Tbilisi Mathematical Journal

Multiplicity result for a stationary fractional reaction-diffusion equations

César E. Torres Ledesma

Abstract

In this paper, we consider the stationary fractional reaction-diffusion equations with Riemann-Liouville boundary conditions \begin{aligned} &{_{x}}D_{T}^{\alpha}({_{0}}D_{x}^{\alpha}u(x)) + {_{0}}D_{x}^{\beta}({_{x}}D_{T}^{\beta}u(x)) = f(x,u(x)),\;\;x\in (0,T),\\ &\lim_{x\to 0} {_{0}}I_{x}^{1-\alpha}u(x) = \lim_{x\to T} {_{x}}I_{T}^{1-\beta}u(x) = 0. \end{aligned}

(0.1)

where $0\lt \alpha , \beta \lt 1$ and $f\in C([0,T] \times \mathbb{R}, \mathbb{R})$. Under suitable conditions on the nonlinearity $f$, we study the multiplicity of weak solutions of (0.1) by using the genus in the critical point theory.

Article information

Source
Tbilisi Math. J., Volume 9, Issue 2 (2016), 115-127.

Dates
Accepted: 7 October 2016
First available in Project Euclid: 12 June 2018

https://projecteuclid.org/euclid.tbilisi/1528769072

Digital Object Identifier
doi:10.1515/tmj-2016-0024

Mathematical Reviews number (MathSciNet)
MR3578791

Zentralblatt MATH identifier
1361.35202

Citation

Ledesma, César E. Torres. Multiplicity result for a stationary fractional reaction-diffusion equations. Tbilisi Math. J. 9 (2016), no. 2, 115--127. doi:10.1515/tmj-2016-0024. https://projecteuclid.org/euclid.tbilisi/1528769072

References

• Adams R. and Fournier J., Sobolev space, second ed., Academic Press, New York-London, 2003.
• Ambrosetti A. and Malchiodi A., Nonlinear analysis and semilinear elliptic problems, Cambridge Stud. Adv. Math. 14, 2007.
• Baleanu D., Güvenc Z. and Machado J. (eds), New trends in nanotechnology and fractional calculus applications, Singapore 2010.
• Belmekki M., Nieto J. and Rodríguez-López R., Existence of periodic solution for a nonlinear fractional differential equation, Bound. Value Probl. 2009, Art. ID 324561, 18 pp. (2009).
• Benchohra M., Cabada A. and Seba D., An existence result for nonlinear fractional differential equations on Banach spaces. Bound. Value Probl. 2009, Article ID 628916, 11 pp. (2009).
• Britton N., Reaction - Diffusion Equations and Their Applications to Biology, Academic Press Inc. London,1986.
• Brezis H., Functional analysis, Sobolev spaces and partial differential equations, Springer, New York, 2011.
• Cantrell R. and Cosner C., Spatial ecology via reaction-diffusion equations, in: Wiley Series in Mathematical and Computational Biology, John Wiley & Sons Ltd., Chichester, 2003.
• Clarke D., A variant of the Lusternik - Schnirelman theory, Indiana Univ. Math. J. 22, 65-74 (1972).
• Ervin V. and Roop J., Variational formulation for the stationary fractional advection dispersion equation, Numer. Meth. Part. Diff. Eqs, 22, 58-76(2006).
• Fazli H. and Bahrami F., On the steady solutions of fractional reaction-diffusion equations, Acceted manuscript in Filomat, 2016
• Jiao F. and Zhou Y., Existence of solution for a class of fractional boundary value problems via critical point theory. Comp. Math. Appl., 62, 1181-1199(2011).
• Jiao F. and Zhou Y., Existence results for fractional boundary value problem via critical point theory, Intern. Journal of Bif. and Chaos, 22(4), 1-17(2012).
• Kilbas A., Srivastava H. and Trujillo J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol 204, Amsterdam, 2006.
• Mawhin J. and Willen M., Critical point theory and Hamiltonian systems, Applied Mathematical Sciences 74, Springer, Berlin, 1989.
• Mendez A. and Torres C., Multiplicity of solutions for fractional Hamiltonian systems with Liouville-Weyl fractional derivarives, Fract. Calc. Appl. Anal., 18, No 4, 875-890, 2015.
• Nyamoradi N., Infinitely Many Solutions for a Class of Fractional Boundary Value Problems with Dirichlet Boundary Conditions, Medit. J. Math., 11(1), 75-87(2014).
• Podlubny I., Fractional differential equations, Academic Press, New York, 1999.
• Rabinowitz P., Minimax method in critical point theory with applications to differential equations, CBMS Amer. Math. Soc., No 65, 1986.
• Rivero M., Trujillo J., Vázquez L. and Velasco M., Fractional dynamics of populations, Appl. Math. Comput, 218, 1089 - 1095(2011).
• Sabatier J., Agrawal O. and Tenreiro Machado J., Advances in fractional calculus. Theoretical developments and applications in physics and engineering, Springer-Verlag, Berlin, 2007.
• Samko S., Kilbas A. and Marichev O. Fractional integrals and derivatives: Theory and applications, Gordon and Breach, New York, 1993.
• Schechter M., Linking methods in critical point theory, Birkhäuser, Boston, 1999.
• Sokolov I. and Klafter J., From diffusion to anomalous diffusion: A century after Einsteins Brownian motion, Chaos, 15, 2, 26-103(2005).
• Torres C., Existence of solution for fractional Hamiltonian systems, Electronic Jour. Diff. Eq. 2013, 259, 1-12(2013).
• Torres C., Mountain pass solution for a fractional boundary value problem, Journal of Fractional Calculus and Applications, 5, 1, 1-10(2014).
• Torres C., Existence of a solution for fractional forced pendulum, Journal of Applied Mathematics and Computational Mechanics, 13, 1, 125-142(2014).
• Torres C. Boundary value problem with fractional $p$-Laplacian operator, Adv. Nonlinear Anal. (2015) (Preprint), doi: 10.1515/anona-2015-0076.
• Torres C., Ground state solution for a class of differential equations with left and right fractional derivatives, Math. Methods Appl. Sci, 38, 5063-5073(2015).
• Torres C., Existence and symmetric result for Liouville-Weyl fractional nonlinear Schrödinger equation, Commun Nonlinear Sci Numer Simulat., 27, 314-327(2015).
• Torres C., Existence of solution for fractional Langevin equation: variational approach, Electron. J. Qual. Theory Differ. Equ. 2014, No 54, 1-14.
• Xu J., O'Regan D. and Zhang K., Multiple solutions for a class of fractional Hamiltonian systems, Fract. Calc. Appl. Anal., 18, No 1, 48-63(2015)
• Zhang Z. and Li J., Variational approach to solutions for a class of fractional boundary value problems, Electron. J. Qual. Theory Differ. Equ. 2015, No 11, 1-10.
• Zhou Y., Basic theory of fractional differential equations, World Scientific Publishing Co. Pte. Ltd. 2014.