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Abstrakty
We study the multiplicity of weak solutions to the following fourth order nonlinear elliptic problem with a p(x)-biharmonic operator [formula] where Ω is a smooth bounded domain in RN, [formula] is the p(x)-biharmonic operator, and λ > 0 is a parameter. We establish sufficient conditions under which there exists a positive number λ* such that the above problem has at least two nontrivial weak solutions for each λ > λ*. Our analysis mainly relies on variational arguments based on the mountain pass lemma and some recent theory on the generalized Lebesgue-Sobolev spaces [formula].
Słowa kluczowe
Czasopismo
Rocznik
Tom
Strony
253--264
Opis fizyczny
Bibliogr. 22 poz.
Twórcy
autor
- Department of Mathematics University of Tennessee at Chattanooga Chattanooga, TN 37403, USA
Bibliografia
- [1] A. Ayoujil, A.R. El Amrouss, On the spectrum of a fourth order elliptic equation with variable exponent, Nonlinear Anal. 71 (2009), 4916-4926.
- [2] A. Ayoujil, A.R. El Amrouss, Continuous spectrum of a fourth order nonhomogenous elliptic equation with variable exponent, Electron. J. Differential Equations 2011 (2011) 24, 12 pp.
- [3] J. Benedikt, P. Drabek, Estimates of the principal eigenvalue of the p-biharmonic operator, Nonlinear Anal. 75 (2012), 5374-5379.
- [4] D. Edmunds, J. Rakosnik, Soblev embeddings with variable exponent, Studia Math. 143 (2000), 267-293.
- [5] A.R. El Amrouss, A. Ourraoui, Existence of solutions for a boundary value problem, involving a p{x)-biharmonic operator, Bol. Soc. Paran. Mat. 31 (2013), 179-192.
- [6] X. Fan, X. Han, Existence and multiplicity of solutions for p{x)-Laplacian equations in RN, Nonlinear Anal. 59 (2004), 173-188.
- [7] X. Fan, D. Zhao, On the spaces Lp(x\n) and Wm'v{x){Cl), J. Math. Anal. Appl. 263 (2001), 424-446.
- [8] J.R. Graef, S. Heidarkhani, L. Kong, Multiple solutions for a class of {pi,... ,Pn)-biharmonic systems, Commun. Pure Appl. Anal. 12 (2013), 1393-1406.
- [9] M. Ghergu, A biharmonic equation with singular nonlinearity, Proc. Edinburgh Math. Soc. 55 (2012), 155-166.
- [10] T.C. Halsey, Electrorheological fluids, Science 258 (1992), 761-766.
- [11] Y. Jabri, The Mountain Pass Theorem,, Variants, Generalizations and some Applications, Encyclopedia of Mathematics and its Applications 95, Cambridge, New York, 2003.
- [12] K. Kefi, p{x)-Laplacian with indefinite weight, Proc. Amer. Math. Soc. 139 (2011), 4351-4360.
- [13] L. Kong, On a fourth, order elliptic problem, with, a p{x)-biharmonic operator, Appl. Math. Lett. 27 (2014), 21-25.
- [14] L. Kong, Eigenvalues for a fourth, order elliptic problem,, Proc. Amer. Math. Soc. 143 (2015), 249-258.
- [15] O. Kovacik, J. Rakosnik, On spaces Lp(x) and Wrn'p(x), Czechoslovak Math. J. 41 (1991), 592-618,
- [16] M. Lazzo, P.G. Schmidt, Oscillatory radial solutions for subcritical biharmonic equations, J. Differential Equations 247 (2009), 1479-1504.
- [17] J. Liu, S. Chen, and X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in RN, J. Math. Anal. Appl. 395 (2012), 608-615.
- [18] M. Mihalescu, V. Radulesu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. A 462 (2006), 2625-2641.
- [19] M. Mihalescu, V. Radulesu, On a nonhom.oge.ne.ous quasilinear eigenvalue problem, in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc. 135 (2007), 2929-2937.
- [20] M. Rauźicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000.
- [21] A. Zang, Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, Nonlinear Anal. 69 (2008), 3629-3636.
- [22] V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSRIzv. 29 (1987), 33-66
Uwagi
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-6cfea1c5-3c64-443b-b9ef-30b57e3409d8