Some 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 unbounded $C^1$ functionals in Banach spaces, establishing existence results for ground states and multiple critical points, with applications to various nonlinear PDE problems.

Contribution

It introduces a novel application of the Nehari manifold method to unbounded $C^1$ functionals in Banach spaces, including new existence results for critical points.

Findings

Existence of ground states for unbounded $C^1$ functionals.

Infinitely many critical points established.

Applications to prescribed energy, affine $p$-Laplacian, and Kirchhoff problems.

Abstract

Given a real Banach space $X$, we show that the Nehari manifold method can be applied to functionals which are $C^1$ in $X \setminus \{0\}$. In particular we deal with functionals that can be unbounded near $0$, and prove the existence of a ground state and infinitely many critical points for such functionals. These results are then applied to three classes of problems: the {\it prescribed energy problem} for a family of functionals depending on a parameter, problems involving the {\it affine} $p$-Laplacian operator, and degenerate Kirchhoff type problems.