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

Abstract

We study applicability conditions of the Nehari manifold method to 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 Rayleigh quotient theory for nonlinear problems. It turns out that the extreme values of parameter $\lambda$ which define intervals of applicability of the Nehari manifold method can be found through the critical values of the corresponding nonlinear generalized Rayleigh quotient. In the main part of this paper, we provide general results on this relationship. Theoretical results are illustrated by considering several examples of nonlinear boundary value problems. Furthermore, we demonstrate that the introduced tool of nonlinear generalized Rayleigh quotient can also be applied to prove new results on the existence of multiple solutions for nonlinear elliptic equations.