Multiplicity solutions of a class fractional Schrödinger equations

Abstract In this paper, we study the existence of nontrivial solutions to a class fractional Schrödinger equations (−Δ)su+V(x)u=λf(x,u)inRN, $$ {( - \Delta )^s}u + V(x)u = \lambda f(x,u)\,\,{\rm in}\,\,{\mathbb{R}^N}, $$ where (−Δ)su(x)=2limε→0∫RN∖Bε(X)u(x)−u(y)|x−y|N+2sdy,x∈RN $ {( - \Delta )^s}u(x) = 2\lim\limits_{\varepsilon \to 0} \int_ {{\mathbb{R}^N}\backslash {B_\varepsilon }(X)} {{u(x) - u(y)} \over {|x - y{|^{N + 2s}}}}dy,\,\,x \in {\mathbb{R}^N} $ is a fractional operator and s ∈ (0, 1). By using variational methods, we prove this problem has at least two nontrivial solutions in a suitable weighted fractional Sobolev space.


Introduction
There are a lot of interesting problems in the standard framework of the Laplacian (and, more generally, of uniformly elliptic operators), widely studied in the literature. A natural question is whether or not the existence results got in this classical context can be extended to the non-local framework of the fractional Laplacian type operators.
First, we focus on the so-called fractional SchrR odinger equation where .x; t / 2 R .0; C1/, and V W R N ! R an external potential function. The fractional Laplacian operator . 4/ s u with 0 < s < 1 of a function 2`is defined by =.. 4/ s /. / D j j 2s =. / ; where`denotes the Schwartz space of rapidly decreasing C 1 functions in R N = is the Fourier transform, i.e., where u solves the elliptic equation x/ j j p 1 D 0: In this paper we study the following fractional SchrR odinger equation where is a parameter. The fractional SchrR odinger equations are an important model in the study of the fractional quantum mechanics. Recently, this has been widely investigated by many authors in the last decades, see [3-15, 18-20, 22-26] and references therein. In most of the papers mentioned above the existence of positive solutions has been considered under different assumptions on V and f . We refer the reader to [16,17] and to the references included for a selfcontained overview of the basic properties of fractional Sobolev spaces.
In [3], the author used the Ekeland variational principle and the mountain pass theorem to obtain a nontrivial solution for (2) with the Ambrosetti-Rabinowitz condition: there is a constant > 2 such that f .x; t /dt Ä sf .x; s/; for all x 2 R N ; s 2 R n f0g: .AR/ In [4][5][6], the authors used variant fountain theorems and the Z 2 version of mountain pass theorem to establish the existence of infinitely many nontrivial high-energy or small-energy solutions for (2). In [7], the authors used the concentration compactness principle to show that (2) (V .x/ D 1) has at least two nontrivial radial solutions without the .AR/ condition. In [25], Bisci and Rǎdulescu studied the following equation where denotes the Lebesgue measure in R N , B.y; r/ denotes the open ball in R N with center y and radius r > 0, and established two existence theorems for two nontrivial solutions when the nonlinearity f and g satisfy where W 2 L 1 .R N / T L 1 .R N / and q 2 .0; 1/. Motivated by the above papers, we shall assume that f .x; t / satisfies the following conditions: jf .x; t /j t D 0: where F .x; u/ D R u 0 f .x; s/ds. And on the potential function V we assume (V 1 ) V 2 C.R N ; R/ is a positive weight and there exists a constant V 0 > 0 such that V .x/ V 0 for all The main purpose of this paper is to generalize the main results of [24,25]. Now we state our main results: This paper is organized as follows. In Section 2, we will give some notation and introduce our main idea. In Section 3, we prove Theorem 1.
does not satisfy .h/, but it satisfies our conditions .f 2 /.

