Electron. J. Diff. Equ., Vol. 2014 (2014), No. 139, pp. 1-11.

Eigenvalue problems with p-Laplacian operators

Yan-Hsiou Cheng

Abstract:
In this article, we study eigenvalue problems with the p-Laplacian operator:
$$ -(|y'|^{p-2}y')'= (p-1)(\lambda\rho(x)-q(x))|y|^{p-2}y \quad \text{on } (0,\pi_{p}), $$
where p>1 and $\pi_{p}\equiv 2\pi/(p\sin(\pi/p))$. We show that if $\rho \equiv 1$ and q is single-well with transition point $a=\pi_{p}/2$, then the second Neumann eigenvalue is greater than or equal to the first Dirichlet eigenvalue; the equality holds if and only if q is constant. The same result also holds for p-Laplacian problem with single-barrier $\rho$ and $q \equiv 0$. Applying these results, we extend and improve a result by [24] by using finitely many eigenvalues and by generalizing the string equation to p-Laplacian problem. Moreover, our results also extend a result of Huang [14] on the estimate of the first instability interval for Hill equation to single-well function q.

Submitted November 7, 2013. Published June 16, 2014.
Math Subject Classifications: 34A55, 34L15.
Key Words: p-Laplacian; inverse spectral problem; instability interval.

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Yan-Hsiou Cheng
Department of Mathematics and Information Education
National Taipei University of Education
Taipei City 106, Taiwan
email: yhcheng@tea.ntue.edu.tw

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