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Article Contents

# Continuous dependence of eigenvalues of $p$-biharmonic problems on $p$

• We are concerned with the Dirichlet and Neumann eigenvalue problem for the ordinary quasilinear fourth-order ($p$-biharmonic) equation \begin{eqnarray} (|u''|^{p-2}u'')''=\lambda|u|^{p-2}u, in \quad [0,1], \quad p>1. \end{eqnarray} It is known that the eigenvalues of the Dirichlet and Neumann $p$-biharmonic problem are positive and nonnegative, respectively, isolated, and form an increasing unbounded sequence. We prove that the eigenvalues depend continuously on $p$, and that they interlace with the eigenvalues of the Navier $p$-biharmonic problem.
Mathematics Subject Classification: Primary: 34B15, 47J10; Secondary: 34L15.

 Citation:

•  [1] J. Benedikt, Uniqueness theorem for $p$-biharmonic equations, Electron. J. Differential Equations, 53 (2002), 1-17. [2] J. Benedikt, Uniqueness theorem for quasilinear $2n$th-order equations, J. Math. Anal. Appl., 293 (2004), 589-604. [3] J. Benedikt, On simplicity of spectra of $p$-biharmonic equations, Nonlinear Anal., 58 (2004), 835-853. [4] J. Benedikt, On the discreteness of the spectra of the Dirichlet and Neumann $p$-biharmonic problem, Abstr. Appl. Anal., 2004 (2004), 777-792. [5] J. Benedikt, Global bifurcation result for Dirichlet and Neumann $p$-biharmonic problem, NoDEA, Nonlinear Differ. Equ. Appl., 14 (2007), 541-558. [6] M.\,A. Del Pino, M. Elgueta and R.\,F. Man\'asevich, A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(|u'|^{p-2} u')'+f(t,u)=0, u(0)=u(T)=0, p>1$, J. Differential Equations, 80 (1989), 1-13. [7] P. Drábek, Ranges of $a$-homogeneous operators and their perturbations, Časopis P\vest. Mat., 105 (1980), 167-183. [8] P. Drábek and M. Ôtani, Global bifurcation result for the $p$-biharmonic operator, Electron. J. Differential Equations, 48 (2001), 1-19. [9] A. El Khalil, S. Kellati and A. Touzani, On the spectrum of the $p$-biharmonic operator, in "2002-Fez conference on Partial Differential Equations,'' Electron. J. Differential Equations, Conference 09 (2002), 161-170. [10] A. Kratochvíl and J. Nečas, The discreteness of the spectrum of a nonlinear Sturm-Liouville equation of fourth order, Comment. Math. Univ. Carolinæ, 12 (1971), 639-653 (in Russian, O diskretnosti spektra nelinenogo uravneni turmaLiuvill etvertogo pordka). [11] A. Pinkus, $n$-widths of Sobolev spaces in $L^p$, Constr. Approx., 1 (1985), 15-62.