Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation

Hikmet Koyunbakan

doi:10.1186/1687-2770-2013-272

Abstract

In this study, the inverse nodal problem is solved for p-Laplacian Schrodinger equation with energy-dependent potential function with the Dirichlet conditions. Asymptotic estimates of eigenvalues, nodal points and nodal lengths are given by using Prufer substitution. Especially, an explicit formula for a potential function is given by using nodal lengths. Results are more general than the classical p-Laplacian Sturm-Liouville problem. For the proofs, methods previously developed by Law et al. and Wang et al., in 2009 and 2011, respectively, are used. In there, they solved an inverse nodal problem for the classical p-Laplacian Sturm-Liouville equation with eigenparameter boundary conditions. MSC:34A55, 34L20.

Highlights

Consider the following p-Laplacian eigenvalue problem for– u (p– ) = (p – ) λ – q(x) u(p– ), < x is a constant, u(p– ) := |u|p– Sgn u and λ is the spectral parameter [ ]
The recent interest is a study by Hald and McLaughlin [, ] wherein the given spectral information consists of a set of nodal points of eigenfunctions for the Sturm-Liouville problems
These results were extended to the case of problems with eigenparameter-dependent boundary conditions by Browne and Sleeman [ ]

Summary

Introduction

Introduction Consider the following p-Laplacian eigenvalue problem for ) is known as a one-dimensional p-Laplacian eigenvalue equation. The recent interest is a study by Hald and McLaughlin [ , ] wherein the given spectral information consists of a set of nodal points of eigenfunctions for the Sturm-Liouville problems. Law et al [ ], Law and Yang [ ] solved the inverse nodal problem of determining the smoothness of the potential function q of the Sturm-Liouville problem by using nodal data.

Results

Conclusion