Details

Mostafa Allaoui ^[a]

Three solutions for elliptic systems involving \(p(x)\)-biharmonic operators

Pages: 211-224
Received: 12 May 2016
Accepted in revised form: 6 Febraury 2017
Mathematics Subject Classification (2010): 35J35, 35J55, 47J30.
Keywords: \(p(x)\)-biharmonic operator, variable exponent Sobolev space, critical point theorem.
Author address:
[a]: University of Mohamed I, FSTH, Department of Mathematics, LANOL, Oujda, 60000, Morocco

Abstract: In this paper, we study the existence of solutions for elliptic systems with variable exponents. Under some suitable conditions and by applying an equivalent variational approach to a recent Ricceris three critical points theorem, we established the existence of at least three weak solutions.

References

[1]: G. A. Afrouzi, S. Ala and A. Niknam, Existence of positive solutions for \((p(x), q(x))\) Laplacian system, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 57(105) (2014), 153-162. MR3236645
[2]: M. Allaoui, Existence of three solutions for variable exponent elliptic systems, Ann. Univ. Ferrara Sez. VII Sci. Mat. 61 (2015), 241-253. MR3421703
[3]: M. Allaoui, A. R. El Amrouss and A. Ourraoui, Existence and multiplicity of solutions for a Steklov problem involving the \(P(X)\)-Laplace operator, Electron. J. Differential Equations 2012 (2012), no. 132, 1-12. MR2967197
[4]: M. Allaoui, A. R. El Amrouss and A. Ourraoui, On superlinear fourth order-PDEs with variable exponents, J. Abstr. Differ. Equ. Appl. 3 (2012), 67-75. MR2944666
[5]: M. Allaoui, A. R. El Amrouss and A. Ourraoui, Infinitely many solutions for a nonlinear Navier boundary systems involving (\(p(x), q(x))\)-biharmonic, Bol. Soc. Parana. Mat (3) 33 (2015), 157-170. MR3267305
[6]: K. Ben Haddouch, Z. El Allali, N. Tsouli, S. El Habib and F. Kissi, Existence of solutions for a fourth order eigenvalue problem with variable exponent under Neumann boundary conditions, Bol. Soc. Parana. Mat. (3) 34 (2016), 253-272. MR3414292
[7]: G. Bonanno, A minimax inequality and its applications to ordinary differential equations, J. Math. Anal. Appl. 270 (2002), 210-229. MR1911762
[8]: P. Candito and R. Livrea, Infintely many solution for a nonlinear Navier boundary value problem involving \(p\)-biharmonic, Stud. Univ. Babeş-Bolyai Math. 55 (2010), 41-51. MR2784993
[9]: P. Candito, L. Li and R. Livrea, Infinitely many solutions for a perturbed nonlinear Navier boundary value problem involving the \(p\)-biharmonic, Nonlinear Anal. 75 (2012), 6360-6369. MR2959811
[10]: G. D'Aguì and A. Sciammetta, Infinitely many solutions to elliptic problems with variable exponent and nonhomogeneous Neumann conditions, Nonlinear Anal. 75 (2012), 5612-5619. MR2942940
[11]: A. El Hamidi, Existence results to elliptic systems with nonstandard growth conditions, J. Math. Anal. Appl. 300 (2004), 30-42. MR2100236
[12]: A. R. El Amrouss, S. El Habib and N. Tsouli, Existence of solutions for an eigenvalue problem with weight, Electron. J. Differential Equations 2010 (2010), no. 45, 1-10. MR2629751
[13]: X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces \(W^{k,p(x)}(\Omega)\), J. Math. Anal. Appl. 262 (2001), 749-760. MR1859337
[14]: X. Fan and D. Zhao, On the spaces \(L^{p(x)}(\Omega)\) and \(W^{m,p(x)}(\Omega)\), J. Math. Anal. Appl. 263 (2001), 424-446. MR1866056
[15]: C. Li and C.-L. Tang, Three solutions for a Navier boundary value problem involving the \(p\)-biharmonic, Nonlinear Anal. 72 (2010), 1339-1347. MR2577535
[16]: L. Li, L. Ding and W.-W. Pan, Existence of multiple solutions for a \(p(x)\)-biharmonic equation, Electron. J. Differential Equations 2013 (2013), no. 139, 1-10. MR3084619
[17]: B. Ricceri, A three critical points theorem revisited, Nonlinear Anal. 70 (2009), 3084-3089. MR2503052
[18]: M. Ružička, Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin, 2000. MR1810360
[19]: X. Xu and Y. An, Existence and multiplicity of solutions for elliptic systems with nonstandard growth condition in \(\mathbf{R}^{N}\), Nonlinear Anal. 68 (2008), 956-968. MR2382312
[20]: A. Zang and Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, Nonlinear Anal. 69 (2008), 3629-3636. MR2450565
[21]: Q. Zhang, Existence and asymptotic behavior of positive solutions for variable exponent elliptic systems, Nonlinear Anal. 70 (2009), 305-316. MR2468238
[22]: Q. Zhang, Existence of solutions for \(p(x)\)-Laplacian equations with singular coefficients in \(\mathbb{R}^{N}\), J. Math. Anal. Appl. 348 (2008), 38-50. MR2449325
[23]: V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 675-710; English transl: Math. USSR-Izv. 29 (1987), 33-66. MR0864171 DOI: 10.1070/IM1987v029n01ABEH000958