Existence of solutions for Kirchhoff-type fractional Dirichlet problem with $p$-Laplacian

In this paper, we investigate the existence of solutions for a class of $p$-Laplacian fractional order Kirchhoff-type system with Riemann-Liouville fractional derivatives and a parameter $\lambda$. By mountain pass theorem, we obtain that system has at least one non-trivial weak solution $u_\lambda$ under some local superquadratic conditions for each given large parameter $\lambda$. We get a concrete lower bound of the parameter $\lambda$, and then obtain two estimates of weak solutions $u_\lambda$. We also obtain that $u_\lambda\to 0$ if $\lambda$ tends to $\infty$. Finally, we present an example as an application of our results.


Introduction and main results
In this paper, we are concerned with the following system where a, b, λ > 0, p > 1 and 1/p < α ≤ 1 are constants, p is an integer, u(t) = (u 1 (t), • • • , u N (t)) τ ∈ R N for a.e.
t ∈ [0, T ], T > 0, and N is a given positive integer, (•) τ denote the transpose of a vector, V (t) ∈ C([0, T ], R) with min t∈[0,T ] V (t) > 0, 0 D α t and t D α T are the left and right Riemann-Liouville fractional derivatives, respectively, φ p (s) := |s| p−2 s, ∇F (t, x) is the gradient of F with respect to x = (x 1 , • • • , x N ) ∈ R N , that is, ∇F (t, x) = ( ∂F ∂x1 , • • • , ∂F ∂xN ) τ , and F : [0, T ] × R N → R satisfies the following condition: (H0) there exists a constant δ > 0 such that F (t, x) is continuously differentiable in x ∈ R N with |x| ≤ δ for a.e.When α = 1, the operator t D α T ( 0 D α t u(t)) reduces to the usual second order differential operator −d 2 /dt 2 .Hence, if α = 1, p = 2, N = 1, λ = 1 and V (t) = 0 for a.e.t ∈ [0, T ], system (1.1) becomes the equation with where f (t, x) = ∂F (t,x) ∂x and F : [0, T ] × R → R. It is well known that equation (1.2) is related to the stationary problem of a classical model introduced by Kirchhoff [1].To be precise, in [1], Kirchhoff introduced the model where 0 ≤ y ≤ L, t ≥ 0, u is the lateral deflection, ρ is the mass density, h is the cross-sectional area, L is the length, E is the Youngs modulus and P 0 is the initial axial tension.(Notations: in model (1.3), (1.7) and (1.8) below, t is time variable and y is spatial variable, which are conventional notations in partial differential equations.One need to distinguish them to t in (1.1), (1.2), (1.4), (1.5) and (1.6) below, where t corresponds to the spatial variable x).The model (1.3) is used to describe small vibrations of an elastic stretched string.
When α < 1, 0 D α t and t D α T are the left and right Riemann-Liouville fractional derivatives, respectively, which has been given some physical interpretations in [20].Moreover, it is also applied to describe the anomalous diffusion, Lévy flights and traps in [21] and [22].Fractional differential equations have been proved to provide a natural framework in the modeling of many real phenomena such as viscoelasticity, neurons, electrochemistry, control, porous media, electromagnetic (the reader can consult [23] in which a collection of references is given).
In [23], Jiao and Zhou considered the system (1.4) They successfully applied critical point theory to investigate the existence of weak solutions for system (1.4).
To be precise, they obtained that system (1.4) has at least one weak solution when F has a quadratic growth or a superquadratic growth by using the least action principle and mountain pass theorem.Subsequently, this topic related to system (1.4) attracted lots of attention, for example, [24], [25], [26], [27], [28], [29] and references therein.It is obvious that system (1.1) is much more complicated than system (1.4) since the appearance of nonlocal term A(u(t)) and p-Laplacian term φ p (s).Recently, in [30], the following fractional Kirchhoff equation with Dirichlet boundary condition was investigated where a, b, λ > 0, f ∈ C([0, T ] × R, R).By using the mountain pass theorem in [42] and the linking theorem in [43], the authors established some existence results of nontrivial solutions for system (1.5) if f satisfies (f1) there exist constants µ > 4, 0 < τ < 2 and a nonnegative function g ∈ L 2 2−τ such that |x| σ < ∞, and some other reasonable conditions.
In [31], Chen-Liu investigated the Kirchhoff-type fractional Dirichlet problem with p-Laplacian where a, b, λ > 0, f ∈ C 1 ([0, T ] × R, R).By using the Nehari method, they established the existence result of ground state solution for system (1.6) if f satisfies and the following well-known Ambrosetti-Rabinowitz (AR for short) condition (AR) there exist two constants µ > p 2 , R > 0 such that where F (t, x) = x 0 f (t, s)ds, and some additional conditions.It is easy to see that all of these conditions (f1), (f2), (f2) ′ and (AR) imply that F (t, x) needs to have a growth near the infinity for x, and (f3) and (f4) imply that F (t, x) needs to have a growth near 0 for x.
In this paper, we investigate the existence and concentration of solutions for system (1.1) under local assumptions only near 0 for the nonlinear term F .Our work is mainly motivated by [32] and [12].In [32], Costa and Wang investigated the multiplicity of both signed and sign-changing solutions for the one-parameter family of elliptic problems where λ > 0 is a parameter, Ω is a bounded smooth domain in R N (N ≥ 3) and f ∈ C 1 (R, R).They assumed that the nonlinearity f (u) has superlinear growth in a neighborhood of u = 0 and then obtained the number of signed and sign-changing solutions which are dependent on the parameter λ.The idea in [32] has been applied to some different problems, for example, [33] and [35] for quasilinear elliptic problems with p-Laplacian operator, [34] for an elliptic problem with fractional Laplacian operator, [36] for Schrödinger equations, [11] for Neumann problem with nonhomogeneous differential operator and critical growth, and [38] for quasilinear Schrödinger equations.Especially, in [12], Li and Su investigated the Kirchhoff-type equations where λ > 0, V, Q are radial functions and f ∈ C((−δ 0 , δ 0 ), R) for some δ 0 > 0. Via the idea in [32], they also established the existence result of solutions when f (u) has superlinear growth in a neighborhood of u = 0.It is worthy to note that λ usually needs to be sufficiently large, that is, λ has a lower bound λ * .However, the concrete values of λ * are not given in these references.Similar to system (1.8), comparing with equation (1.5) and equation (1.6), we add a nonlocal term where min t∈[0,T ] V (t) > 0, and multiply V (t)φ p (u(t)) by the nonlocal part A(u(t)).Moreover, we consider the high-dimensional case, that is, 2), (1.5), (1.6) and system (1.4).
More importantly, we present a concrete value of the lower bound λ * for system (1.1) and then obtain two estimates of the solutions family {u λ } for all λ > λ * .Next, we make some assumptions for F .

