Eigenvalue problems with $p$-Laplacian operators

Cheng, Yan-Hsiou

Electronic Journal of Differential Equations, Tome 2014 (2014).

Résumé

Summary: In this article, we study eigenvalue problems with the p-Laplacian operator: $$ -(|y'|^{p-2}y')'= (p-1)(\lambda\rho(x)-q(x))|y|^{p-2}y \quad {on } (0,\pi_{p}), $$ 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.

Détail

Classification : 34A55, 34L15
Keywords: p-Laplacian, inverse spectral problem, instability interval