On a class of a boundary value problems involving the p(x)-Biharmonic operator

Authors

DOI:

https://doi.org/10.22199/issn.0717-6279-2019-05-0061

Keywords:

p(x)-biharmonic, Topological degree, Variational methods

Abstract

Our aim is to establish the existence of weak solution for a class of Robin problems involving fourth order operator. The nonlinearity is superlinear but does not satisfy the usual Ambrosetti-Rabinowitz condition.The proof is made with and without variational structure.

Author Biography

Anass Ourraoui, University of Mohamed First.

Faculty of sciences oujda, departament of maths and computer sciences.

References

M. Allaoui, A. El Amrouss and A. Ourraoui, “Three solutions for a quasi-linear elliptic problem”, Applied mathematics e-notes, vol. 13, pp. 51-59, 2013. [On line]. Available: https://bit.ly/34lMJXS

R. Ayazoglu, G. Alisoy and I. Ekincioglu, “Existence of one weak solution for p(x)−Biharmonic equations involving a concave-convex nonlinearity”, Matematički vesnik, vol. 69, no. 4, pp. 296-307, Dec. 2017. [On line]. Available: https://bit.ly/2RWAhLI

A. Ayoujil and A. Elamrouss, “Multiple solutions for a Robin problem involving the p(x)−biharmonic operator”, Analele Universităţii din Craiova. Seria matematică, informatică vol. 44, no. 1, pp. 87-93, 2017. [On line]. Available: https://bit.ly/35qjbcW

M. Alimohammady and F. Fattahi, “Existence of solutions to hemivariational inequalities involving the p(x)−Biharmonic operator”, Electronic journal of differential equations, Vol. 2015, Art. ID 79, 2015. [On line]. Available: https://bit.ly/35pW9CP

M. Boureanua, V. Rǎdulescu and B. Repovš, “On a p(·)−biharmonic problem with no-flux boundary condition”, Computers & mathematics with applications, vol. 72, no. 9, pp. 2505-2515, Nov. 2016, doi: 10.1016/j.camwa.2016.09.017.

Y. Chen, S. Levine and M. Rao, “Variable exponent, linear growth functionals in image restoration”, SIAM Journal on applied mathematics, vol. 66, no. 4, pp. 1383-1406, 2006, doi: 10.1137/050624522.

S. Deng, “Positive solutions for Robin problem involving the p(x)−Laplacian”, Journal of mathematical analysis and applications, vol. 360, no. 2, pp. 548-560, Dec. 2009, doi: 10.1016/j.jmaa.2009.06.032.

A. El Amrouss and A. Ourraoui, “Existence of solutions for a boundary problem involving p(x)-biharmonic operator”, Boletim da sociedade paranaense de matemática, vol. 31, no. 1, pp. 179-192, 2013, doi: 10.5269/bspm.v31i1.15148.

X. Fan and Q. Zhang, “Existence of solutions for p(x)-laplacian Dirichlet problems”, Nonlinear analysis, vol. 52, no. 8, pp. 1843-1852, Dec. 2003, doi: 10.1016/S0362-546X(02)00150-5.

X. Fan and D. Zhao, “On the spaces Lp(x)(Ω) and Wm,p(x)(Ω)”, Journal of mathematical analysis and applications, vol. 263, no. 2, pp. 424-446, Nov. 2001, doi: 10.1006/jmaa.2000.7617.

B. Ge and Q. Zhou, “Continuous spectrum of a fourth order nonhomogeneous differential operators with variable exponent”, Electronic journal of qualitative theory of differential equations, no. 18, pp. 1-11, 2013, doi: 10.14232/ejqtde.2013.1.18.

K. Kefi, V. Rǎdulescu, “On a ????(????)-biharmonic problem with singular weights”, Zeitschrift für angewandte mathematik und physik, vol. 68, Art. ID 80, Jun. 2017, doi: 10.1007/s00033-017-0827-3.

L. Kong, “Eigenvalues for a fourth order elliptic problem”, Proceeding of the American mathematical society, vol. 143, no. 1, pp. 249-258, 2015, doi: 10.1090/S0002-9939-2014-12213-1.

L. Martinson and K. Pavlov, “Unsteady shear flows of a conducting fluid with a rheological power law”, Magnitnaya gidrodinamika, vol. 7, no. 2, pp. 50-58, 1971. [On line]. Available: https://bit.ly/2tnETQF

A. Ourraoui, “Multiplicity results for Steklov problem with variable exponent”, Applied mathematics and computation, vol. 277, pp. 34-43, Mar. 2016, doi: 10.1016/j.amc.2015.12.043.

M. Růžička, Electrorheological fluids: modeling and mathematical theory, Berlin: Springer, 2000, doi: 10.1007/BFb0104029.

F. Yasuhiro and K. Takasi , “A supersolution-subsolution method for nonlinear biharmonic equations in ℝN” , Czechoslovak mathematical journal, vol. 47, no. 4, pp. 749-768, Dec. 1997, doi: 10.1023/A:1022830920903.

V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory”, Mathematics of the USSR-Izvestiya, vol. 29, no. 1, pp. 33-36, 1987, doi: 10.1070/IM1987v029n01ABEH000958.

Q. Zhang, Y. Guo and G. Chen. “Existence and multiple solutions for a variable exponent system”, Nonlinear analysis: theory, methods & applications, vol. 73, no. 12, pp. 3788-3804, Dec. 2010, doi: 10.1016/j.na.2010.08.005.

Published

2019-12-16

How to Cite

[1]
A. Ourraoui, “On a class of a boundary value problems involving the p(x)-Biharmonic operator”, Proyecciones (Antofagasta, On line), vol. 38, no. 5, pp. 955-967, Dec. 2019.

Issue

Section

Artículos