Energy dependent potential problems for the one dimensional p-Laplacian operator

Abstract

In this work we analyze a nonlinear eigenvalue problem for the p-Laplacian operator with zero Dirichlet boundary conditions. We assume that the problem has a potential which depends on the eigenvalue parameter, and we show that, for n big enough, there exists a real eigenvalue λn, and their corresponding eigenfunctions have exactly n nodal domains. We characterize the asymptotic behavior of these eigenvalues, obtaining two terms in the asymptotic expansion of λn in powers of n. Finally, we study the inverse nodal problem in the case of energy dependent potentials, showing that some subset of the zeros of the corresponding eigenfunctions is enough to determine the main term of the potential.