INFINITELY MANY POSITIVE AND SIGN-CHANGING SOLUTIONS FOR NONLINEAR FRACTIONAL SCALAR FIELD EQUATIONS

. We consider the following nonlinear fractional scalar ﬁeld equation where K ( | x | ) is a positive radial function, N ≥ 2, 0 < s < 1, and 1 < p < N +2 s N − 2 s . Under various asymptotic assumptions on K ( x ) at inﬁnity, we show that this problem has inﬁnitely many non-radial positive solutions and sign-changing solutions, whose energy can be made arbitrarily large.

Here, the fractional Laplacian of a function f : R N → R is expressed by the formula where C N,s is some normalization constant (See Sect.2).
Problem (1) arises from looking for standing waves Ψ(t, x) = exp(iEt)u(x) for the following nonlinear equations where i is the imaginary unit, E ∈ R. This equation is of particular interest in fractional quantum mechanics for the study of particles on stochastic fields modelled by Lévy processes. A path integral over the Lévy flights paths and a fractional Schrödinger equation of fractional quantum mechanics are formulated by Laskin [24] from the idea of Feynman and Hibbs's path integrals (see also [25]). The Lévy processes occur widely in physics, chemistry and biology. The stable Lévy processes that give rise to equations with the fractional Laplacians have recently attracted much research interest, and there are a lot of results in the literature on the existence of such solutions, e.g., [4,7,18,35,5,1,40,26,30,10,11] and the references therein.
A partner problem of (1) is the following Schrödinger equation In the sequel, we will assume that V, K is bounded, and V (x) ≥ V 0 > 0, K(x) ≥ K 0 > 0 in R N . It is well known, but not completely trivial, that (−∆) s reduces to the standard Laplacian −∆ as s → 1. When s = 1, the classical nonlinear Schrödinger equation and scalar field equation have been extensively studied in the last thirty years. Moreover, if then, using the concentration compactness principle [31,32], one can show that (1) (resp. (4)) has a least energy solution (See for example [17,31,32,15,40]). But if (5) does not hold, (1) or (4)  then it is easy to see that problem (4) has no least energy solutions. So, in this case, one needs to find solutions with higher energy. Recently, Cerami et al. [8] showed that problem (4) with s = 1 has infinitely many sign-changing solutions if V (x) goes to its limit at infinity from below at a suitable rate. In [38], Wei and Yan gave a surprising result which says that (1) or (4) with s = 1 and V (x) or K(x) being radial has solutions with large number of bumps near infinity and the energy of this solutions can be very large. This kind of results was generalized by Ao and Wei in [2] very recently to the case in which V (x) or K(x) does not have any symmetry assumption. We should also mention another interesting paper [9], where infinitely many positive solutions to (4) with s = 1 were verified to exist by Cerami and Passaseo by using mini-max argument and no symmetry assumption was imposed on V (x). For more results on (1) and (4) with s = 1, we can refer to [16,17] and the references therein. For results on this aspect when 0 < s < 1, the readers can refer [18,27,34] and the references therein. In recent years, the singularly perturbed problem of (4) with s = 1 has been widely researched, such as [3,6,13,22,28,29,12,14]. When 0 < s < 1, Chen and Zheng [12] studied the following singularly perturbed problem They showed that when N = 1, 2, 3, ε is sufficiently small, max{ 1 2 , n 4 } < s < 1 and V satisfies some smoothness and boundedness assumptions, equation (6) has a nontrivial solution u ε concentrated to some single point as ε → 0. Very recently, in [14], Dávila, del Pino and Wei generalized various existence results known for (6) with s = 1 to the case of fractional Laplacian. For results which are not for singularly perturbed type of (1) and (4) with 0 < s < 1, the readers can refer to [4,18,36] and the references therein.
As far as we know, it seems that there is no result on the existence of multiple solutions of equation (1) which is not a singularly perturbed problem. The aim of this paper is to obtain infinitely many non-radial positive solutions and sing-changing solutions for (1) whose functional energy are very large, under some assumptions for K(x) = K(|x|) > 0 near the infinity. For convenience, in this paper, we consider the following nonlinear fractional scalar field equation Let N + 2s N + 2s + 1 < m < N + 2s.
We assume that 0 < K(|x|) ∈ C(R N ) satisfies the following conditions at infinity for some a > 0, θ > 0. Without loss of generality, we may assume that K 0 = 1.
Remark 1. The radial symmetry can be replaced by the following weaker symmetry assumption: after suitably rotating the coordinated system, where a > 0, θ > 0 and K 0 > 0 are some constants.
Remark 2. Using the same argument, we can prove that if where V 0 > 0, θ > 0, then problem (4) has infinitely many positive non-radial solutions if a > 0 and has infinitely many non-radial sign-changing solutions if a < 0. Now let us outline the main idea to prove our main results. We will use the unique ground state U of to build up the approximate solutions for (7). It is well known that when s = 1, the ground state solution of (9) decays exponentially at infinity. But from [19,20], we see that when s ∈ (0, 1), the unique ground solution of (9) decays like 1 |x| N +2s when |x| → ∞.

