Elliptic Regularization for a Class of Strongly Nonlinear Degenerate Eigenvalue Problems on Orlicz-Sobolev Spaces. I: The Ode Case
This paper is devoted to proving sharp Holder-Lipschitz regularity estimates for the eigensolutione to a class of degenerate Sturm-Liouville eigenvalue problems with polynomial and exponential nonlinearities. The general method of proof represents as blending of a suitable regularization procedure along with a new method of auxiliary solutions, which we develop in detail. it allows one to exhibit a phenomenon of screening to regularity due to degeneracy and a new phenomenon of breakdown of regularity at short distances: while all of the eigensolutions are Lipschitz-continuous functions of their argument, their first derivatives exhibit a transition from Lipschitz-continuity to Hoder-continuity when the argument becomes sufficiently smell. The nature of thin transition is completely described in term of three constants, which all characterize both the shape and the rate of growth of the nonliterary in the principal part of the equation. Several examples and open problems are discussed.