Existence of infinitely many solutions for a nonlocal problem

Abstract: In this paper, we deal with a class of fractional Hénon equation and by using the LyapunovSchmidt reduction method, under some suitable assumptions, we derive the existence of infinitely many solutions, whose energy can be made arbitrarily large. Compared to the previous works, we encounter some new challenges because of the nonlocal property for fractional Laplacian. But by doing some delicate estimates for the nonlocal term we overcome the difficulty and find infinitely many nonradial solutions.


Introduction
In this paper, we consider the following nonlocal Hénon equation with critical growth, where α > 0 is a positive constant, p = n+2s n−2s , n ≥ 2+2s, 1 2 < s < 1, B 1 (0) is the unit ball in R n and A s stands for the fractional Laplacian operator in B 1 (0) with zero Dirichlet boundary values on ∂B 1 (0).
Here, to define the fractional Laplacian operator A s in B 1 (0), let {λ k , ϕ k } be the eigenvalues and corresponding eigenfunctions of the Laplacian operator −∆ in B 1 (0) with zero Dirichlet boundary values on ∂B 1 (0), namely, {λ k , ϕ k } satisfies on ∂B 1 (0) with ϕ k L 2 (B 1 (0)) = 1. Then we can define the fractional Laplacian operator A s : H s 0 (B 1 (0)) → H −s 0 (B 1 (0)) as where the fractional Sobolev space H s 0 (B 1 (0))(0 < s < 1) is given by It is well known that the nonlinear fractional equations appear in diverse areas including physics, biological modeling and mathematical finances and have attracted the considerable attention in the recent period. Also in recent years, there have been many investigations for the related fractional problem A s u = f (u), where f : R n → R is a certain function. But a complete review of the available results in this context goes beyond the aim of this paper. Here we just mention some very recent papers which study fractional equations involving the critical sobolev exponent (cf. [3,7,19,20,21]).
On the other hand, our main interest in the present paper is motivated by some works that have appeared in recent years related to the classical local Hénon equation of this kind, Among pioneer works we mention Ni [13], where the author established a compactness result of H 1 0,rad (B 1 (0)) → L p+1 (B 1 (0)) and thus got the existence of one positive radial solution for (1.2) if p ∈ (1, n+2+2α n−2 ). Later, in [18], Smets, Su and Willem established some symmetry breaking phenomenon and obtained the non-radial property of the ground state solution of (1.2) if 1 < p < n+2 n−2 and α is large enough. When n ≥ 3 and p = n+2 n−2 − σ, Cao and Peng [1] verified that the ground state solutions of (1.2) are non-radial and blow up as σ → 0. Meanwhile, when p = n+2 n−2 , Serra [17] showed that (1.2) has a non-radial solution if n ≥ 4 and α is large enough. More recently, Wei and Yan [23] proved the existence of infinitely many non-radial solutions for (1.2) for any α > 0. For other results related to the Hénon problem (1.2), one can refer to [2,11,14,15] and the references therein.
Up to our knowledge, not much is obtained for the existence of multiple solutions of equation (1.2) with fractional operator. Motivated by [23] and [12], we want to exploit the finite dimensional reduction method to investigate the existence of infinitely many non-radial solutions for (1.1). To achieve our aim, we will study the following more general problem where Φ(r) is a bounded function defined in [0, 1]. It is easy to check that a necessary condition for the existence of a solution of (1.3) is that Φ(r) is positive somewhere from Pohozaev identity (see [16]). At this point we call attention to the recent work of [12], where we studied (1.3) in R n and proved that if n > 2 + 2s, 0 < s < 1 and Φ(|x|) satisfies 3) has infinitely many non-radial solutions. It is worth mentioning that from assumption (1.4), r 0 is a local maximum point of Φ(r) and then a critical point of Φ(r). Also the function r α achieves its maximum on [0, 1] at r 0 = 1 but r 0 = 1 is not a critical point of r α . However we will verify that if Φ(r) is increasing near r 0 = 1, through r 0 = 1 is not a critical point of Φ(r), the zero Dirichlet boundary condition makes it possible to construct infinitely many solutions of (1.3). Now we state our main result as follows: 3) has infinitely many non-radial solutions. Particularly, the Hénon equation (1.1) has infinitely many non-radial solutions.
In the end of this part, let us outline the main idea in the proof of Theorem 1.1. Given any ε > 0 and y 0 ∈ R n , let for x ∈ R n and σ n,s = 2 4s . In 1983, Lieb [10] (also see [6,7,8,9]) proved that U ε,y 0 (x) solves the following critical fractional equation Also, very recently, J. DÁvila, M. del Pino and Y. Sire [5] obtained the non-degeneracy of U ε,y 0 (x). More precisely, if we define the corresponding functional of (1.5) as then I 0 possesses a finite-dimensional manifold Z of least energy critical points, given by Now let us fix a positive integer k ≥ k 0 , where k 0 is large, which is to be determined later and set to be the scaling parameter. Using the transformation u(x) → ν − n−2s Since U ε,y 0 is not zero on ∂B ν (0), we define PU ε,y 0 as the solution of the following problem and we will use the solution PU ε,y 0 to build up the approximate solutions for (1.6).
Also, we denote with 0 is the zero vector in R n−2 . And throughout this paper, we always assume that 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 , (1.6) has a solution u k of the form where ω k ∈ H k , and as k → +∞, ω k L ∞ (B ν (0)) → 0, r k ∈ ν(1 − r 0 k ), ν(1 − r 1 k ) , ε 0 ≤ ε ≤ ε 1 . We want to point out that compared with [23], due to the fact that the fractional Laplacian operator is nonlocal and very few things on this topic are known about the fractional Laplacian, we have to face much difficulties in the reduction process and need some more delicate estimates in the proof of our results.
The rest of the paper is organized as follows. In Section 2, we will carry out a reduction procedure and we prove our main result in Section 3. Finally, in Appendix, some basic estimates and an energy expansion for the functional corresponding to problem (1.6) will be established.

