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Ground state solutions of Kirchhoff-type fractional Dirichlet problem with p-Laplacian
Advances in Difference Equations volume 2018, Article number: 436 (2018)
Abstract
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.
1 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, \({}_{0}D_{t}^{\alpha}\) and \({}_{t}D_{T}^{\alpha}\) are the left and right Riemann–Liouville fractional derivatives of order \(\alpha\in(1/p,1]\), respectively, \(\phi_{p}:\mathbb{R}\rightarrow\mathbb {R}\) is the p-Laplacian defined by
and \(f\in C^{1}([0,T]\times\mathbb{R},\mathbb{R})\).
The Kirchhoff equation [21] 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 fact that the fractional derivatives offer a wonderful tool to describe the memory and hereditary properties of a great deal of processes and materials [12, 15, 16, 22, 25]. Moreover, the p-Laplacian [23] 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 \({}_{t}D_{T}^{\alpha}{{}_{0}D_{t}^{\alpha}}\), and further reduces to the local operator \(-d^{2}/dt^{2}\) when \(\alpha=1\).
In recent years, there have been 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, 24, 26], and to obtain numerous important results. In addition, the models containing left and right fractional derivatives have been recently gaining more attention [5, 9, 13, 14, 18, 19, 28] because of the applications in physical phenomena exhibiting anomalous diffusion.
Motivated by the above works, in this paper, we discuss 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.
- (H1):
-
The mapping \(x\rightarrow f(t,x)/|x|^{p^{2}-1}\) is strictly increasing on \(\mathbb{R}\setminus\{0\}\) for \(\forall t\in[0,T]\).
- (H2):
-
\(f(t,x)=o(|x|^{p-1})\) as \(|x|\rightarrow0\) uniformly for \(\forall t\in[0,T]\).
- (H3):
-
There exist two constants \(\mu>p^{2}\), \(R>0\) such that
$$ 0< \mu F(t,x)\leq xf(t,x), \quad \forall t\in[0,T], x\in\mathbb{R} \mbox{ with } |x|\geq R, $$where \(F(t,x)=\int_{0}^{x}f(t,s)\,ds\).
Now we state our main result.
Theorem 1.1
Let (H1)–(H3) be satisfied. Then BVP (1) possesses at least one nontrivial ground state solution.
The rest of this paper is organized as follows. Some preliminary results are presented in Sect. 2. Section 3 is devoted to proving Theorem 1.1.
2 Preliminaries
In this section, we present some basic definitions and notations of the fractional calculus [20, 27]. Moreover we introduce a fractional Sobolev space and some properties of this space [19].
Definition 2.1
For \(\gamma>0\), the left and right Riemann–Liouville fractional integrals of order γ of a function \(u:[a,b]\rightarrow\mathbb {R}\) are given by
provided that the right-hand side integrals are pointwise defined on \([a,b]\), where \(\Gamma(\cdot)\) is the gamma function.
Definition 2.2
For \(n-1\leq\gamma< n\) (\(n\in\mathbb{N}\)), the left and right Riemann–Liouville fractional derivatives of order γ of a function \(u:[a,b]\rightarrow\mathbb{R}\) are given by
Remark 2.3
When \(\gamma=1\), one can obtain from Definitions 2.1 and 2.2 that
where \(u'\) is the usual first-order derivative of u.
Definition 2.4
For \(0<\alpha\leq1\) and \(1< p<\infty\), the fractional derivative space \(E{{}_{0}^{\alpha,p}}\) is defined by the closure of \(C_{0}^{\infty}((0,T),\mathbb{R})\) with respect to the following norm:
where \(\|u\|_{L^{p}}= (\int_{0}^{T}|u(t)|^{p}\,dt )^{1/p}\) is the norm of \(L^{p}((0,T),\mathbb{R})\).
Remark 2.5
It is obvious that, for \(u\in E{{}_{0}^{\alpha,p}}\), one has
Lemma 2.6
(see [19])
Let \(0<\alpha\leq1\) and \(1< p<\infty\). The fractional derivative space \(E{{}_{0}^{\alpha,p}}\) is a reflexive and separable Banach space.
Lemma 2.7
(see [19])
Let \(0<\alpha\leq1\) and \(1< p<\infty\). For \(u\in E_{0}^{\alpha,p}\), one has
where
is a constant. Moreover, if \(\alpha>1/p\), then
where \(\|u\|_{\infty}=\max_{t\in[0,T]}|u(t)|\) is the norm of \(C([0,T],\mathbb{R})\) and
are two constants.
Remark 2.8
By (2), we can consider the space \(E_{0}^{\alpha,p}\) with the norm
in what follows.
Lemma 2.9
(see [19])
Let \(1/p<\alpha\leq1\) and \(1< p<\infty\). The imbedding of \(E_{0}^{\alpha ,p}\) in \(C([0,T],\mathbb{R})\) is compact.