WEI LONG, SHUANGJIE PENG AND JING YANG
For any integer k > 0, define To prove Theorem 1.1, it suffices to verify the following result: Under the assumption of Theorem (1.1), there is an integer k 0 > 0, such that for any integer k ≥ k 0 , (7) has a solution u k of the form N +2s−m ] for some constants r 1 > r 0 > 0 and as k → +∞, To consider the sign-changing solutions, for any integer k > 0, we definē N +2s−m ] for somer 1 >r 0 > 0. WriteŪ To prove Theorem 1.2, we only need to show that: Then there is an integer k > 0, such that for any integer k ≥k, (7) has a solutionū k of the form N +2s−m ] for some constantsr 1 >r 0 > 0 and as k → +∞, The idea of our proof is inspired by that of [38] where infinitely many positive non-radial solutions to a nonlinear Schrödinger equations (3) with s = 1 are obtained when the potential approaches to a positive constant algebraically at infinity. We will use the well-known Lyapunov-Schmidt reduction scheme to transfer our problem to a maximization problem of a one-dimensional function in a suitable range. Compared with the operator −∆, which is local, the operator (−∆) s with 0 < s < 1 on R N is nonlocal. So it is expected that the standard techniques for −∆ do not work directly. In particular, when we try to find spike solutions for (7) with 0 < s < 1, (−∆) s may kill bumps by averaging on the whole R N . For example, the ground state for (7) with 0 < s < 1 decays algebraically at infinity, which is a contrast to the fact that the ground state for −∆ decays exponentially at infinity. This kind of property requires us to establish some new basic estimates and give a precise estimate on the energy of the approximate solutions.
This paper is organized as follows. In Sect.2, we will give some preliminary properties related to the fractional Laplacian operator. In Sect.3, we will establish some preliminary estimate. We will carry out a reduction procedure and study the reduced one dimensional problem to prove Theorems 1.3 and 1.4 in Sect.4. In Appendix, some basic estimates and an energy expansion for the functional corresponding to problem (7) will be established.
2. Basic theory on fractional Laplacian operator. In this section, we recall some properties of the fractional order Sobolev space and the ground state solution U of the limit equation (9).
Let 0 < s < 1. Various definitions of the fractional Laplacian (−∆) s f (x) of a function f defined in R N are available, depending on its regularity and growth properties.
It is well known that it can be defined as a pseudo-differential operator where is Fourier transform. When f have some sufficiently regular, the fractional Laplacian of a function f : R N → R is expressed by the formular . This integral makes sense directly when s < 1 In the remarkable work of Caffarelli and Silvestre [5], this nonlocal operator was expressed as a generalized Dirichlet-Neumann map for a certain elliptic boundary value problem with nonlocal differential operator defined on the upper half-space R N +1 and The equation (11) can also be written as div(y 1−2s ∇u) = 0, which is clearly the Euler-Lagarnge equation for the functional Then, it follows from (10)-(12) that and the norm is Here the term is the so-called Gagliardo (semi) norm of u. The following identity yields the relation between the fractional operator (−∆) s and the fractional Laplacian Sobolev space for a suitable positive constant C depending only s and N . Clearly, · H s (R N ) is a Hilbertian norm induced by the inner product On the Sobolev inequality and the compactness of embedding, one has Theorem 2.1. [33] The following imbeddings are continuous: Now, we recall some known results for the limit equation (9). If s = 1, the uniqueness and non-degeneracy of the ground state U for (9) is due to [23]. In the celebrated paper [19], Frank and Lenzemann proved the uniqueness of ground state solution U (x) = U (|x|) ≥ 0 for N = 1, 0 < s < 1, 1 < p < 2 * (s) − 1. Very recently, Frank, Lenzemann and Silvestre [20] obtained the non-degeneracy of ground state solutions for (9) in arbitrary dimension N ≥ 1 and any admissible exponent 1 < p < 2 * (s) − 1.
For convenience, we summarize the properties of the ground state U of (9) which can be found in [19,20].
Then the following hold.
(i) (Uniqueness) The ground state solution U ∈ H s (R N ) for equation (9) is unique.
(ii)(Symmetry, regularity and decay) U (x) is radial, positive and strictly decreasing in |x|. Moreover, the function U belongs to H 2s+1(R N ) ∩ C ∞ (R N ) and satisfies , its kernel is given by By Lemma C.2 of [20], it holds that, for j = 1, · · · , N, ∂ xj U has the following decay estimate, From Theorem 2.2, the ground bound state solution like 1 |x| N +2s when |x| → +∞. Fortunately, this polynomial decay is enough for us in the estimates of our proof.
3. Some preliminaries. In the sequel, we mainly concentrate on the existence of positive solutions. For the existence of sign-changing solutions, we will only give a sketch at the end of Section 4. Let for some r 1 > r 0 > 0. Define Note that the variational functional corresponding to (7) is

