MULTIPLICITY AND CONCENTRATION OF SOLUTIONS FOR NONLINEAR FRACTIONAL ELLIPTIC EQUATIONS WITH STEEP POTENTIAL

. In this article, we prove the existence, multiplicity and concentration of non-trivial solutions for the following indeﬁnite fractional elliptic equation with concave-convex nonlinearities: where 0 < α < 1, N > 2 α , 1 < q < 2 < p < 2 ∗ α with 2 ∗ α = 2 N/ ( N − 2 α ), the potential V λ ( x ) = λV + ( x ) − V − ( x ) with V ± = max {± V, 0 } and the parameter λ > 0. Our multiplicity results are based on studying the decomposition of the Nehari manifold.


1.
Introduction. The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics. It was discovered by Nick Laskin [18,19] as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantumn mechanical paths.
Remark 1.1. By [7], there exists a sharp constant S α > 0 such that for any u ∈ H α (R N ). Then by the condition (V 1) − (V 3) and the Hölder and Sobolev inequalities, we have , for all λ ≥ 0, which implies that if V − L N/2 ≤ S 2 α , then Therefore, the condition (V 4) holds.
This type of assumptions was first introduced by Bartsch and Wang [4] and they considered a nonlinear Schrödinger equation. The potential V λ with V satisfies (V 1) − (V 3) is called as the steep well potential.
In recent years, problem (1.1) with α = 1 (that is , the Laplace case) has been widely studied under variant assumptions on V (X) and b(x). Most of the literature has focused on the problem for V being a positive constant and b being a positive weight function. Existence and multiplicity results have been obtained in many papers, see for example [4,5] and references therein. Recently, Cheng and Wu [8] consider the potential V λ which can be sign-changed, that is, V λ = λV + − V − , and they obtained the multiplicity and concentration results with concave-convex nonlinearity terms. In our paper, we would like to extend Cheng and Wu's results to fractional Laplacian case, that is, equation (1.1).
For the multiplicity result, we need more conditions on b(x). Therefore, our second main result is Theorem 1.2. Assume that 2 < p < 2 * α and the functions a, b and V satisfy the assumptions (A1), (B1) and (V1)-(V5) as well as the following condition: Then there existsΛ 0 > 0 such that for every λ >Λ 0 , equation (1.1) has at least two non-trivial non-negative solutions u + λ and u − λ . Recently, concentration phenomena for the fractional Schrödinger equation has been attracted many attentions, see for example [9,10,16,27] and references therein. When V λ = λV + , Torres [27] consider the following problem When f (x, u) satisfies some sub-quadratic assumptions as |u| → ∞, they proved existence and concentration results. In our paper, we consider equation (1.1) with a more general potential V and convex-concave nonlinearity. Furthermore, we also obtain the following concentration result.
We use variational methods to find non-trivial non-negative solutions of equation (1.1). As it is well known, when one uses the variational methods to find the critical points of the functional, some geometry structures are needed such as the mountain pass structure, the linking structures and so on. For problem (1.1), the main difficulty lies in the functional may not posses such structures since the signchanging weight. In order to overcome this difficulty, we turn to another approach, that is, the Nehari manifold, which was introduced by Nehari in [20] and has been widely used in the literature, for example [26,1,28,29,6,30,11] and references therein. The main idea of these articles lies in dividing the Nehari manifold into three parts and considering the infima of the functional on each part. Precisely, the Nehari manifold for Φ λ (u) which is the functional of equation (1.1) (see Section 2), is defined as It is clear that all critical points of Φ λ must be lie on N λ , as we will see below, local minimizers on N λ are usually critical points of Φ λ . By consider the fibering map h u (t) = Φ λ (tu), we can divide that N λ into three subsets N + λ , N − λ and N 0 λ which correspond to local minima, local maxima and points of inflexion of fibbering maps. Then we can find that N 0 λ = ∅ if λ ∈ (Λ 0 , ∞) and meanwhile there exists at least one non-trivial non-negative solution in N + λ and N − λ respectively. This article is organized as follows. In Section 2 we give some notations and preliminaries for the Nehari manifold. Sections 3 and 4 are devoted to prove the existence and multiplicity of non-trivial non-negative solutions of equation (1.1), Theorem 1.1 and Theorem 1.2, respectively. We prove Theorem 1.3 in Section 5.