3 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 \(I:E_{0}^{\alpha,p}\rightarrow\mathbb{R}\) 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 \(f\in C^{1}([0,T]\times \mathbb{R},\mathbb{R})\) that the functional I is well defined on \(E_{0}^{\alpha,p}\) and is second-order continuously Fréchet differentiable, that is, \(I\in C^{2}(E_{0}^{\alpha,p},\mathbb{R})\). Furthermore, we have
which yields
Now let us define
where
Thus we know that any non-zero critical point of I must be on \(\mathcal{N}\). In the following, for simplicity, let
From (H1), one has
where \(f'_{2}(t,x)=\frac{\partial f(t,x)}{\partial x}\). Then, for \(u\in \mathcal{N}\), we have
which means that \(\mathcal{N}\) has a \(C^{1}\) structure and is a manifold.
Lemma 3.1
Assume that (H1) holds. If \(u\in\mathcal{N}\) is a critical point of \(I|_{\mathcal{N}}\), then \(I'(u)=0\), that is, \(\mathcal{N}\) is a natural constraint for I.
Proof
If \(u\in\mathcal{N}\) is a critical point of \(I|_{\mathcal{N}}\), then there exists a Lagrange multiplier \(\lambda\in\mathbb{R}\) such that
Then we get
which together with (6) yields \(\lambda=0\). So we have \(I'(u)=0\). □
In order to discuss the critical points of \(I|_{\mathcal{N}}\), we need to investigate the structure of \(\mathcal{N}\).
Lemma 3.2
Assume that (H1)–(H3) hold. For each \(u\in E_{0}^{\alpha ,p}\setminus\{0\}\), there is unique \(s=s(u)\in\mathbb{R}^{+}\) such that \(su\in\mathcal{N}\).
Proof
First, we claim that there exist constants \(\rho,\sigma>0\) such that
where \(B_{\rho}(0)\) is an open ball in \(E_{0}^{\alpha,p}\) with the radius ρ and centered at 0, and \(\partial B_{\rho}(0)\) denotes its boundary. That is, by \(I(0)=0\), 0 is a strict local minimizer of I. In fact, from (H2), there are two constants \(0<\varepsilon<1\), \(\delta >0\) such that
where \(C_{p}>0\) is a constant defined in (2). Let \(\rho=\delta /C_{\infty}\) and \(\sigma=\varepsilon a^{p-1}\rho^{p}/p\), where \(C_{\infty}>0\) is a constant defined in (3). Then, by (3) and (4), one has
which together with (2), (4), and (8) yields
Second, we claim that \(I(\xi u)\rightarrow-\infty\) as \(\xi\rightarrow \infty\). In fact, from (H3), a simple argument can show that there are two constants \(c_{1},c_{2}>0\) such that
Thus, for each \(u\in E_{0}^{\alpha,p}\setminus\{0\}\), \(\xi\in\mathbb{R}^{+}\), we obtain from \(\mu>p^{2}\) that
Let
Then, from what we have proved, \(g_{u}\) has at least one maximum point \(s(u)\) with maximum value greater than \(\sigma>0\). Next, we prove that \(g_{u}\) has a unique critical point for \(s\in\mathbb{R}^{+}\), which then must be the global maximum point. Considering 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 a critical point of \(g_{u}\). Finally, from
we obtain that, if s is a critical point of \(g_{u}\), then \(su\in\mathcal {N}\). □
Let us define
Then we get from (7) that
Lemma 3.3
Assume that (H1)–(H3) hold. Then there exists \(u^{*}\in\mathcal {N}\) such that \(I(u^{*})=m\).
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 \(E_{0}^{\alpha,p}\).
Let \(\{u_{k}\}\subset\mathcal{N}\) be a minimizing sequence of I, then one has
Next, we prove that \(\{u_{k}\}\) is bounded in \(E_{0}^{\alpha,p}\). Based on the continuity of \(\mu F(t,x)-xf(t,x)\) and (H3), we see that there exists a constant \(c>0\) such that
Thus, from (11), we have
Hence it follows from \(\mu>p^{2}\) that \(\{u_{k}\}\) is bounded in \(E_{0}^{\alpha,p}\).
Since \(E_{0}^{\alpha,p}\) is a reflexive Banach space (see Lemma 2.6), up to a subsequence, we can assume \(u_{k}\rightharpoonup u\) in \(E_{0}^{\alpha,p}\). Moreover, from Lemma 2.9, one has \(u_{k}\rightarrow u\) in \(C([0,T],\mathbb{R})\). Next, we prove \(u\neq0\). By (H2), we get that, for \(\forall\varepsilon>0\), there exists a constant \(\delta>0\) such that
Then, assume \(\|u_{k}\|_{\infty}\leq\delta\), we obtain from (3), (4), and \(u_{k}\in\mathcal{N}\) that
which is a contradiction from the arbitrariness of ε. Hence we have
and then \(u\neq0\). Thus, by Lemma 3.2, there exists \(s\in\mathbb {R}^{+}\) such that \(su\in\mathcal{N}\). Therefore, together with the fact that I is weakly lower semi-continuous, we obtain
Finally, for \(\forall u_{k}\in\mathcal{N}\), we see from (9) and (10) that \(s=1\) is the global maximum point of \(g_{u_{k}}\). So one has
which together with (12) implies
That is, m is achieved at \(su\in\mathcal{N}\). □
Now we give the proof of our main result.
