Further applications of the Nehari manifold method to functionals in $C^1(X \setminus \{0\})$

TL;DR

This paper extends the Nehari manifold method to certain functionals in Banach spaces, identifying critical points and values, and applies these results to prescribed energy problems and affine p-Laplacian equations.

Contribution

It develops the Nehari manifold approach for functionals with two critical points in Banach spaces and applies it to specific nonlinear problems.

Findings

Identified ground state levels and critical value sequences.

Applied the method to prescribed energy and affine p-Laplacian problems.

Extended the Nehari manifold technique to new classes of functionals.

Abstract

We proceed with the study of the Nehari manifold method for functionals in $C^1(X \setminus \{0\})$, where $X$ is a Banach space. We deal now with functionals whose fibering maps have two critical points (a minimiser followed by a maximiser). Under some additional conditions we show that the Nehari manifold method provides us with the ground state level and two sequences of critical values for these functionals. These results are applied to the class of {\it prescribed energy problems} as well as to the concave-convex problem for the {\it affine} $p$-Laplacian operator.