Nontrivial Solutions for Time Fractional Nonlinear Schrödinger-Kirchhoff Type Equations

In recent years, there has been a great interest in studying problems involving fractional Schrödinger equations [1– 5], Kirchhoff type equations [6–8], fractional Navier-Stokes equations [9, 10], and fractional ordinary differential equations and Hamiltonian systems [11–17], and so forth. For further details and applications, we refer the reader to [18, 19] and the references cited therein. On the other hand, the integer-order SchrödingerKirchhoff type equations have also been investigated by many authors; for example, see [20–23]. In fact, SchrödingerKirchhoff type equations play an important role in modelling several physical and biological systems. However, to the best of our knowledge, the existence of solutions to the time fractional Schrödinger-Kirchhoff type equations has yet to be addressed. The objective of the present paper is to study time fractional Schrödinger-Kirchhoff type equation of the form

On the other hand, the integer-order Schrödinger-Kirchhoff type equations have also been investigated by many authors; for example, see [20][21][22][23]. In fact, Schrödinger-Kirchhoff type equations play an important role in modelling several physical and biological systems. However, to the best of our knowledge, the existence of solutions to the time fractional Schrödinger-Kirchhoff type equations has yet to be addressed.
The objective of the present paper is to study time fractional Schrödinger-Kirchhoff type equation of the form where ∈ (1/2, 1], −∞ and ∞ , respectively, denote left and right Liouville-Weyl fractional derivatives of order on R, , > 0 are constants, > 0 is parameter, > 1, ∈ (R × R, R), and : R → R + is a potential function. The rest of the paper is organized as follows. Section 2 contains preliminary concepts of fractional calculus and fractional Sobolev space, while some important lemmas, which are needed in the proof of main results, are obtained in Section 3. We present our main results in Section 4.

Preliminaries
In this section, we recall important definitions and concepts of fractional calculus and then prove certain results about fractional Sobolev space (R) related to our study of the problem at hand.
Definition 1 (see [24]). The left and right Liouville-Weyl fractional integrals of order ∈ (0, 1) on R are defined by respectively, where ∈ R. The left and right Liouville-Weyl fractional derivatives of order ∈ (0, 1) on R are defined by The definitions (3) may be written in an alternative form as follows: Also, we define the Fourier transform F( )( ) of ( ) as For any > 0, we define the seminorm and norm, respectively, as [16] and let the space −∞ (R) denote the completion of ∞ 0 (R) with respect to the norm ‖ ⋅ ‖ −∞ . Next, for 0 < < 1, we give the relationship between classical fractional Sobolev space (R) and −∞ (R), where (R) is defined by with the norm and seminorm Observe that the spaces (R) and −∞ (R) are equal and have equivalent norms (see [16]). Therefore, we define The space is a reflexive and separable Hilbert space with the inner product and the corresponding norm Define the space with the norm
Discrete Dynamics in Nature and Society 3 In the sequel, we need the following assumptions.

Moreover, if (V1) and (V2) hold, then the embedding
Proof. Clearly, the chain of embeddings → → 2 (R) is continuous and consequently one can obtain (23). Also in view of (V1), (V2), and following the method of proof similar to that of Lemma 2.2 in [15], the embedding → 2 (R) is compact.
Proof. The proof is similar to that of Theorem 2.1 in [16], so we omit it.
In order to establish the main results, we need the following known Theorems. (ii) there is an ∈ \ (0) such that ( ) ≤ 0.
Then possesses a critical value ≥ . Moreover can be characterized as

Some Lemmas
Recall that ∈ is said to be a weak solution of problem (1) if and the energy functional , : → R is given by the formula where ( , ) = ∫ 0 ( , ) .
In view of assumptions (V1) and (F1), the functional , is of class 1 ( , R) and by similar method in Theorem 4.1 in [27] and the definition of Gâteaux derivative, one can get (37)
Discrete Dynamics in Nature and Society 7 Let { } be a total orthonormal basis of 2 (R) and define = R , ∈ N, (51) Lemma 12. Assume that (V1) holds. Then, for 2 < < +∞, Proof. The proof is similar to that of Lemma 3.8 in [28]. So it is omitted.

Existence of Weak Solutions
In this section, we present our main results.
Proof. We know that , (0) = 0, and it is even by (F5). Let = and and be as defined in Section 2. By Lemmas 11, 13, and 14, it follows that , satisfies all the condition of the Theorem 10. Therefore, problem (1) has infinitely many nontrivial weak solutions whenever > 0 is sufficiently large.