On extreme values of Nehari manifold method via nonlinear Rayleigh's quotient

TL;DR

This paper develops a nonlinear Rayleigh quotient framework to determine extreme parameter values for the Nehari manifold method, enhancing understanding of nonlinear elliptic equations and systems.

Contribution

It introduces a novel approach linking the Nehari manifold method with nonlinear Rayleigh quotients, providing general results and applications to elliptic equations.

Findings

Extreme parameter values are identified via critical points of nonlinear Rayleigh quotients.

Theoretical results connect Nehari manifold extrema with quotient critical values.

Applications demonstrate the method's effectiveness for nonlinear elliptic problems.

Abstract

We study applicability conditions of the Nehari manifold method for the equation of the form $ D_u T(u)-\lambda D_u F(u)=0 $ in a Banach space $W$, where $\lambda$ is a real parameter. Our study is based on the development of the theory Rayleigh's quotient for nonlinear problems. It turns out that the extreme values of parameter $\lambda$ for the Nehari manifold method can be found through the critical values of a corresponding nonlinear generalized Rayleigh's quotient. In the main part of the paper, we provide some general results on this relationship. Applications are given to several types of nonlinear elliptic equations and systems of equations.