Abstract
In this article, we consider the nonlinear Schrödinger equation \begin{equation} -\Delta u + V(x)u=|u|^{p-1}u \quad \text{in} \quad \mathbb{R}^N. \end{equation} Here $V$ is invariant under an orthogonal involution. The basic tool employed here is the concentration--compactness principle. A theorem on existence of a solution which changes sign exactly once is given.
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