A note on existence of antisymmetric solutions for a class of nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\triangle u + V(x)u= f(u)\quad {\rm in}\quad \mathbb{R}^N.$$\end{document}We assume that V is invariant under an orthogonal involution and show the existence of a particular type of sign changing solution. The basic tool employed here is the Concentration–Compactness Principle.