Infinitely many solutions for a class of sublinear fractional Schrödinger equations with indefinite potentials

The nonlinear field Eq. (1.2) reflects the stable diffusion process of Lévy particles in random field. Later, people found that this stable diffusion of Lévy process has also a very important application in the mechanical system, flame propagation, chemical reactions in the liquid, and the anomalous diffusion of physics in the plasma. For more details, readers can refer to [5, 25, 26, 45] and the references therein.

For fractional equations on the whole space R N , the main difficulty one may face is that the Sobolev embedding H s (R N ) → L q (R N ) is not compact for q ∈ [2, 2 * s ). To overcome this difficulty, some authors [8,10,24,31,38,50] considered fractional equations with the potential V satisfying the following conditions: where V 0 is a constant and meas denotes Lebesgue measure in R N . Due to condition (V ), the subspace of H s (R N ) embeds compactly into L q (R N ) for q ∈ [2, 2 * s ), which is crucial in their paper. In fact, condition (V ) is certain coercive condition. In the case of coercive condition lim |x|→+∞ V (x) = +∞, some authors, for example [12,33], considered fractional equations on the whole space R N .
To overcome the difficulties caused by the lack of compactness, on the other hand, some authors restricted the energy functional to a subspace for H s (R N ) of radially symmetric functions, which embeds compactly into L s (R N ), for example, [9,21,34,44,54].
However, in this paper, we do not need some conditions like (V ) or radially symmetric. That is, our paper does not use any compact embedding on the whole space R N .
It is worth noting that, for fractional equations on the whole space R N , most results need condition V (x) ≥ 0 (see [1, 8-10, 12, 13, 16, 18, 20-22, 24, 28, 33, 34, 36-38, 44, 50, 52-54], in which some results were obtained in case of V (x) = 1 [16,18,21,28,44]). To the best of our knowledge, there are few results on the existence of solutions for fractional equations with a sign-changing potential except [11,51]. In fact, replaced inf x∈R N V (x) ≥ V 0 > 0 with inf x∈R N V (x) > -∞, condition similar to (V ) is needed in [11]. In [51], Xu, Wei, and Dong considered the following p-Laplacian equation with positive nonlinearity: where N, p ≥ 2, s ∈ (0, 1), λ is a parameter, (-) s p is the fractional p-Laplacian, and f : R N × R → R is a Carathéodory function. In the case of λ = 0, they obtained the existence of a nontrivial solution to this equation. Furthermore, they proved that this equation has infinitely many nontrivial solutions when λ ≤ 0 or λ > 0 is small enough.
In this article, we are interested in the existence of infinitely many solutions for problem (1.1) with potential function V (x) changing sign in R N . Moreover, nonlinearity can be allowed to change sign. To state our main result, we assume the following: and S is the constant of Sobolev: N -2s .
(K) K ∈ L ∞ (R N ) and there exist β > 0, R 1 > R 2 > 0, y 0 = (y 1 , . . . , y N ) ∈ R N such that Our main result of this paper can be stated as follows.
Remark 1.1 The ideas in this article come from the paper [3], where Schrödinger equations were considered. However, our proof is nontrivial since we present a simplified proof for the PS condition by comparing to that in [3]. In fact, the PS condition was proved in [3] by concentration compactness principle. It is noticed that the PS condition plays important role in the proof of the main results in [3].

Notations and preliminaries
In this paper, we use the following notations. Let Let E be a Banach space and ϕ : E → R be a functional of class C 1 . The Fréchet derivative of ϕ at u, ϕ (u) is an element of the dual space E * , and we denote ϕ and endowed with the natural norm 1 2 is the so-called Gagliardo (semi) norm of u. Using Fourier transform, the space H s (R N ) can also be defined by where F u denotes the Fourier transform of u. Let be the Schwartz space of rapidly decreasing C ∞ function on R N , u ∈ , one has From the results of [15], we have Then, by Proposition 3.4 and Proposition 3.6 of [15], we have From the above facts, the norms on H s (R N ) defined as follows are all equivalent.
for every u ∈ H s (R N ). Moreover, the embedding H s (R N ) ⊂ L p (R N ) is continuous for any p ∈ [2, 2 * s ] and locally compact whenever p ∈ [2, 2 * s ).

