On a class of a boundary value problems involving the p(x)-Biharmonic operator
Keywords:p(x)-biharmonic, Topological degree, Variational methods
AbstractOur 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.
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.