Proof of Theorem 1.1
By Lemma 3.3, we get \(u^{*}\in\mathcal{N}\) such that \(I(u^{*})=m=\inf_{\mathcal{N}}I>0\), that is, \(u^{*}\) is a non-zero critical point of \(I|_{\mathcal{N}}\). Then, from Lemma 3.1, we know \(I'(u^{*})=0\), and so \(u^{*}\) is a nontrivial ground state solution of BVP (1). □
References
Agarwal, R.P., O’Regan, D., Stanek, S.: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371, 57–68 (2010)
Arosio, A., Panizzi, S.: On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348, 305–330 (1996)
Bai, Z., Lü, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495–505 (2005)
Benchohra, M., Hamani, S., Ntouyas, S.K.: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 71, 2391–2396 (2009)
Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional-order governing equation of Lévy motion. Water Resour. Res. 36, 1413–1423 (2000)
Bernstein, S.: Sur une classe d’équations fonctionnelles aux déivés partielles. Bull. Acad. Sci. URSS Sér. Math. 4, 17–26 (1940)
Bisci, G.M., Repovs, D.: Higher nonlocal problems with bounded potential. J. Math. Anal. Appl. 420, 167–176 (2014)
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.: Global existence and uniform decay rates for the Kirchhoff–Carrier equation with nonlinear dissipation. Adv. Differ. Equ. 6, 701–730 (2001)
Cresson, J.: Inverse problem of fractional calculus of variations for partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 15, 987–996 (2010)
D’Ancona, P., Spagnolo, S.: Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 108, 247–262 (1992)
Darwish, M.A., Ntouyas, S.K.: On initial and boundary value problems for fractional order mixed type functional differential inclusions. Comput. Math. Appl. 59, 1253–1265 (2010)
Diethelm, K., Freed, A.D.: On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity. In: Keil, F., Mackens, W., Voss, H., Werther, J. (eds.) Scientific Computing in Chemical Engineering II—Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, pp. 217–224. Springer, Heidelberg (1999)
Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, 558–576 (2006)
Fix, G.J., Roop, J.P.: Least squares finite-element solution of a fractional order two-point boundary value problem. Comput. Math. Appl. 48, 1017–1033 (2004)
Glockle, W.G., Nonnenmacher, T.F.: A fractional calculus approach of self-similar protein dynamics. Biophys. J. 68, 46–53 (1995)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Jiang, W.: The existence of solutions to boundary value problems of fractional differential equations at resonance. Nonlinear Anal. 74, 1987–1994 (2011)
Jiao, F., Zhou, Y.: Existence of solutions for a class of fractional boundary value problems via critical point theory. Comput. Math. Appl. 62, 1181–1199 (2011)
Jiao, F., Zhou, Y.: Existence results for fractional boundary value problem via critical point theory. Int. J. Bifurc. Chaos 22, Article ID 1250086 (2012)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)
Kirchner, J.W., Feng, X., Neal, C.: Fractal stream chemistry and its implications for contaminant transport in catchments. Nature 403, 524–526 (2000)
Leibenson, L.S.: General problem of the movement of a compressible fluid in a porous medium. Izv. Akad. Nauk Kirg. SSSR 9, 7–10 (1983)
Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations. Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977. North-Holland Math. Stud., vol. 30, pp. 284–346. North-Holland, Amsterdam (1978)
Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348. Springer, Wien (1997)
Pohozăev, S.I.: A certain class of quasilinear hyperbolic equations. Mat. Sb. 96, 152–166 (1975)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, New York (1993)
Zhang, Z., Yuan, R.: Infinitely-many solutions for subquadratic fractional Hamiltonian systems with potential changing sign. Adv. Nonlinear Anal. 4, 59–72 (2015)
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The authors sincerely thank the editors and anonymous referees for the careful reading of the original manuscript and for valuable comments, which have improved the quality of our work.
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This work was supported by the Fundamental Research Funds for the Central Universities (2017XKQY089).
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Chen, T., Liu, W. Ground state solutions of Kirchhoff-type fractional Dirichlet problem with p-Laplacian. Adv Differ Equ 2018, 436 (2018). https://doi.org/10.1186/s13662-018-1902-6
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DOI: https://doi.org/10.1186/s13662-018-1902-6