Preliminaries
In this section, we mainly recall some basic definitions and results.
From the definition of E α,p 0 , it is apparent that the fractional derivative space E α,p 0 is the space of functions Remark 2.1.It is easy to see that u V defined by (1.9) is also a norm on E α,p 0 and u V and u are equivalent and is a reflexive and separable Banach space.
Lemma 2.2.( [23]) Let 0 < α ≤ 1 and 1 < p < ∞.For all u ∈ E α,p 0 , there has where Let X be a Banach space.ϕ ∈ C 1 (X, R) and c ∈ R. A sequence {u n } ⊂ X is called (PS) c sequence (named after R. Palais and S. Smale) if the sequence {u n } satisfies Then there exists a (PS) c sequence with As in [31], for each λ > 0, we can define the functional It is easy to see that the assumption (H0)-(H2) can not ensure that I λ is well defined on E α,p 0 .So we follow the idea in [32] and simply sketch the outline of proof here.We use Lemma 2.4 to complete the proof.Since F satisfies the growth condition only near 0 by (H0)-(H2), in order to use the conditions globally, we modify and extend F to F defined in section 3, and the corresponding functional is defined as Īλ .Next we prove that Īλ has mountain pass geometry on E α,p 0 .Then Lemma 2.4 implies that Īλ has a (PS) c λ sequence.Then by a standard analysis, a convergent subsequence of the (PS) c λ sequence is obtained to ensure that c λ is the critical value of Īλ .Finally, by an estimate about u λ ∞ , we obtain that the critical point u λ of Īλ with u λ ∞ ≤ δ/2 is just right the solution of system (1.1) for all λ > λ * for some concrete λ * .

Proofs
Define m(s) ∈ C 1 (R, [0, 1]) as an even cut-off function satisfying sm ′ (s) ≤ 0 and We define the variational functional corresponding to F as for all u ∈ E α,p 0 .By (H0) and the definition of F , it is easy to obtain that F satisfies (H0) ′ F (t, x) is continuously differentiable in R N for a.e.t ∈ [0, T ], measurable in t for every x ∈ R N , and for all u ∈ E α,p 0 .
Proof.By virtue of Lemma 3.1, (3.8) and θ = min{q 2 , β} > p 2 , there exists a positive constant M > 0 such that for n large enough, which shows that {u n } is bounded in E α,p 0 by p > 1.By Lemma 2.1, we can assume that, up to a subsequence, for some u λ ∈ E α,p 0 , by By the uniform convexity of E α,p 0 and u n ⇀ u λ , it follows from the Kadec-Klee property (see [41]) and (2.14), By the continuity of Īλ , we obtain that Īλ (u) = c λ , where c λ is defined by (3.9).Then (3.3) implies that Next, we show that u λ precisely is the nontrivial weak solution of system (1.1) for any given λ > λ * .In order to get this, we need to make an estimate for the critical level c λ .We introduce the functional J where C * is defined by (1.13) which is obviously independent of λ.

Definition 2 . 1 .
(Left and Right Riemann-Liouville Fractional Integrals [39, 24]) Let f be a function defined on [a, b].The left and right Riemann-Liouville fractional integrals of order γ > 0 for function f denoted by a D −γ t f (t) and t D −γ b f (t) , respectively, are defined by provided the right-hand sides are pointwise defined on [a, b], where Γ > 0 is the Gamma function.Definition 2.2.(Left and Right Riemann-Liouville Fractional Derivatives[39, 24]) Let f be a function defined on [a, b].The left and right Riemann-Liouville fractional derivatives of order γ > 0 for function f denoted by a D γ t f (t) and t D γ b f (t), respectively, are defined by is absolutely continuous and has an α-order left and right Riemann-Liouville fractional derivative 0 D α t u ∈ L p ([0, T ], R N ) and u(0) = u(T ) = 0 and one can define the norm on L p ([0, T ], R N ) as u L p = by the uniform convexity of L p .