Opuscula Math. 36, no. 2 (2016), 253-264
http://dx.doi.org/10.7494/OpMath.2016.36.2.253

Opuscula Mathematica

# Multiple solutions for fourth order elliptic problems with p(x)-biharmonic operators

Lingju Kong

Abstract. We study the multiplicity of weak solutions to the following fourth order nonlinear elliptic problem with a $$p(x)$$-biharmonic operator $\begin{cases}\Delta^2_{p(x)}u+a(x)|u|^{p(x)-2}u=\lambda f(x,u)\quad\text{ in }\Omega,\\ u=\Delta u=0\quad\text{ on }\partial\Omega,\end{cases}$ where $$\Omega$$ is a smooth bounded domain in $$\mathbb{R}^N$$, $$p\in C(\overline{\Omega})$$, $$\Delta^2_{p(x)}u=\Delta(|\Delta u|^{p(x)-2}\Delta u)$$ is the $$p(x)$$-biharmonic operator, and $$\lambda\gt 0$$ is a parameter. We establish sufficient conditions under which there exists a positive number $$\lambda^{*}$$ such that the above problem has at least two nontrivial weak solutions for each $$\lambda\gt\lambda^{*}$$. Our analysis mainly relies on variational arguments based on the mountain pass lemma and some recent theory on the generalized Lebesgue-Sobolev spaces $$L^{p(x)}(\Omega)$$ and $$W^{k,p(x)}(\Omega)$$.

Keywords: critical points, $$p(x)$$-biharmonic operator, weak solutions, mountain pass lemma.

Mathematics Subject Classification: 35J66, 35J40, 35J92, 47J10.

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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. Drábek, Estimates of the principal eigenvalue of the $$p$$-biharmonic operator, Nonlinear Anal. 75 (2012), 5374-5379.
4. D. Edmunds, J. Rákosník, 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 $$R^N$$, Nonlinear Anal. 59 (2004), 173-188.
7. X. Fan, D. Zhao, On the spaces $$L^{p(x)}(\Omega)$$ and $$W^{m,p(x)}(\Omega)$$, J. Math. Anal. Appl. 263 (2001), 424-446.
8. J.R. Graef, S. Heidarkhani, L. Kong, Multiple solutions for a class of $$(p_1,\ldots,p_n)$$-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. Kováčik, J. Rákosník, On spaces $$L^{p(x)}$$ and $$W^{m,p(x)}$$, 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 $$R^N$$, J. Math. Anal. Appl. 395 (2012), 608-615.
18. M. Mihălescu, V. Rădulesu, 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. Mihălescu, V. Rădulesu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc. 135 (2007), 2929-2937.
20. M. Råužička, 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. USSR Izv. 29 (1987), 33-66.
• Lingju Kong
• Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA
• Communicated by Marius Ghergu.
• Accepted: 2015-08-14.
• Published online: 2015-12-18.