2.
Variational setting and preliminaries. In this section, we give a few results that we are later going to use for the proofs of the main results. For α ∈ (0, 1), we define D α,2 (R N ) as the completion of C ∞ 0 (R N ) with respect to

Now, let us introduce the fractional Sobolev space
Next, we give the variational setting for equation (1.1) following [14], and we establish the compactness conditions. Let be equipped with the inner product and norm For λ > 0, we also need the following scalar product and norm We use the variational methods to find solutions of equation (1.1). Associated with the equation (1.1), we consider the energy functional Φ λ : X → R, We refer reader to [13,24] for more details of fractional Sobolev space and variational methods. Since the energy functional Φ λ is not bounded below on X, it is useful to consider the functional on the Nehari manifold We notice that N λ contain every non-zero solution of equation (1.1). Next, we define the Palais-Smale (PS)-sequences and (PS)-condition in X for Φ λ as follows.
contains a strongly convergent subsequence.
Since the energy functional Φ λ is not bounded below on X, it is useful to consider the functional on the Nehari manifold N λ . Set then we have the following estimate.
Proof. By the Hölder inequality, for any p ∈ [2, 2 * α ), we have, Moreover, using the conditions (V 1) and (V 2), and by Hölder and Sobolev inequalities, we also have Therefore, We have the following results.
Lemma 2.2. The energy functional Φ λ is coercive and bounded below on N λ . Moreover, we have Proof. If u ∈ N λ , then, by Hölder inequality, (2.1) and Lemma 2.1, we have Thus Φ λ is coercive and bounded below on N λ . The Nehari manifold N λ is closely related to the behaviour of the function of the form h u : t → Φ λ (tu) for t > 0. Such map are know as fibering maps that dates back to the fundamental works [21,23,12]. If u ∈ X, we have We observe that and thus, for u ∈ X \ {0} and t > 0, h u (t) = 0 if and only if tu ∈ N λ , that is, positive critical points of h u correspond points on the Nehari manifold. In particular, h u (1) = 0 if and only if u ∈ N λ . So it is natural to split N λ into three parts corresponding local minimal, local maximum and points of inflection. Accordingly, we define Next, we establish some basic properties of N + λ , N 0 λ , and N − λ .
Proof. If u 0 is a local minimizer for Φ λ on N λ , then u 0 is a solution of the optimization problem Hence, by the theory of Lagrange multipliers, there exists µ ∈ R such that Φ λ (u 0 ) = µJ (u 0 ). Thus we have So, if u 0 ∈ N 0 λ , J (u 0 ), u 0 = 0 and thus µ = 0 by (2.4). Hence, we complete the proof.
For each u ∈ N λ , we know that Then we have following result.
Proof. By the definitions of N + λ and N 0 λ , it is easy to get that R N a|u| q dx > 0 from (2.5). Similarly, the definition of N − λ and (2.6) imply that R N b|u| p dx > 0. Let Λ 0 be as in (2.2). Then we have the following result. .
In order to get a better understanding of the Nehari manifold and the fibering maps, we considering the function m u : R + → R defined by It is clear that tu ∈ N λ if and only if m u (t) = R N a|u| q . Moreover, and it is easy to see that, if tu ∈ N λ , then t q−1 m u (t) = h u (t). Hence tu ∈ N + λ (or (2.11) which leads the following lemma.
Proof. By (2.10), we know t max,λ is the unique critical point of m u and m u is strictly increasing on (0, t max,λ ) and strictly decreasing on (t max,λ , ∞) with lim t→∞ m u (t) = −∞.
Moreover, by (2.1), Lemma 2.1, Hölder inequality and the same arguments as in the proof of Lemma 2.5, we have that Next, we fix u ∈ X \ {0}. Suppose that R N a|u| q dx ≤ 0. Then m u (t) = R N a|u| q has unique solution t − > t max,λ and m u (t − ) < 0. Hence h u has a unique turning point at t = t − and h (t − ) < 0. Thus t − u ∈ N − λ and (2.12) holds. Suppose R N a|u| q dx > 0. Since m u (t max,λ ) > R N a|u| q dx, the equation m u (t) = R N a|u| q has exactly two solutions 0 < t + < t max,λ (u) < t − such that m u (t + ) > 0 and m u (t − ) < 0. Hence, there are two multiplies of u lying in N λ , that is, t + u ∈ N + λ and t − u ∈ N − λ . Thus h u has turning points at t = t + and t = t − with h (t + ) < 0 and h (t − ) < 0. Thus, h u is decreasing on (0, t + ), increasing on (t − , t + ) and decreasing on (t − , ∞). Hence (2.13) holds.
3. Proof of Theorem 1.1. In this section, we prove Theorem 1.1 by variational methods. We establish the existence of a local minimum for Φ λ on N + λ . Theorem 3.1. Suppose that 2 < p < 2 * α and the functions a, b and V satisfy the conditions (A1), (B1) and (V 1) − (V 5). Then, for each λ ≥ Λ 0 , the functional Φ λ has a minimizer u + λ in N + λ satisfying that (2) u + λ is a non-trivial non-negative solution of (1.1). Proof. By Lemma 2.7 and the Ekeland variational principle [15], there exists {u n } ⊂ N + λ such that it is a (P S) c + λ -sequence for Φ λ . Moreover, {u n } is bounded in X by Lemma 2.2. Therefore, there exists a subsequence of {u n } (we still denote as {u n }) and u + λ in X such that u n u in X and u n → u in L r loc (R N ) with 2 ≤ r < 2 * α . Moreover, Φ λ (u + λ ) = 0. Next, we show that u + λ ≡ 0. Suppose the contrary, then by (2.1), the condition (A1), the Egoroff theorem and the Hölder inequality, we have this contradicts lim n→∞ Φ λ (u n ) = c + λ < 0. Hence R N a|u n |dx ≡ 0. In particular, u + λ is a nontrivial solution of equation (1.1). Now, we prove that u n → u + λ in X. Suppose the contrary, then by (2.1), [25]) and |u + λ | ∈ N + λ , by Lemma 2.3, we may assume u + λ is a non-trivial non-negative solution of (1.1).
Proof of Theorem 1.1. Theorem 1.1 is a direct conclusion of Theorem 3.1.

