Eigenvalues for a fourth order elliptic problem
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- by Lingju Kong
- Proc. Amer. Math. Soc. 143 (2015), 249-258
- DOI: https://doi.org/10.1090/S0002-9939-2014-12213-1
- Published electronically: August 28, 2014
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Abstract:
We study the fourth order nonlinear eigenvalue problem with a $p(x)$-biharmonic operator \begin{equation*} \left \{\begin {array}{l} \Delta ^2_{p(x)}u+a(x)|u|^{p(x)-2}u=\lambda w(x)f(u)\quad \text {in}\ \Omega , \\ u=\Delta u=0\quad \text {on}\ \partial \Omega , \end{array} \right . \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb {R}^N$, $p\in C(\overline {\Omega })$ with $p(x)>1$ on $\overline {\Omega }$, $\Delta ^2_{p(x)}u=\Delta (|\Delta u|^{p(x)-2}\Delta u)$ 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
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Bibliographic Information
- Lingju Kong
- Affiliation: Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, Tennessee 37403
- Email: Lingju-Kong@utc.edu
- 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
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3272750