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

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  • 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.

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