INFINITELY MANY SOLUTIONS FOR CRITICAL FRACTIONAL EQUATION WITH SIGN-CHANGING WEIGHT FUNCTION

In this work, we consider the fractional Schrödinger type equations with critical exponent, concave nonlinearity and sign-changing weight function on R . With the aid of the symmetric Mountain Pass Theorem, we prove this problem has infinitely many small energy solutions.


Introduction and main result
In this paper, we are concerned the infinitely many solutions to the following fractional Schrödinger equation with critical exponent where µ > 0 is a parameter, 1 < q < 2, 0 < s < 1, N > 2s and 2 * s = 2N N −2s is a non-local fractional Sobolev exponent. Here, (− ) s is the fractional Laplace operator (see [2]), which is defined as (− ) s u(x) := C N,s P.V.
The symbol P.V. represents the Cauchy principal value, C N,s is a normalization constant that depends on N and s. The weight function h(x) satisfies the following condition: s −q and h + = max{h, 0} = 0. The fractional Schrödinger equation is a class of fundamental equation in fractional quantum mechanics. It reflects the stable diffusion of particles of Lévy processes, which was first discovered by Laskin [15,16]. Through the equivalent definition of fractional operator, the authors obtained the corresponding variational principle and proved the existence of solutions in [3,11,[19][20][21]. Specially, for the concave-convex nonlinearity, this type of problems has currently been actively studied, see [5,8,13,22] and their references. Here, we are interested in the case of the fractional Schrödinger equations involving concave-convex nonlinearities with critical exponent.
For Schrödinger equations, Chabrowski and Drabek [10] studied the following nonlinear elliptic problem: where ε is a positive constant, 1 < q < 2 and N ≥ 3. Under the condition that h is a nonnegative and nonzero function in L 2 * 2 * −q−1 (R N ) ∩ C(R N ), they obtained infinitely many solutions of equation (1.2) for ε small.
For fractional Schrödinger equations, if the weight functions h(x) = 1, Barrios etc [4] dealt with the following problem: where Ω ⊂ R N is a smooth bounded domain, λ > 0, 0 < s < 1, N > 2s and 0 < q < 1. They obtained that there exist at least two positive solutions for every 0 < λ < Λ, at least one positive solution if λ = Λ, no positive solution if λ > Λ.
In [23], Zhang etc proved the existence of a nontrivial radially symmetric weak solution to the following problem: 4) where N ≥ 2, λ is a positive real parameter, V (x) and k(x) are positive and bounded functions satisfying some suitable conditions. Motivated by above papers, we consider the fractional problem (1.1) with critical exponent on R N . The main difficulty is how to recover the compactness. To the best of our knowledge, few papers deal with this problem with sign-changing weight function up to now. Inspired by [9], the main purpose of this paper is to study the existence of infinitely many small energy solutions of (1.1) for µ sufficiently small via a new version of the symmetric Mountain Pass Theorem due to Kajikiya [14].
The main result of this paper is the following.
Theorem 1.1. Assume 1 < q < 2 and the condition (H) is fulfilled. There exists µ * > 0 such that for all µ ∈ (0, µ * ), problem (1.1) possesses infinitely many nontrivial solutions {u k } ∞ k=1 satisfying The rest of this paper is organized as follows. In Section 2 we will introduce some knowledge of dealing with the fractional Laplacian operator and get some helpful results. We will finish the proof of Theorem 1.1 in Section 3.

Preliminaries
In this part we first recall some results on Sobolev spaces of fractional order. For a deeper introduction to fractional Sobolev spaces can be found in [6,18] and references therein.