Finite-dimension reduction
In this section, we perform a finite-dimensional reduction. Let where τ = n−2s n−2s+1 . For this choice of τ, we find that

Now we consider
3) for g = g k . If g k * * goes to zero as k goes to infinity, so does ϕ k * .
Note that we can rewrite (2.3) as Now we estimate each terms in (2.4). Analogous to Lemma A.3, we have On the other hand, Define Observe that for y ∈ Ω 1 , |y − x i | ≥ |y − x 1 | and then As a result, from (2.9), we have (2.10) Using the same argument used for proving (A.7), it follows from (2.9) and (2.10) that Hence using (2.8), we get So, combining (2.4)-(2.7) and (2.12), one has (2.13) Being ϕ * = 1, we obtain from (2.13) that there is R > 0 such that for some i. But, by using (2.3),φ(x) = ϕ(x − x i ) converges, uniformly in any compact set, to a solution φ of the following equation Due to the non-degeneracy of U ε,0 , we can infer that φ = 0, which yields a contradiction with (2.14) and then this proof has been proved.
From Lemma 2.1, arguing as proving Proposition 4.1 in [6] or Proposition 2.2 in [12], we can show the following result.
On the other hand, since .
Taking into account that Thus B is a contraction map. Therefore, Applying the contraction mapping theorem, we can find a unique ϕ = ϕ(r, ε) ∈ N such that ϕ = B(ϕ) and Moreover, we get the estimate of c l from (2.12).
there is t ∈ (0, 1) such that F( , ε) = I(U ε,r + ϕ) Firstly, Therefore, we see On the other hand, by Hölder inequality, we have Combining the estimates above and applying Proposition A.5, we have proved where A 1 , A 2 are the same constants as in Proposition 3.1, and > 0 is a small constant.
There is a small > 0 and some constant C > 0 such that Proof. We can find this proof in [12]. Here we just need to use Let G(x, y) be the Green function of A s in B 1 (0) with the Dirichlet boundary condition (see [4]), namely, G(x, y) solves A s G(·, y) = δ y in B 1 (0), G(·, y) = 0 on B 1 (0), and the regular part of G is given by where α n,s = 1 Denote byG(x, y) the Green function of A s in B ν (0) with the Dirichlet boundary condition and bỹ H(x, y) the regular part ofG(x, y). So we can find which, together with (A.2), yields that on ∂B ν (0).

As a result,
for some constant C > 0. Using (A.3), we have proved Proposition A.5. There holds where A, A 1 , A 2 are some positive constants, and is a small constant.
Proof. Note that Using the symmetry and (A.1), we have we can obtain that ε,x 1 k i=2 U ε,x i (p+1)/2 .