THE ROTATION NUMBER APPROACH TO EIGENVALUES OF THE ONE-DIMENSIONAL p-LAPLACIAN WITH PERIODIC POTENTIALS

Abstract

The paper studies the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic potential. After a rotation number function ρ(λ) has been introduced, it is proved that for any non-negative integer n, the endpoints of the interval ρ −1 (n/2) in R yield the corresponding periodic or anti-periodic eigenvalues. However, as in the Dirichlet problem of the higher dimensional p-Laplacian, it remains open if these eigenvalues represent all periodic and anti-periodic eigenvalues. The result obtained is a partial generalization of the spectrum theory of the one-dimensional Schrödinger operators with periodic potentials.

Continuity in weak topology and extremal problems of eigenvalues of the $p$-Laplacian

“…These results were proved in [29] using the comparison result for solutions of the first-order ODEs (2.3). They can also be obtained from (2.15) because the differentials are positive functionals.…”

“…The complete proofs of Theorems 1.1 and 1.2 will be given in Section 3. In this section, by considering q ∈ L γ as 1-periodic potentials, we will also prove that the rotation number (q) of (1.5) (see [29]) and the variational 1-periodic and 1-anti-periodic eigenvalues of (1.1) (see [5,6]) are also continuous in q ∈ (L γ , w γ ). See Theorem 3.5 and Theorem 3.6, respectively.…”

“…For the detailed constructions, one can refer to [29]. In the following we first consider equation (2.3) for the argument θ.…”

“…In the present case, (q) ∈ [0, ∞). See [26,29]. Due to the 1-periodicity of q(t), we know that θ(t; ϑ, q) satisfies θ(t + n; ϑ, q) = θ(t; θ(n; ϑ, q), q), θ(t; ϑ + 2nπ p , q) = θ(t; ϑ, q) + 2nπ p for any n ∈ Z.…”

“…See, for example, [4,28,29]. The eigenvalues λ m (q) are also dependent on the exponent p and the boundary data c ij .…”