WEI LONG, SHUANGJIE PENG AND JING YANG
Letting We can expand J(ϕ) as follows: where and In order to find a critical point for J(ϕ), we need to discuss each term in the expansion (13). Now we come to the main result in this section.
Proof. By direct calculation, we know that Firstly, we deal with the case p > 2. Since On the other hand, it follows from Lemma A.2 that U r is bounded. We obtain So, R (ϕ) ≤ C ϕ s . Using the same argument, for the case 1 < p ≤ 2, we also can obtain that 4. The finite-dimensional reduction and proof of the main results. In this section, we intend to prove the main theorem by the Lyapunov-Schmidt reduction. Associated to the quadratic form L(ϕ), we define L to be a bounded linear map from E to E, such that Now, we assume that r ∈ S k and define where α > 0 is a small constant, B 0 and B 1 are defined in Proposition A.1. Next, we show the invertibility of L in E.
Proposition 4.1. There exists an integer k 0 > 0, such that for k ≥ k 0 , there is a constant ρ > 0 independent of k, satisfying that for any r ∈ S k , Lu ≥ ρ u s , u ∈ E.
Proof. We argue it by contradiction. Suppose that there are n → +∞, r k ∈ S k , and u n ∈ E, such that By symmetry, we have In particular, where o n (1) → 0 as n → +∞. Set u n (x) = u n (x − x 1 ). Then for any R > 0, since One can obtain So we suppose that there is a u ∈ H s (R N ), such that as n → +∞, and u n → u, in L 2 loc (R N ). Since u n is even in x j , j = 2, · · · , N, it is easy to see that u is even in x j , j = 2, · · · , N. On the other hand, from Now, we claim that u satisfies Indeed, we set For any R > 0, let ϕ ∈ C ∞ 0 (B R (0)) ∩ E be any function, satisfying that ϕ is even in x j , j = 2, · · · , N. Then ϕ 1 (x) = ϕ(x − x 1 ) ∈ C ∞ 0 (B R (x 1 )). We may identify ϕ 1 (x) as elements in E by redefining the values outside Ω 1 with the symmetry. By using (15) and Lemma A.2, we find On the other hand, since u is even in x j , j = 2, · · · , N, (18) holds for any ϕ ∈ C ∞ 0 (B R (0)) ∩ E. By the density of C ∞ 0 (R N ) in H s (R N ), it is easy to show that We know ϕ = ∂U ∂x1 is a solution of (19), thus (19) is true for any ϕ ∈ H s (R N ). One see that u = 0 because u is even in x j , j = 2, · · · , N and (17). As a result, Now, using the Lemma A.2, we obtained that for any 1 < η ≤ N + 2s, there is a constant C > 0, such that which is impossible for large R.
As a result, we get a contradiction.

