Complete continuity and Fréchet derivatives of nodes in potentials for one-dimensional p-Laplacian

Abstract

The aim of this paper is to study the dependence of all nodes on integrable potentials, for one-dimensional p-Laplacian with separated boundary conditions, including the complete continuity of nodes in potentials with the weak topology, and the continuous Fréchet differentiability of nodes in potentials. We present the precise formula for the Fréchet derivatives of nodes in potentials. These results are natural but nontrivial generalizations of those for Sturm-Liouville operators, with quite different proofs due to the nonlinearity of the p-Laplacian.