Preliminary
In this section, for the reader's convenience, we collect some basic results that will be used in the forthcoming sections. In the following, we denote the N -dimensional Lebesgue measure of a set A R N by meas.A/. We use " * " and " ! " to denote weak and strong convergence in the related function space. For any Eucliden space .R N ; j j/ we will denote B D fu 2 R N W ju u 0 j Ä g.u 0 2 R N ; > 0/:

Variational formulation of the problem
First we introduce a variational setting for problem (2). The Gagliardo seminorm is defined for all measurable function u W R N ! R by ju.x/ u.y/j 2 jx yj N C2s dxdy/ we can define the fractional Sobolev space The space H s .R N / can be described by means of the Fourier transform. Indeed, it is defined by In this case, the inner product and the norm are defined as In order to give the relationship of the above two norms, we introduce the definition of Schwartz function`, that is, the rapidly decreasing C 1 function on R N . If u 2`, the fractional Laplacian . 4/ s acts on u as where C.N; s/ is the following constant In [17], it is proved that . 4/ s u D = 1 .j j 2s =u/; and that As a consequence, the norms on H s .R N / defined above are all equivalent.
Moreover, it is easy to see that .H s .R N /; kuk 2;s / is a uniformly convex Banach space and the embedding H s .R N / ,! L # .R N / is continuous for any # 2 OE2; 2 s by Theorem 6.7 of [17], that is, there exists a positive constant C such that kuk L # .R N / Ä C kuk 2;s for all u 2 H s .R N /: We will work in the following linear subspace We also know that .E; k k E / is a uniformly convex Banach space, see ( [27] Lemma 10). The dual space of .E; k k E / is denoted by .E ; k k E /. h ; i denotes the pairing between E and its dual space E . We define the nonlinear operatorˆ: E ! E as It can be seen that a weak solution of problem (2) is a function u 2 E such that for all v 2 E. Clearly, for all u 2 E , hˆ.u/; ui D kuk 2 E . Now we introduce the minimal hypotheses on the reaction term of (2): a.e. in R N and for all t 2 R (a > 0).
We set for all u 2 E defined on E, and for any u 2 E, it holds that : By hypotheses .H /, we have I 2 C 1 .E/. We denote by K.I / the set of all critical points of I . If u 2 K.I /, then (4) hold for all v 2 E, i.e., u is a weak solution to (2).

Some preliminary lemmas
We first recall some embedding results related to the fractional Sobolev space E, for more details, see [27]. Proof.
Since v j is bounded in E, by lemma 2.1 we have v j is bounded in L # .R N /. Then by the reflexivity of E, up to a subsequence, we get that v j * v weakly in E \ L # .R N / as j ! 1. Next we prove that v j ! v st rongly i n L # .R N /: Now, for any " > 0, there exists R 1 > 0 such that for all R R 1 , since 1 V .x/ 2 L 1 .R N / by assumption .V 2 /. Then, by H R older inequality , we can get that for all R R 1 .
by Theorem 6.7 of [17]. Since 2 Ä # < 2 s , by Corollary 7.2 of [17], we obtain v j ! v strongly in L # .B R 1 .0//, i.e. for above " > 0, there exists N 1 > 0 such that for all j N 1 . Combining (7) and (8), for all j N 1 , by interpolation inequality we have where C denotes various positive constants, and Â 2 .0; 1/ such that Proposition 2.3. Let .E; k k E / be a Banach space and its dual space .E , k k E / and I 2 C 1 .E; R 1 / (1) For c 2 R 1 , we say that I satisfies the C c condition if for any sequence fx n g E with I.x n / ! c; kI 0 .x n /k E .1 C kx n k E / ! 0: (2) For c 2 R 1 , we say that I satisfies the .PS / condition if for any sequence fx n g E with I.x n / ! c; I 0 .x n / ! 0 i n E : The following critical points theorem was established in [1,2]. Then there is an open interval ƒ OE0; N a and a number Ä > 0 such that for each 2 ƒ, the equation J 0 .z/ D 0 admits at least three solutions in E having norm less than Ä.

