In this article, we study eigenvalue problems with the p-Laplacian operator:
where p>1 and . We show that if and q is single-well with transition point , 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 and . Applying these results, we extend and improve a result by  by using finitely many eigenvalues and by generalizing the string equation to p-Laplacian problem. Moreover, our results also extend a result of Huang  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
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