0. Proof. If not, we assume that there exist 1 ≤ i0 ≤ mj and εn > 0 such that limn→∞ εn = 0 and lim n→∞ dist(εny i0 j,εn , A) > 0. Without loss of generality, we assume that for every i, limε→∞ εny i j,εn exists. By the condition of (A2), we deduce that there exists δ′ > 0 such that, for every y ∈ Λδ ′ , inf x∈B(y,δ′)\Λ ∇V (y) · ∇ dist(x, ∂Λ) > 0. (4.42) Since yi0j = limn→∞ εny i0 j,εn /∈ A, we infer that there exists δ′′ > 0 such that, for sufficiently large n, inf x∈B(y i0 j,εn ,δ′′ε−1 n ) ∇V (εnx) · ∇V (εny i0 j,εn ) ≥ 1 2 |∇V (yi0j )|2 > 0. (4.43) Let 0 < δ0 < min{δ′, δ′′, ϑ∗}. To abbreviate notation, let wn = vj,εn and B̃ = B(yi0j,εn , δ0ε −1 n ). Because 0 < δ0 < ϑ∗ and Lemma 4.5, there exist constants c, C > 0 independent of n such that |∇wn(x)|+ |wn(x)| ≤ Cexp(−cε−1 n ), x ∈ ∂B̃, (4.44) for sufficiently large n. From Lemma 4.1, we infer that there exists C > 0 indepen- dent of n such that 0 ≤ ξn ≤ C ∀n. (4.45) We denote ~tn = ∇V (εny i0 j,εn ). Since wn solves (2.4) and the coefficients of (2.4) are all C1 functions, we infer that wn is a C2 function. Multiplying both sides of (2.4) by ~tn · ∇wn and integrating in B̃, we obtain the local Pohozaev type identity 1 2 ∫ B̃ ( εn~tn · (∇V )(εnx) + ξn∇χεn~tn ) w2 n dx = ( a+ b ∫ R3 |∇wn|2dx )∫ ∂B̃ 1 2 |∇wn|2~tn · ν − ( a+ b ∫ R3 |∇wn|2dx )∫ ∂B̃ (∇wn · ν)(∇wn · ~tn) ds − 1 p ∫ ∂B̃ |wn|p(~tn · ν) ds, (4.46) where ν denotes the unit outward normal to the boundary of B̃. EJDE-2022/57 LOCALIZED NODAL SOLUTIONS FOR KIRCHHOFF EQUATIONS 19 From (4.43) and w(·+ yi0 j,ε−1 n ) ⇀ ṽi0 6= 0 in H1(R3), we obtain that εn ∫ B̃ (~tn · (∇V )(εnx))w2 n dx ≥ εn 2 |∇V (yi0j )|2 ∫ B(0,,δ0ε −1 n ) w2 n(·+ yi0j,εn) dx ≥ Cεn, (4.47) where C = 1 4 |∇V (yi0j )|2 ∫ R3 ṽ2 i0 > 0. By (4.42), we obtain that, for any x ∈ B̃ \ Λεn , ~tn · ∇χεn(x) ≥ 0. (4.48) Furthermore, by (4.44) and (4.45), there exist two positive constants C, c indepen- dent of n such that, for sufficiently large n,( a+ b ∫ R3 |∇wn|2dx )∫ ∂B̃ 1 2 |∇wn|2~tn · ν − ( a+ b ∫ R3 |∇wn|2dx )∫ ∂B̃ (∇wn · ν)(∇wn · ~tn) ds − 1 p ∫ ∂B̃ |wn|p(~tn · ν) ds ≤ C exp(−cε−1 n ). (4.49) This contradicts (4.46). The proof is complete. � Lemma 4.7. For any δ > 0, there exist two positive constants C = C(δ,N) and c = c(δ,N)0 independent of ε such that for every 1 ≤ j ≤ N , |vj,ε(x)| ≤ Cexp(−cdist(x, (Aε) δ)), x ∈ R3. Proof. By (4.14), (4.35), and Lemma 4.6, we infer that there is R0 > 0 independent of ε such that, for sufficiently small ε > 0, |vj,ε(x)|p−2 < m0/2if dist(x, (Aε) δ) ≥ R0. (4.50) To prove the result, we only need to show that |vj,ε(x)| ≤ C exp(−cdist(x, (Aε) δ)), if dist(x, (Aε) δ) ≥ R0. (4.51) For m ∈ N, let Bm = {x ∈ R3 : dist(x, (Aε)δ ≥ R0 −m + 1}. Let ρm be a cut-off function satisfying 0 ≤ ρm(t) ≤ 