EIGENVALUE PROBLEMS WITH p-LAPLACIAN OPERATORS

In this article, we study eigenvalue problems with the p-Laplacian operator: −(|y′|p−2y′)′ = (p− 1)(λρ(x)− q(x))|y|p−2y on (0, πp), where p > 1 and πp ≡ 2π/(p sin(π/p)). We show that if ρ ≡ 1 and q is singlewell with transition point a = πp/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 ρ and q ≡ 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.