Proposition 4.2.
There is an integer k 0 > 0, such that for each k ≥ k 0 , there is a C 1 map from S k to H : ω = ω(r), r = |x 1 |, satisfying ω ∈ E, and J (ω) Moreover, there exists a constant C > 0 independent of k such that ω s ≤ Ck where τ > 0 is a small constant.
Proof. We will use the contraction theorem to prove it. By the following Lemma 4.1, l(ω) is a bounded linear functional in E. We know by Reisz representation theorem that there is an l k ∈ E, such that l(ω) = l k , ω .

WEI LONG, SHUANGJIE PENG AND JING YANG
We shall verify that A is a contraction mapping fromS k to itself. In fact, on one hand, for any ω ∈S k , by Lemmas 4.1 and 3.1, we obtain On the other hand, for any ω 1 , ω 2 ∈S k , Then the result follows from the contraction mapping theorem. The estimate (20) follows Lemma 4.1.
On the other hand, we have for some small constant τ > 0, where the last inequality is due to the assumption N +2s N +2s+1 < m < N + 2s. Inserting (24)- (25) into (22), we can complete the proof. Now, we are ready to prove our main theorem. Let ω = ω(r) be the map obtained in Proposition 4.2. Define It follows from Lemma 6.1 in [14] that if r is a critical point of F (r), then U r + ω is a solution of (7).
Proof of Theorem 1.3. It follows from Propositions 4.2 and A.1 that

WEI LONG, SHUANGJIE PENG AND JING YANG
Define We consider the following maximization problem Suppose thatr is a maximizer, we will prove thatr is an interior point of S k .
We can check that the function By direct computation, we deduce that On the other hand, we find and similarly where we have used the fact that the function f (t) = t − N +2s N +2s−m (t − 1) attains its maximum at t 0 = N +2s The above estimates implies thatr is indeed an interior point of S k . Thus ur = Ur + ωr is a solution of (7).
At last, we claim that ur > 0. Indeed, since ωr s → 0 as k → ∞, noticing the fact (see [14] for example) that whereũr(x, y) is the s-harmonic extension of ur satisfyingũr(x, 0) = ur(x), we can use the standard argument to verify that (ur) − = 0 and hence ur ≥ 0. Since ur solves (12), we conclude by using the strong maximum principle that ur > 0.
The sketch of proof of Theorem 1.4. Set where B 0 , B 1 are given in Proposition A.2, α > 0 is a small constant. For r ∈S k , letŪ We will find a solution for equation (7) of the formŪ r (x) +ω with To this end, we should also perform the same procedure as the proof of Theorem 1.3. Proceeding as we prove Proposition 4.2, we conclude that for any r ∈S k , there is uniqueω r ∈ C 1 (S k ,Ē), such thatŪ r (x)+ω r is a critical point of J onĒ. Moreover, ω s ≤ Ck Now to prove thatŪ r (x) +ω is a critical point of J on H s (R N ) can be reduced to finding a minimum point of functionF (r) inS k , which can be realized exactly as we have done in the proof of Theorem 1.3.
Then, we have the following basic estimate: For any x ∈ Ω 1 , and η ∈ (1, N + 2s], there is a constant C > 0, such that Proof. The proof of this lemma is similar to Lemma A.1 in [38], we sketch the proof below for the sake of completeness.
For any x ∈ Ω 1 , we have for i = 1, So, we find Since So, there is a constant B > 0, such that There is a small constant τ > 0, such that Proof. Using the symmetry, we have It follows from Lemma A.1 that Hence, there exists B 0 (which may depend on k) in [C 2 , C 1 ], where C 1 and C 2 are independent of k, such that Now, by symmetry, we see where κ > 0 satisfies min{ p+1 2 (N + 2s − κ), 2(N + 2s − κ)} > N + 2s. Hence, we get .
So, we have proved Now, inserting (27)- (29) into I(U r ), we complete the proof.