1, |ρ′m(t)| ≤ 4 for all t ∈ R and ρm(t) = { 0, if t ≤ R0 +m− 1, 1, if t ≤ R0 +m. For x ∈ R3, set φm(x) = ρm(dist(x,Aδε)). Multiplying both sides of (2.4) by φ2 mvj,ε and integrating on R3, we have( a+ b ∫ R3 |∇vj,ε|2dx )∫ Bm |∇vj,ε|2φ2 m dx+ ∫ Bm V (εx)v2 j,εφ 2 m dx + ξε ∫ Bm χεv 2 j,εφ 2 m dx− ∫ Bm |vj,ε|pφ2 m dx ≤ 8(a+ bAj) ∫ Bm\Bm+1 (|∇vj,ε|2 + v2 j,ε) dx, (4.52) 20 L. WANG EJDE-2022/57 and by (4.50), we obtain( a+ b ∫ R3 |∇vj,ε|2dx )∫ Bm+1 |∇vj,ε|2φ2 m dx+ ∫ Bm+1 V (εx)v2 j,εφ 2 m dx + ξε ∫ Bm+1 χεv 2 j,εφ 2 m dx− ∫ Bm+1 |vj,ε|pψ2 m dx ≥ min{a+ b Aj 2 , m0 2 } ∫ Bm+1 (|∇vj,ε|2 + v2 j,ε) dx, (4.53) where ξε is defined by (4.15). From the above two inequalities, we have( a+ b ∫ R3 |∇vj,ε|2dx )∫ Bm |∇vj,ε|2φ2 m dx ≤ C ∫ Bm\Bm+1 (|∇vj,ε|2 +v2 j,ε) dx, (4.54) where C = 8/min{a+ bAj/2,m0/2}. Then similar to the proof of Lemma 4.4, we can obtain (4.51). The proof is complete. � Lemma 4.8. There exist εN > 0 such that if 0 < ε < εN , then for every 1 ≤ j ≤ N, vj,ε is a solution of (2.4). Proof. Since A is a compact subset of Λ, dist(A, ∂Λ) > 0. By choosing 0 < δ < dist(A, ∂Λ), from Lemma 4.7, we obtain that, for every 1 ≤ j ≤ N , lim ε→0 ∫ R3 χεv 2 j,εdx = 0. (4.55) It follows from that Qε(vj,ε) = 0 if ε > 0 is small enough. Hence there exists εN > 0 such that if 0 < ε < εN , then for every 1 ≤ j ≤ N, vj,ε is a solution of (2.4). The proof is complete. � Proof of Theorem 1.1. By Proposition 3.4 and Lemmas 4.7 and 4.8, we can obtain the results for Theorem 1.1. � 5. Appendix In this section, we give the proof of Proposition 3.3. Let G is an operator on H1(R3). For u ∈ H1(R3), we define w = G(u) by − ( a+b ∫ R3 |∇u|2dx ) ∆w+V (εx)w+2β (∫ R3 χεu 2dx−1 )β−1 + χεw = |u|p−2u, (5.1) where w ∈ H1(R3). We can check that G is odd on H1(R3). Lemma 5.1. G is well defined and continuous on H1(R3). Proof. Since ξ(u) := 2β (∫ R3 χεu 2dx− 1 )β−1 + (5.2) is non-negative, G is well defined and continuous on H1(R3). If un → u in H1(R3), we can obtain that min { a+ b ∫ R3 |∇u|2dx, a0 } ‖A(un)−A(u)‖2 ≤ ∫ R3 ∣∣|un|p−2un − |u|p−2u ∣∣ |A(un)−A(u)|dx + |ξ(un)− ξ(u)| ∫ R3 χε|A(un)−A(u)||A(u)|dx. EJDE-2022/57 LOCALIZED NODAL SOLUTIONS FOR KIRCHHOFF EQUATIONS 21 Since |ξ(un)− ξ(u)| → 0 is obvious, by Sobolev embedding, we can get the conclu- sion. The proof is complete. � Lemma 5.2. For any u ∈ H1(R3), 〈Γ′ε(u), u−A(u)〉 = ( a+ b ∫ R3 |∇u|2dx )∫ R3 |∇(u−A(u))|2dx + ∫ R3 V (εx)(u−A(u))2dx+ ξ(u) ∫ R3 χε(u−A(u))2dx. (5.3) and for any u ∈ H1(R3), there exists a positive constant C such that ‖Γ′ε(u)‖ ≤ ‖u−A(u)‖ ( max { a+ b ∫ R3 |∇u|2dx, 1 } + C‖u‖2β−2 ) . (5.4) Proof. By a direct computation, we can get (5.3). In the following, we only need to show (5.4). For any ψ ∈ H1(R3), 〈Γ′ε(u), ψ〉 = ( a+ b ∫ R3 |∇u|2dx )∫ R3 ∇u∇ψ dx+ ∫ R3 V (εx)uψ dx + ξ(u) ∫ R3 χεuψ dx− ∫ R3 |u|p−2uψ dx (5.5) Multiplying (5.1) by ψ, and then then integrating on both sides, we obtain( a+ b ∫ R3 |∇u|2dx )∫ R3 ∇w∇ψ dx+ ∫ R3 V (εx)wψ dx+ ξ(u) ∫ R3 χεwψ dx = ∫ R3 |u|p−2uψ dx. (5.6) By (5.5) and (5.6), we have 〈Γ′ε(u), ψ〉 = ( a+ b ∫ R3 |∇u|2dx )∫ R3 ∇(u− w)∇ψ dx+ ∫ R3 V (εx)(u− w)ψ dx + ξ(u) ∫ R3 χε(u− w)ψ dx. Then |〈Γ′ε(u), ψ〉| ≤ max { a+ b ∫ R3 |∇u|2dx, 1 } ‖u−A(u)‖‖ψ‖ + C‖u‖2β−2‖u−A(u)‖‖ψ‖ that is for any u ∈ H1(R3), we obtain that ‖Γ′ε(u)‖ ≤ ‖u−A(u)‖ ( max { a+ b ∫ R3 |∇u|2dx, 1 } + C‖u‖2β−2 ) . The proof is complete. � Lemma 5.3. There exists σ0 > 0 such that for σ ∈ (0, σ0), G(∂(Pσ−)) ⊂ Pσ−, G(∂(Pσ+)) ⊂ Pσ+. Proof. We only proof G(∂(Pσ−)) ⊂ Pσ−. For u ∈ H1(R3), let w = G(u), C1 := (min{a+ b ∫ R3 |∇u|2dx,m0})−1. We obtain distH1(w,P−)‖w+‖ ≤ C1‖w+‖2 22 L. WANG EJDE-2022/57 ≤ {( a+ b ∫ R3 |∇u|2dx )∫ R3 |∇w|2dx+ ∫ R3 V (εx)w2dx } = C1 {( a+ b ∫ R3 |∇u|2dx )∫ R3 ww+dx+ ∫ R3 V (εx)ww+dx } = C1 ∫ R3 |u|p−2uw+ dx− C1ξ(u) ∫ R3 χεww +ψ dx ≤ C1 ∫ R3 |u|p−2uw+ dx ≤ C1 ∫ R3 |u|p−2u+w+ dx ≤ C1‖u+‖p−1 p ‖w+‖p = C1(distLp(u, P−))p−1‖w+‖p ≤ C1C(distH1(u, P−))p−1‖w+‖. Then we can infer that distH1(w,P−) ≤ Cσp−1. For σ > 0 small enough, we can get the conclusion. The proof is complete. � We need to have a locally Lipschitz perturbation of G, here G may be only continuous. We E0 = H1(R3) \K, where K is the set of fixed points of G, that is, the set of critical points of Γε. Lemma 5.4. There exists a locally Lipschitz continuous operator B : E0 → H1(R3) such that (1) B(∂(Pσ+)) ⊂ Pσ+ and B(∂(Pσ−)) ⊂ Pσ− for σ ∈ (0, σ0); (2) 1 2‖u−B(u)‖ ≤ ‖u−G(u)‖ ≤ 2‖u−B(u)‖ for u ∈ E0; (3) 〈Γε(u), u−B(u)〉 ≥ 1 2‖u−G(u)‖2 for u ∈ E0; (4) B is odd. Since the proof of the above lemma is similar to the proofs of [1, Lemma 4.1] and [2, Lemma 7], we omit it here. Note that Γε satisfies (PS)c condition for c < L if 0 < ε < εL, by using the map B and similar argument of [14, Lemma 3.5], we can obtain the following lemma. Lemma 5.5. Assume that 0 < ε < εL, c < L, and N is a symmetric closed neighborhood of Kc. Then there exist a positive constant ι0 such that for 0 < ι < ι′ < ι0, there exists a continuous map ζ : [0, 1]×H1(R3)→ H1(R3) satisfying (1) ζ(0, u) = u for all u ∈ H1(R3); (2) ζ(t, u) = u for t ∈ [0, 1], Γε(u) /∈ [c− ι′, c+ ι′]; (3) ζ(t,−u) = −ζ(t, u) for all t ∈ [0, 1] and u ∈ H1(R3); (4) ζ(1, (Γε) c+ι) ⊂ (Γε) c−ι; (5) ζ(t, ∂(Pσ+)) ⊂ Pσ+, ζ(t, ∂(Pσ−)) ⊂ Pσ−, ζ(t, Pσ+) ⊂ Pσ+, ζ(t, Pσ−) ⊂ Pσ−, t ∈ [0, 1]. Proof of Proposition 3.3. Set D is a closed symmetric neighborhood of Kc \ W . Notice that N = D ∪ P̄σ+ ⋃ P̄σ− is a closed symmetric neighborhood of Kc. By Lemma 5.5, we can choose η = ζ(1, ·) in Definition 3.1. � Acknowledgments. 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