Existence of weak solution for a non-linear problem with fractional p-Laplacian


  • Raúl Sánchez A. Departamento Académico de Matemática. Universidad Nacional de Tumbes
  • Cesar Torres L. Departamento de Matemáticas, Universidad Nacional de Trujillo-Perú




Fractional Calculus, Nehari Manifold, Fibering Maps


We study the existence of weak solution for a non-linear problem with fractional p-Laplacian operator for the case where the order of the fractional derivative is 1/p
p < alfa < 1, 1 < q < p-1, with 2 < p <Infinito, then using the minimization method called Nehari Manifold and its important relationship with the Fibering Maps, which
is defined in the form t-->J(tu), where J is the functional associated to the non-linear problem to be studied, the main result is obtained.


Abeliuk Roberto and Howard S. Wheater, Parameter identification of solute transport models for unsaturated soils, Journal of Hydrology, Vol 117:9-18 (1990). Available online 27 March(2003).

Anatoly A. Kilbas, Hari M. Srivastava and Juan J. Trujillo Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam,ED. Jan Van Mill(2006).

Chen T. and Liu W., Solvability of fractional boundary value problem with p-Laplacian via critical point theory,J. Boundary Value Problems. 75 (2016).

Claudianor Oliveira A. Romildo N. de Lima, Introducão à Teoria dos Pontos Críticos. Unidade Acadêmica de Matematica, Universidade Federal de Campinas Grande. PB/Brasil, Marzo(2018).

Drabek P. and Pohozaev S.I. Positive solutions for the P-Laplacian application of the fibering method Problems, Proceedings of the Royal Socieety of Edinburgh A: Mathematics. Vol127, pags 703 - 726. (1997).

Ervin V. J. and Roop J. P. Variational solution of fractional advection dispersion equations on bounded domains in Rd,Methods Partial Differential Equations. Vol 23, pag. 256-381 (2007).

Gabriel S., Lau R. and Gabriel C. The dielectric properties of biological tissues: III. parametric models for the dielectric spectrum of tissues,Physics in Medicine and Biology, Vol 42(1996),no 11, http://stacks.iop.org/0031-9155/41/i=11/a=003, pag. 2271.

Haim Brezis Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer New York Dordrecht Heidelberg London.pag 58. (2010).

Hartnett M. and Cawley A. M. Mathematical modelling of the effects of marine aquaculture developments on certain water quality parameters, Springer Netherlands, ED. Wrobel, pag. 279-295, (1991), https://doi.org/10.1007/978-94-011-3694-5-20.

Hubert H.G.Savenije, Salt intrusion model for high-water slack, low-water slack, and mean tide on spread sheet, Journal of Hydrology, Vol 107(1-4):9-18, (1989). Available online 1 April 2003.

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. Inter. Journal of BiF. and Chaos, 22(4), 1-17(2012).

Kenneth E. Bencala, Diane M. McKnight and Gary W. Zellweger, Characterization of transport in an acidic and metal-rich mountain stream based on a lithium tracer injection and simulations of transient storage. Water Resouces Research, Vol 26(5):989-1000, (1990).

Martha E. Londono L., Principio fenomenológico del comportamiento dieléctrico de un hidrogel de alcohol polivinílico - Phenomenological principle dielectrical behaviour of poly (vinyl alcohol) hidrogel, Universidad Nacional de Colombia, Tesis, Medellin(2011).

Matthew F. Causley. Asymptotic and numerical analysis of time-dependent wave propagation in dispersive dielectric media that exhibit fractional relaxation, The State University of New Jersey, (2011).

Meilan Q. and Liquan M. Existence of Weak Solutions for Nonlinear Time-Fractional p-Laplace Problems, Journal of Applied Mathematics. Vol 2014, 9 pages, (2014).

Nick Laskin, Fractional Schrodinger equation, Physical Review E., Vol 66, art. no. 056108(2002).

Navarrina F., Colominas I., Casteleiro M., Cueto-Felgueroso L., Gómez, H., Fe, J. and Soage, A. Analysis of hydrodynamic and transport phenomena in the R?a de Arousa: a numerical model for high environmental impact estuaries Proceedings of the 8th Congress on Moving Boundary Problems, Vol84, ISSN 1743-3509, (2005), https://www.witpress.com/Secure/elibrary/papers/FSI05/FSI05056FU.pdf

Patyn J., Ledoux E. and Bonne A. Geohydrological research in relation to radioactive waste disposal in an argillaceous formation Journal of Hydrology, Vol 109(3-4): 267-285 (1989).

Podlubny I. Fractional Differential Equations, Academic Press, New York(1999).

Podlubny I. Generalized viscoelastic models: Their fractional equations with solutions, Academic press, New York(1998).

Ross B. Fractional calculus and its aplications: Lecture notas in mathematics. Springer-Verlag, Vol 457, New York(1975).

Ross B. The Development of fractional calculus, Academic Press, Historia mathematica, Vol 4,75-89, New York(1975).

Samko S., Kilbas A. and Marichev O. Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers. Primera Edición. New York(1993).

Shiessel H., R. Metzler, A. Blumen and T. Nonnenmacher. Generalized viscoelastic models: their fractional equations with solutions. Physics A: Mathematical and General. Vol 28, pp. 6567-6584(1995).

Torres C. Boundary value problem with fractional p-Laplacian operator, DE GRUYTER. Advances in Nonlinear Analysis. Vol 5(2015).

Torres C. Existencia y unicidad de la solución de ecuaciones diferenciales ordinarias de orden fraccionario, Tesis Universidad Nacional de Trujillo, Perú, Pages 59-61(2009).

Torres C. and Nyamoradi N. Existence and multiplicity result for a fractional p-Laplacian equation with combined fractional derivates, Mathematics Subject Classification., 26A33; 35A15; 35B38(2010).

Torres C. and Nyamoradi N. Impulsive fractional boundary value problem with p-Laplacian operator, ED Korean Society for Computational and Applied Mathematics. Pages 22(2016).

Willem Michel Minimax Theorem, Université et Marie Curie. Berlin (1996).

Zhao C Z.,Werner M., Taylor S., Chalker P. R., Jones A C and Chun Zhao. Dielectric relaxation of la-doped zirconia caused by annealing ambient, Nanoscale Research Letters. Vol 6(1):48, (2010).

Zhou Y., Basic Theory of Fractional Differential Equations, World Scientifi. Xiangtan University, China. (2014).



How to Cite

Sánchez A., R., & Torres L., C. (2018). Existence of weak solution for a non-linear problem with fractional p-Laplacian. Selecciones Matemáticas, 5(02), 154-163. https://doi.org/10.17268/sel.mat.2018.02.03