The main results and its proofs
In this section, we are ready to prove the Theorem 1.1. In the sequel, for the sake of clarity, we divide the proof of the theorem into several steps. We write the functional J as follows: J.u/ Dˆ.u/ ‰.u/; whereˆ.
F .x; u/dx: We first give two preliminary lemmas.
Proof. By .f 1 / and .f 2 /, there is a positive constants ı such that for a fixed # 2 OE2; 2 s / and for all t 2 R N .
Step 1. We prove for every 2 R, the functional J is coercive and satisfies the compactness .PS/ condition. Let us fix 2 R. By .f 2 /, there is a positive constant ı such that for every jtj ı. So, we get for every u 2 E. Then the functional J is bounded from below and J.u/ ! C1 when kuk E ! C1. Hence J is coercive. Now we prove that J satisfies the .PS / condition. Let fu n g E be a .PS / sequence for J.u/, that is Taking into account the coercivity of J , the sequence fu n g is necessarily bounded in E. Assume without loss of generality that fu n g converges to u weakly in E, and by Lemma 2.2, we may assume that where # 2 OE2; 2 s /.
To prove that fu n g converges strongly to u in E, we first introduce a simple notation. Let ' 2 E be fixed and denote by B ' the linear functional on E defined by Due to .f 1 / and .f 2 /, there exists C " > 0 such that Then, by (10), we get ."ju n j C C " ju n j # 1 C "juj C C " juj # 1 /.u n u/dx Ä"ku n k 2 ku n uk 2 C "kuk 2 ku n uk 2 C C " ku n k # 1 # ku n uk # C C " kuk # 1 # ku n uk # !0: Obviously, hJ 0 .u n / J 0 .u/; u n ui ! 0 as n ! 1, since u n * u in E and J 0 .u n / ! 0 in E : Hence, (10) and (11) give as n ! 1 o.1/ DhJ 0 .u n / J 0 .u/; u n ui That is as n ! 1. Therefore, J satisfies the .PS / condition.
Step 2. We claim that the functional J.u/ is weakly lower semicontinuous on E.
The functional is sequentially weakly lower semicontinuous on E.
Thus it is enough to prove that the map is sequentially weakly continuous on E. To this aim fix u n 2 E and u 2 E such that u n * u in E as n ! 1. Then, by Lemma 2.2, without loss of generality, we can assume that u n ! u strongly in L # .R N / for 2 Ä # < 2 s and a.e. in R N . It is dominated by some function h # 2 L # .R N /, that is, ju n j Ä h # .x/ a:e: x 2 R N for any n 2 N and for any # 2 OE2; 2 s /.
By .f 3 /, there exists t 0 2 R N such that F .t 0 / > 0. Further, let 0 2 OE0; 1 be such that Indeed, since (recalled the definition of u t 0 ), it follows that where ! N denotes the volume of the unit ball in R N . Then, we have that By Lemma 3.2, we haveˆ. u t 0 / Ä C; By Lemma 3.1, there exists % > 0 such that the function u t 0 2 E verifies the following conditions: and sup Then by (14), we have % <ˆ.u t 0 /; as well as, by (12) and (13), it follows that: .u t 0 / : Hence, (15) and (16) give sup u2S %ˆ.
u/ Ä % ‰.u t 0 / .u t 0 / : By choosing u 0 D u t 0 , we get that % <ˆ.u 0 /, and sup .u/<% ‰.u/ < % ‰.u 0 / .u 0 / . Set Clearly, 1 C % > 1, and N a is a positive constant. It's easy to see that inf a and a number Ä > 0 such for all 2 ƒ, the functional J admits three solutions in E having norm less than Ä. Since one of them may be a trivial one, we have at least two distinct, nontrivial weak solutions of problem (2). Remark 3.3. Our hypotheses are similar to those employed by Bisci-R M adulescu [24] of (2) on bounded subset of R N . Moreover, .f 2 / is weaker than the condition .h/. So Theorem 1.1 extends Theorem 1 of [24] and Theorem 2 of [25].