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May  2013, 12(3): 1469-1486. doi: 10.3934/cpaa.2013.12.1469

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

Jiří Benedikt 1,

1. 

Department of mathematics, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 22, 306,14 Plzeň, Czech Republic

Received  April 2012 Revised  May 2012 Published  September 2012

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.
Keywords: interlacing property., $p$-biharmonic operator, nonlinear spectral theory.
Mathematics Subject Classification: Primary: 34B15, 47J10; Secondary: 34L1.
Citation: Jiří Benedikt. Continuous dependence of eigenvalues of $p$-biharmonic problems on $p$. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1469-1486. doi: 10.3934/cpaa.2013.12.1469
References:
[1]

J. Benedikt, Uniqueness theorem for $p$-biharmonic equations, Electron. J. Differential Equations, 53 (2002), 1-17.  Google Scholar

[2]

J. Benedikt, Uniqueness theorem for quasilinear $2n$th-order equations, J. Math. Anal. Appl., 293 (2004), 589-604.  Google Scholar

[3]

J. Benedikt, On simplicity of spectra of $p$-biharmonic equations, Nonlinear Anal., 58 (2004), 835-853.  Google Scholar

[4]

J. Benedikt, On the discreteness of the spectra of the Dirichlet and Neumann $p$-biharmonic problem, Abstr. Appl. Anal., 2004 (2004), 777-792.  Google Scholar

[5]

J. Benedikt, Global bifurcation result for Dirichlet and Neumann $p$-biharmonic problem, NoDEA, Nonlinear Differ. Equ. Appl., 14 (2007), 541-558.  Google Scholar

[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.  Google Scholar

[7]

P. Drábek, Ranges of $a$-homogeneous operators and their perturbations, Časopis P\vest. Mat., 105 (1980), 167-183.  Google Scholar

[8]

P. Drábek and M. Ôtani, Global bifurcation result for the $p$-biharmonic operator, Electron. J. Differential Equations, 48 (2001), 1-19.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[11]

A. Pinkus, $n$-widths of Sobolev spaces in $L^p$, Constr. Approx., 1 (1985), 15-62.  Google Scholar

show all references

References:
[1]

J. Benedikt, Uniqueness theorem for $p$-biharmonic equations, Electron. J. Differential Equations, 53 (2002), 1-17.  Google Scholar

[2]

J. Benedikt, Uniqueness theorem for quasilinear $2n$th-order equations, J. Math. Anal. Appl., 293 (2004), 589-604.  Google Scholar

[3]

J. Benedikt, On simplicity of spectra of $p$-biharmonic equations, Nonlinear Anal., 58 (2004), 835-853.  Google Scholar

[4]

J. Benedikt, On the discreteness of the spectra of the Dirichlet and Neumann $p$-biharmonic problem, Abstr. Appl. Anal., 2004 (2004), 777-792.  Google Scholar

[5]

J. Benedikt, Global bifurcation result for Dirichlet and Neumann $p$-biharmonic problem, NoDEA, Nonlinear Differ. Equ. Appl., 14 (2007), 541-558.  Google Scholar

[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.  Google Scholar

[7]

P. Drábek, Ranges of $a$-homogeneous operators and their perturbations, Časopis P\vest. Mat., 105 (1980), 167-183.  Google Scholar

[8]

P. Drábek and M. Ôtani, Global bifurcation result for the $p$-biharmonic operator, Electron. J. Differential Equations, 48 (2001), 1-19.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[11]

A. Pinkus, $n$-widths of Sobolev spaces in $L^p$, Constr. Approx., 1 (1985), 15-62.  Google Scholar

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