Ground State Solutions of Kirchhoff-type Fractional Dirichlet Problem with $p$-Laplacian

We consider the Kirchhoff-type $p$-Laplacian Dirichlet problem containing the left and right fractional derivative operators. By using the Nehari method in critical point theory, we obtain the existence theorem of ground state solutions for such Dirichlet problem.


Introduction
In the present paper, we discuss the existence of ground state solutions for the Kirchhoff-type fractional Dirichlet problem with p-Laplacian of the form where a, b > 0, p > 1 are constants, 0 D α t and t D α T are the left and right Riemann-Liouville fractional derivatives of order α ∈ (1/p, 1] respectively, 1 Corresponding author.  The Kirchhoff equation ( [20]) is an extension of the wave equation which comes from the free vibrations of elastic strings, and takes into account the changes in length of the string produced by transverse vibrations. In addition, the fractional order models are more appropriate than the integer order models in real world owing to the fractional derivatives offer an wonderful tool to describe the memory and hereditary properties of a great deal of processes and materials ( [12,15,16,21,24]). Moreover the p-Laplacian ( [22]) often appears in non-Newtonian fluid theory, nonlinear elastic mechanics and so on.
Notice that, when a = 1, b = 0 and p = 2, the left hand side of equation of BVP (1), which is nonlinear and nonlocal, reduces to the linear operator t D α T 0 D α t , and further reduces to the local operator −d 2 /dt 2 when α = 1. In recent years, there are many authors to study the fractional boundary value problems (BVPs for short) ( [1,3,4,7,11,17]) and the Kirchhoff equations ( [2,6,8,10,23,25]), and obtain numerous important results. In addition, the models containing left and right fractional derivatives are recently gaining more attention ( [5,9,13,14,18,27]) because of the applications in physical phenomena exhibiting anomalous diffusion.
Motivated by the above works, in this paper, we dicuss the existence of nontrivial ground state solutions for BVP (1). The main tool used here is the Nehari method.
For the nonlinearity f , we make the following assumptions throughout this paper. ( where F (t, x) =  The rest of this paper is organized as follows. Some preliminary results are presented in Section 2. Section 3 is devoted to prove Theorem 1.1.

Preliminaries
In this section, we present some basic definitions and notations of the fractional calculus ( [19,26]). Moreover we introduce a fractional Sobolev space and some properties of this space ( [18]).
provided that the right-hand side integrals are pointwise defined on [a, b], where Γ(·) is the Gamma function.
, the left and right Riemann-Liouville fractional derivatives of order γ of a function u : [a, b] → R are given by Remark 2.3. When γ = 1, one can obtain from Definition 2.1 and 2.2 that Lemma 2.6 (see [18]). Let 0 < α ≤ 1 and 1 < p < ∞. The fractional derivative space E α,p 0 is a reflexive and separable Banach space.
Remark 2.8. By (2), we can consider the space E α,p 0 with norm in what follows.

Ground state solutions of BVP (1)
The purpose of this section is to prove our main result via the Nehari method. To this end, we are going to set up the corresponding variational framework of BVP (1).
Define the functional I : E α,p 0 → R by Then there is one-to-one correspondence between the critical points of energy functional I and the weak solutions of BVP (1). It is easy to check from (3), (4) and f ∈ C 1 ([0, T ]×R, R) that the functional I is well defined on E α,p 0 and is second-order continuously Fréchet differentiable, that is, I ∈ C 2 (E α,p 0 , R). Furthermore we have which yields

Now let us define
Thus we know that any non-zero critical point of I must be on N . In the following, for simplicity, let where f ′ 2 (t, x) = ∂f (t,x) ∂x . Then, for u ∈ N , we have which means that N has a C 1 structure and is a manifold.
Lemma 3.1. Assume (H 1 ) holds. If u ∈ N is a critical point of I| N , then I ′ (u) = 0, that is, N is a natural constraint for I.
Proof. If u ∈ N is a critical point of I| N , then there exists a Lagrange multiplier λ ∈ R such that I ′ (u) = λG ′ (u).
In order to discuss the critical points of I| N , we need to investigate the structure of N .
Then, from what we have proved, g u has at least one maximum point s(u) with maximum value greater than σ > 0. Next, we prove that g u has a unique critical point for s ∈ R + , which then must be the global maximum point. Consider a critical point of g u , one has which together with (5) yields Hence, if s is a critical point of g u , then it must be a strict local maximum point. This ensures the uniqueness of critical point of g u . Finally, from we obtain that, if s is a critical point of g u , then su ∈ N .
Let us define Then we get from (7) that Proof. By Lemma 2.9, we obtain that the functional is weakly continuous. Thus, as the sum of a convex continuous functional and a weakly continuous one, I is weakly lower semi-continuous on E α,p 0 . Let {u k } ⊂ N be a minimizing sequence of I, then one has Next, we prove that {u k } is bounded in E α,p 0 . Based on the continuity of µF (t, x) − xf (t, x) and (H 3 ), we see that there exists a constant c > 0 such that Thus, from (11), we have Hence it follows from µ > p 2 that {u k } is bounded in E α,p 0 . Since E α,p 0 is a reflexive Banach space (see Lemma 2.6), up to a subsequence, we can assume u k ⇀ u in E α,p 0 . Moreover, from Lemma 2.9, one has u k → u in C([0, T ], R). Next, we prove u = 0. By (H 2 ), we get that for ∀ε > 0, there exists a constant δ > 0 such that Then, assume u k ∞ ≤ δ, we obtain from (3), (4) and u k ∈ N that which is a contradiction. Hence we have u ∞ = lim k→∞ u k ∞ ≥ δ > 0, and then u = 0. Thus, by Lemma 3.2, there exists s ∈ R + such that su ∈ N . Therefore, together with the fact that I is weakly lower semi-continuous, we obtain m ≤ I(su) ≤ lim k→∞ I(su k ) ≤ lim k→∞ I(su k ).
Finally, for ∀u k ∈ N , we see from (9) and (10)  That is, m is achieved at su ∈ N . Now we give the proof of our main result.
Proof of Theorem 1.1. By Lemma 3.3, we get u * ∈ N such that I(u * ) = m = inf N I > 0, that is, u * is a non-zero critical point of I| N . Then, from Lemma 3.1, we know I ′ (u * ) = 0, and so u * is a nontrivial ground state solution of BVP (1).