4.
Proof of Theorem 1.2. This section is devoted to proof of Theorem 1.2. We begin this section by proving the following proposition provides a precise description for the (PS)-sequence of Φ λ .
Proof. Let {u n } be a (P S) c -sequence with c < β. By Lemma 2.2, there exists constant C(λ) such that u n λ ≤ C(λ). Hence, there has a subsequence which still denoted as {u n } and u 0 ∈ X such that u n u 0 in X and u n → u 0 in L r loc (R N ) with 2 ≤ r < 2 * α . Moreover, Φ λ (u 0 ) = 0. Then, by (A1), we have Next, we claim that u n → u 0 in X. In fact, let v n = u n − u 0 . Using (V2), we have Then, by the Hölder and Sobolev inequalities, we have By (A1), (B1) and the Brezis-Lieb Lemma, we have Hence, this together with (4.1) and Lemma 2.1, we have (1), it follows from (2.1), (4.2) and (4.3) that Thus, there existsΛ 0 =Λ 0 (β) > 0 such that v n → 0 in X for λ >Λ 0 . This completes the proof.
We observe that if the functions a, b and V satisfy the assumptions in Theorem 1.2, then by Lemma 2.6, we may choose ψ ∈ C ∞ 0 (Ω b ) , which the function have t 0 > 0 and β 0 are independent of λ such that t 0 ψ ∈ N − λ for all λ ≥ Λ 0 and sup which implies c − λ ≤ β 0 for λ ≥ Λ 0 . Moreover, we have the following result. Theorem 4.1. Suppose that the functions a, b and V satisfy the conditions as in the Theorem 1.2. Then for each λ >Λ 0 , the function Φ λ has a minimizer u − λ in N − λ and it satisfies λ is a non-trivial non-negative solution of (1.1). Proof. By Lemma 2.7 and the Ekeland variational principle [15], there exists {u n } ⊂ N + λ such that it is a (P S) c − λ -sequence for Φ λ . Thus, by Proposition 4.1, there exists a subsequence {u n } and u − λ ∈ N λ is a non-trivial solution of equation (1.1), such that u n → u − λ in X and Φ λ ( [25]) and |u − λ | ∈ N + λ , by Lemma 2.3, we may assume u − λ is a non-trivial non-negative solution of (1.1).
This completes the proof of Theorem 1.2.

5.
Proof of Theorem 1.3. In this section we consider the concentration of solutions and give the proof of Theorem 1.3.