Let the homogeneous Sobolev space
This space can be equivalently defined as the completion of C ∞ 0 (R N ) under the norm Obviously, E is a reflexive Banach space. The energy functional ϕ : E → R corresponding to problem (1.1) is defined by Under our conditions, ϕ ∈ C 1 (E) and its critical points are solutions of problem (1.1).

Definition 2.1 ([32]) Let E be a Banach space and A be a subset of E. Set
A is said to be symmetric if u ∈ E implies -u ∈ E. For a closed symmetric set A which does not contain the origin, we define a genus γ (A) of A by the smallest integer k such that there exists an odd continuous mapping from A to R k \ {0}. If there does not exist such k, we define γ (A) = ∞. We set γ (∅) = 0. Let Γ k denote the family of closed symmetric subsets A of E such that 0 / ∈ A and γ (A) ≥ k.
The following result is a version of the classical symmetric mountain pass theorem [2,32]. For the proof, please see [23]. That is, there exists C > 0 such that ϕ(u n ) ≤ C. So, according to Hölder's inequality and Sobolev's inequality, one has that Since 0 < p < 1, there exists η > 0 such that On the other hand, we have that where · denotes the norm in E. Thanks to (K), we have that Then, by K ∈ L ∞ (R N ), we get Hence, by Hölder's inequality and Sobolev's inequality, we have that Using (K) again, we know that K -(x) ≥ β for all |x| > R 1 . Then we have that Since 0 < p < 1, there exists a constant C 2 > 0 such that Hence, it follows from (3.1) and (3.4) that {u n } is bounded in E.

Lemma 3.2 Suppose that (V 1 )-(V 2 ) and (K) hold. Then ϕ satisfies the PS condition on E.
Proof Let {u n } ⊂ E be such that ϕ(u n ) is bounded and ϕ (u n ) → 0 as n → ∞.
Going if necessary to a subsequence, from Lemma 2.1 we can assume that By u n → u in L p+1 (supp(ψ)) [15,30] and Lebesgue's dominated convergence theorem, one has that Hence, we have Then ϕ (u), u = 0.
Let v n = u nu, then u n = v n + u, we have that Thanks to (3.5) and Lemma 4.2 in [3], we have that So, we have that In fact, by (V 1 ), we have that In fact, thanks to (K), we have that K + (x) = 0 for all |x| > R 1 . So, by K ∈ L ∞ (R N ) and v n → 0 in L q loc (R N ), 2 ≤ q < 2 * s , we get That is, v n → 0 in E. The proof is complete.
Proof The proof is based on some ideas of Kajikiya [23] and is very similar to the one contained in [3]. For readers' convenience, we give the proof. Let R 2 and y 0 be fixed as in (K) and denote Let k ∈ N be an arbitrary number and define n = min{n ∈ N : n N ≥ k}. By planes parallel to each face of D(R 2 ), let D(R 2 ) be equally divided into n N small parts D i with 1 ≤ i ≤ n N . In fact, the length a of the edge D i is R 2 n . Let F i ⊂ D i be new cubes such that F i has the same center as that of D i . The faces of F i and D i are parallel, and the length of the edge of According to the fact that the mapping (t 1 , . . . , t k ) → k i=1 t i φ i from S k-1 to W k is odd and homeomorphic, so γ (W k ) = γ (S k-1 ) = k. Since W k is compact in E, then ∃α k > 0 such that On the other hand, by Hölder's inequality and Sobolev's embedding, we have that where r = 2 * s (1-p) 2(2 * s -p-1) . According to the above facts, there exists c k > 0 such that According to φ j (x) = 1 for x ∈ F j and |t j | = 1, one has that (3.10) By (K), one has that According to (3.8), (3.9), (3.10), and (3.11), we have that Hence, we can fix t small enough such that sup{ϕ(u), u ∈ A k } < 0, where A k = tW k ∈ Γ k .
Proof By (K), Hölder's inequality and Sobolev's embedding, as in the proof of Lemma 3.1, we have that Since 0 < p < 1, we conclude the proof.