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

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.