Browse > Article
http://dx.doi.org/10.4134/BKMS.b181172

NEHARI MANIFOLD AND MULTIPLICITY RESULTS FOR A CLASS OF FRACTIONAL BOUNDARY VALUE PROBLEMS WITH p-LAPLACIAN  

Ghanmi, Abdeljabbar (Department of Mathematics Faculty of Science and Arts Khulais University of Jeddah)
Zhang, Ziheng (School of Mathematical Sciences Tianjin Polytechnic University)
Publication Information
Bulletin of the Korean Mathematical Society / v.56, no.5, 2019 , pp. 1297-1314 More about this Journal
Abstract
In this work, we investigate the following fractional boundary value problems $$\{_tD^{\alpha}_T({\mid}_0D^{\alpha}_t(u(t)){\mid}^{p-2}_0D^{\alpha}_tu(t))\\={\nabla}W(t,u(t))+{\lambda}g(t){\mid}u(t){\mid}^{q-2}u(t),\;t{\in}(0,T),\\u(0)=u(T)=0,$$ where ${\nabla}W(t,u)$ is the gradient of W(t, u) at u and $W{\in}C([0,T]{\times}{\mathbb{R}}^n,{\mathbb{R}})$ is homogeneous of degree r, ${\lambda}$ is a positive parameter, $g{\in}C([0,T])$, 1 < r < p < q and ${\frac{1}{p}}<{\alpha}<1$. Using the Fibering map and Nehari manifold, for some positive constant ${\lambda}_0$ such that $0<{\lambda}<{\lambda}_0$, we prove the existence of at least two non-trivial solutions
Keywords
nonlinear fractional differential equations; boundary value problem; existence of solutions; Nehari manifold;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 S. Zhang, Solutions for a class of fractional boundary value problem with mixed nonlinearities, Bull. Korean Math. Soc. 53 (2016), no. 5, 1585-1596. https://doi.org/10.4134/BKMS.b150857   DOI
2 Z. Zhang and J. Li, Variational approach to solutions for a class of fractional boundary value problems, Electron. J. Qual. Theory Differ. Equ. 2015 (2015), No. 11, 10 pp. https://doi.org/10.14232/ejqtde.2015.1.11   DOI
3 Y. Zhao and L. Tang, Multiplicity results for impulsive fractional differential equations with p-Laplacian via variational methods, Bound. Value Probl. 2017 (2017), Paper No. 123, 15 pp. https://doi.org/10.1186/s13661-017-0855-0   DOI
4 Z. Bai and H. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005), no. 2, 495-505. https://doi.org/10.1016/j.jmaa.2005.02.052   DOI
5 L. Bourdin, Existence of a weak solution for fractional Euler-Lagrange equations, J. Math. Anal. Appl. 399 (2013), no. 1, 239-251. https://doi.org/10.1016/j.jmaa.2012.10.008   DOI
6 M. Chamekh, A. Ghanmi, and S. Horrigue, Iterative approximation of positive solutions for fractional boundary value problem on the half-line, Filomat 32 (2018), no. 18, 6177-6187.   DOI
7 T. Chen and W. Liu, Solvability of fractional boundary value problem with p-Laplacian via critical point theory, Bound. Value Probl. 2016 (2016), Paper No. 75, 12 pp. https://doi.org/10.1186/s13661-016-0583-x   DOI
8 T. Chen, W. Liu, and H. Jin, Infinitely many weak solutions for fractional dirichlet problem with p-Laplacian, arXiv:1605.09238.
