Multiple non-spherically symmetric solutions for a nonlinear Schrödinger equation

In this paper, we are concerned with a nonlinear Schrödinger equation as follows: $$ - {\Delta u} + {\lambda u} = g\left ( u\right ) \text{ in }{\mathbb{R}}^{N}, $$ where g satisfies the Berestycki–Lions conditions. We are going to prove the equation mentioned above has at least $\kappa \left ( N\right )$ non-spherically symmetric solutions with $\kappa \left ( N\right ) \rightarrow \infty $ as $N \rightarrow \infty $. The proof is based on a property of invariant subspace under group action.