Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Eigenvalues for a fourth order elliptic problem


Author: Lingju Kong
Journal: Proc. Amer. Math. Soc. 143 (2015), 249-258
MSC (2010): Primary 35J66, 35J40, 35J92, 47J10
DOI: https://doi.org/10.1090/S0002-9939-2014-12213-1
Published electronically: August 28, 2014
MathSciNet review: 3272750
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the fourth order nonlinear eigenvalue problem with a $ p(x)$-biharmonic operator

$\displaystyle \left \{\begin {array}{l} \Delta ^2_{p(x)}u+a(x)\vert u\vert^{p(x... ...\medskip } u=\Delta u=0\quad \text {on}\ \partial \Omega , \end{array} \right .$    

where $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^N$, $ p\in C(\overline {\Omega })$ with $ p(x)>1$ on $ \overline {\Omega }$, $ \Delta ^2_{p(x)}u=\Delta (\vert\Delta u\vert^{p(x)-2}\Delta u)$ is the $ p(x)$-biharmonic operator, and $ \lambda >0$ is a parameter. Under some appropriate conditions on the functions $ p, a, w, f$, we prove that there exists $ \overline {\lambda }>0$ such that any $ \lambda \in (0,\overline {\lambda })$ is an eigenvalue of the above problem. Our analysis mainly relies on variational arguments based on Ekeland's variational principle and some recent theory on the generalized Lebesgue-Sobolev spaces $ L^{p(x)}(\Omega )$ and $ W^{k,p(x)}(\Omega )$.

References [Enhancements On Off] (What's this?)

  • [1] A. Ayoujil and A. R. El Amrouss, On the spectrum of a fourth order elliptic equation with variable exponent, Nonlinear Anal. 71 (2009), no. 10, 4916-4926. MR 2548723 (2010k:35100), https://doi.org/10.1016/j.na.2009.03.074
  • [2] Abdesslem Ayoujil and Abdel Rachid El Amrouss, Continuous spectrum of a fourth order nonhomogeneous elliptic equation with variable exponent, Electron. J. Differential Equations (2011), No. 24, 12. MR 2781059 (2012c:35098)
  • [3] Jiří Benedikt and Pavel Drábek, Estimates of the principal eigenvalue of the $ p$-biharmonic operator, Nonlinear Anal. 75 (2012), no. 13, 5374-5379. MR 2927595, https://doi.org/10.1016/j.na.2012.04.055
  • [4] David E. Edmunds and Jiří Rákosník, Sobolev embeddings with variable exponent, Studia Math. 143 (2000), no. 3, 267-293. MR 1815935 (2001m:46072)
  • [5] Abdel Rachid El Amrouss and Anass Ourraoui, Existence of solutions for a boundary problem involving $ p(x)$-biharmonic operator, Bol. Soc. Parana. Mat. (3) 31 (2013), no. 1, 179-192. MR 2990539
  • [6] Xianling Fan and Shao-Gao Deng, Remarks on Ricceri's variational principle and applications to the $ p(x)$-Laplacian equations, Nonlinear Anal. 67 (2007), no. 11, 3064-3075. MR 2347599 (2008f:35070), https://doi.org/10.1016/j.na.2006.09.060
  • [7] Xianling Fan and Xiaoyou Han, Existence and multiplicity of solutions for $ p(x)$-Laplacian equations in $ \mathbf {R}^N$, Nonlinear Anal. 59 (2004), no. 1-2, 173-188. MR 2092084 (2005h:35092), https://doi.org/10.1016/j.na.2004.07.009
  • [8] Xianling Fan and Dun Zhao, On the spaces $ L^{p(x)}(\Omega )$ and $ W^{m,p(x)}(\Omega )$, J. Math. Anal. Appl. 263 (2001), no. 2, 424-446. MR 1866056 (2003a:46051), https://doi.org/10.1006/jmaa.2000.7617
  • [9] John R. Graef, Shapour Heidarkhani, and Lingju Kong, Multiple solutions for a class of $ (p_1,\dots ,p_n)$-biharmonic systems, Commun. Pure Appl. Anal. 12 (2013), no. 3, 1393-1406. MR 2989695, https://doi.org/10.3934/cpaa.2013.12.1393
  • [10] Marius Ghergu, A biharmonic equation with singular nonlinearity, Proc. Edinb. Math. Soc. (2) 55 (2012), no. 1, 155-166. MR 2888446, https://doi.org/10.1017/S0013091510000234
  • [11] T. C. Halsey, Electrorheological fluids, Science 258 (1992), 761-766.
  • [12] Khaled Kefi, $ p(x)$-Laplacian with indefinite weight, Proc. Amer. Math. Soc. 139 (2011), no. 12, 4351-4360. MR 2823080 (2012f:35150), https://doi.org/10.1090/S0002-9939-2011-10850-5
  • [13] L. Kong, On a fourth order elliptic problem with a $ p(x)$-biharmonic operator, Appl. Math. Lett. 27 (2014), 21-25. MR 3111602
  • [14] Ondrej Kováčik and Jiří Rákosník, On spaces $ L^{p(x)}$ and $ W^{k,p(x)}$, Czechoslovak Math. J. 41(116) (1991), no. 4, 592-618. MR 1134951 (92m:46047)
  • [15] M. Lazzo and P. G. Schmidt, Oscillatory radial solutions for subcritical biharmonic equations, J. Differential Equations 247 (2009), no. 5, 1479-1504. MR 2541418 (2010j:35112), https://doi.org/10.1016/j.jde.2009.05.005
  • [16] Jiu Liu, ShaoXiong Chen, and Xian Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in $ R^N$, J. Math. Anal. Appl. 395 (2012), no. 2, 608-615. MR 2948252, https://doi.org/10.1016/j.jmaa.2012.05.063
  • [17] Jean Mawhin and Michel Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, vol. 74, Springer-Verlag, New York, 1989. MR 982267 (90e:58016)
  • [18] Mihai Mihăilescu and Vicenţiu Rădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006), no. 2073, 2625-2641. MR 2253555 (2007i:35081), https://doi.org/10.1098/rspa.2005.1633
  • [19] Mihai Mihăilescu and Vicenţiu Rădulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2929-2937. MR 2317971 (2008i:35085), https://doi.org/10.1090/S0002-9939-07-08815-6
  • [20] Michael Růžička, Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000. MR 1810360 (2002a:76004)
  • [21] Biagio Ricceri, Sublevel sets and global minima of coercive functionals and local minima of their perturbations, J. Nonlinear Convex Anal. 5 (2004), no. 2, 157-168. MR 2083908 (2005d:49021)
  • [22] Aibin Zang and Yong Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, Nonlinear Anal. 69 (2008), no. 10, 3629-3636. MR 2450565 (2009i:26025), https://doi.org/10.1016/j.na.2007.10.001
  • [23] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675-710, 877 (Russian). MR 864171 (88a:49026)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35J66, 35J40, 35J92, 47J10

Retrieve articles in all journals with MSC (2010): 35J66, 35J40, 35J92, 47J10


Additional Information

Lingju Kong
Affiliation: Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, Tennessee 37403
Email: Lingju-Kong@utc.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12213-1
Keywords: Critical points, $p(x)$-biharmonic operator, eigenvalues, weak solutions, Ekeland's variational principle
Received by editor(s): February 4, 2013
Received by editor(s) in revised form: March 25, 2013
Published electronically: August 28, 2014
Communicated by: Joachim Krieger
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society