9 A. Ghanmi, M. Kratou, and K. Saoudi, A multiplicity results for a singular problem involving a Riemann-Liouville fractional derivative, Filomat 32 (2018), no. 2, 653-669. https://doi.org/10.2298/fil1802653g   DOI
10 P. Drabek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 4, 703-726. https://doi.org/10.1017/S0308210500023787   DOI
11 R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. https://doi.org/10.1142/9789812817747
12 W. Jiang, The existence of solutions to boundary value problems of fractional differential equations at resonance, Nonlinear Anal. 74 (2011), no. 5, 1987-1994. https://doi.org/10.1016/j.na.2010.11.005   DOI
13 W. Liu, M. Wang, and T. Shen, Analysis of a class of nonlinear fractional differential models generated by impulsive effects, Bound. Value Probl. 2017 (2017), Paper No. 175, 18 pp. https://doi.org/10.1186/s13661-017-0909-3   DOI
14 F. Jiao and Y. Zhou, Existence results for fractional boundary value problem via critical point theory, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22 (2012), no. 4, 1250086, 17 pp. https://doi.org/10.1142/S0218127412500861   DOI
15 F. Jiao and Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl. 62 (2011), no. 3, 1181-1199. https://doi.org/10.1016/j.camwa.2011.03.086   DOI
16 A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
17 P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1986. https://doi.org/10.1090/cbms/065   DOI
18 J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989. https://doi.org/10.1007/978-1-4757-2061-7
19 K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993.
20 I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.
21 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, translated from the 1987 Russian original, Gordon and Breach Science Publishers, Yverdon, 1993.
22 C. Torres Ledesma, Boundary value problem with fractional p-Laplacian operator, Adv. Nonlinear Anal. 5 (2016), no. 2, 133-146. https://doi.org/10.1515/anona-2015-0076   DOI
23 M. Schechter, Linking Methods in Critical Point Theory, Birkhauser Boston, Inc., Boston, MA, 1999. https://doi.org/10.1007/978-1-4612-1596-7
24 J. Vanterler da Costa Sousa, E. Capelas de Oliveira, and L. A. Magna, Fractional calculus and the ESR test, AIMS Mathematics 2 (2017), 692-705. https://doi.org/10.3934/Math.2017.4.692   DOI
25 J. Vanterler da C. Sousa, Magun N. N. dos Santos, L. A. Magna, and E. Capelas de Oliveira, Validation of a fractional model for erythrocyte sedimentation rate, Comput. Appl. Math. 37 (2018), no. 5, 6903-6919. https://doi.org/10.1007/s40314-018-0717-0   DOI
26 D. Tavares, R. Almeida, and D. F. M. Torres, Combined fractional variational problems of variable order and some computational aspects, J. Comput. Appl. Math. 339 (2018), 374-388. https://doi.org/10.1016/j.cam.2017.04.042   DOI
27 C. Torres Ledesma, Mountain pass solution for a fractional boundary value problem, J. Fract. Calc. Appl. 5 (2014), no. 1, 1-10.
28 C. E. Torres Ledesma and N. Nyamoradi, Impulsive fractional boundary value problem with p-Laplace operator, J. Appl. Math. Comput. 55 (2017), no. 1-2, 257-278. https://doi.org/10.1007/s12190-016-1035-6   DOI
29 W. Xie, J. Xiao, and Z. Luo, Existence of solutions for fractional boundary value problem with nonlinear derivative dependence, Abstr. Appl. Anal. 2014 (2014), Art. ID 812910, 8 pp. https://doi.org/10.1155/2014/812910   DOI
30 S. Zhang, Existence of a solution for the fractional differential equation with nonlinear boundary conditions, Comput. Math. Appl. 61 (2011), no. 4, 1202-1208. https://doi.org/10.1016/j.camwa.2010.12.071   DOI
31 R. Almeida, Analysis of a fractional SEIR model with treatment, Appl. Math. Lett. 84 (2018), 56-62. https://doi.org/10.1016/j.aml.2018.04.015   DOI
32 R. P. Agarwal, M. Benchohra, and S. Hamani, Boundary value problems for fractional differential equations, Georgian Math. J. 16 (2009), no. 3, 401-411.   DOI
33 O. P. Agrawal, J. A. Tenreiro Machado, and J. Sabatier, Fractional Derivatives and Their Application: Nonlinear Dynamics, Springer-Verlag, Berlin, 2004,