An inverse problem related to a half-linear eigenvalue problem

We study an inverse problem on the half-linear Dirichlet eigenvalue problem −(|y′(x)|p−2y′(x))′=(p−1)λr(x)|y(x)|p−2y(x), where p>1 with p≠2 and r is a positive function defined on [0,1]. Using eigenvalues and nodal data (the lengths of two consecutive zeros of solutions), we reconstruct r−1/p(x) and its derivatives. Our method is based on (Law and Yang in Inverse Probl. 14:299-312, 779-780, 1998; Shen and Tsai in Inverse Probl. 11:1113-1123, 1995), and our result extends the result in (Shen and Tsai in Inverse Probl. 11:1113-1123, 1995) for the linear case to the half-linear case.