Existence Results for the Kirchhoff Type Equation with a General Nonlinear Term

This paper is mainly concerned with existence and nonexistence results for solutions to the Kirchhoff type equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ - \left({a + b\,\int_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}}} \right)\Delta u + V\left(x \right)u = f\left(u \right)\,\,{\rm{in}}\,\,{\mathbb{R}^3}$$\end{document}, with the general hypotheses on the nonlinearity f being as introduced by Berestycki and Lions. Our analysis introduces variational techniques to the analysis of the effect of the nonlinearity, especially for those cases when the concentration-compactness principle cannot be applied in terms of obtaining the compactness of the bounded Palais-Smale sequences and a minimizing problem related to the existence of a ground state on the Pohozaev manifold rather than the Nehari manifold associated with the equation.