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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
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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 )$.

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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