Consider the fractional order Sobolev space
is the Gallardo semi-norm. Observe Proposition 3.6 in [18], we can get and the corresponding inner product is Throughout this paper, we will use · to represent the norm of H s (R N ). As usual, for 1 ≤ ν < ∞, we let In order to discuss the weak solutions of (1.1), we need to find the critical points of the energy functional I : H s (R N ) → R defined by Under the condition (H), we can get the energy functional I is well-defined by the Sobolev embedding theorem. It's not hard to prove I ∈ C 1 (H s (R N ), R) and its derivative is given by Lemma 2.2. Assume 1 < q < 2 and the condition (H) holds. If u n u in H s (R N ), then Proof. The proof is similar to Lemma 3.4 in [12], we omit it. Lemma 2.3. If there is a convergent subsequence for any sequence {u n } ⊂ H s (R N ) satisfying I(u n ) → c and I (u n ) → 0, we say that I satisfies the (PS) c condition. Assume the condition (H) holds, the functional I for any λ > 0 satisfies the (PS) c condition with Proof. Let {u n } be a sequence in H s (R N ) and satisfy First we prove that {u n } is bounded in H s (R N ). Arguing by contradiction, suppose that u n → ∞ as n → ∞. By the Hölder inequality and (2.5) for sufficiently large n ∈ N, we obtain This implies {u n } is bounded in H s (R N ). Therefore, up to a subsequence, for some u ∈ H s (R N ), we get Since I is C 1 , we have I (u), u = 0, which implies that and By the Hölder and Young inequalities, we have From (2.7) and (2.8), we obtain Taking w n = u n − u, by the Brezis-Lieb lemma (see [7]) yields Hence by (2.10), (2.11), (2.12) and Lemma 2.2, one has In view of I (u n ), u n = o(1) and I (u), u = 0, we derive that (2.14) Now, we may assume that w n 2 → L ≥ 0. By (2.14), it follows that R N |w n | 2 * s dx → L.
Let us suppose that L > 0. By applying the Sobolev inequality we know that Hence we can deduce that L ≥ S N 2s * . This fact combining with (2.9) and (2.13) yield this contradicts the definition of c. Hence, w n → 0, i.e. u n − u → 0. Thus, we prove that {u n } converges strongly to u in H s (R N ).

Proof of the main results
In this section, we will use some knowledge of genus, but it will not be listed here. Detailed definition and properties of genus can be seen in [1]. In [17], the author first established the following new version of the symmetric mountain-pass lemma based on R. Kajikiya [14].
(B1) I is even, bounded from below, I(0) = 0 and I satisfies the (PS) c condition.
(B2) For each k ∈ N, there exists an A k ∈ Γ k such that sup u∈A k I(u) < 0, where Γ k signifies a family of closed symmetric subsets A of X with 0 ∈ A and γ(A) ≥ k.
Then I admits a sequence of critical points {u k } such that I(u k ) ≤ 0, u k = 0 and lim k→∞ u k = 0.
Proof of Theorem 1.1. If we can prove that G(u) has a sequence of nontrivial weak solutions {u n } satisfying u n → 0 as n → ∞ in H s (R N ), Theorem 1.1 holds. In fact, in this case, it is clear that I(u) = G(u) for u < t 0 . Next we just need to verify that G(u) satisfies the conditions of Theorem 3.1. Let E k be a k-dimensional subspace of H s (R N ). By the equivalence of any norm in finite dimensional space, we obtain We take u ∈ E k with norm u = 1 and ρ > 0 small enough, we get G(ρu) =I(ρu) By the Hölder and Young inequalities, we have where C 2 = 1 qq * |h − | q * q * ≥ 0. From (3.4) and (3.5), we get G(ρu) ≤ ρ 2 2 − µ q α k ρ q + C 2 µ q * := −β(k) < 0.
Therefore, letting A k = u ∈ H s (R N ) : G(u) ≤ −β(k) . By genus proposition, we have γ(A k ) ≥ γ ({u ∈ E k : u = ρ}) ≥ k. We can also see A k ∈ Γ k and sup u∈A k G(u) ≤ −β(k) < 0. We define c k = inf A∈Γ k sup u∈A G(u), then −∞ < c k < 0. This shows G(u) satisfies assumptions (B1) and (B2) of Theorem 3.1. This means that G has a sequence of solutions u k converging to zero.