Ground states to a quasilinear Schrödinger equation with Berestycki-Lions type nonlinearities

In this paper, we study the following quasilinear {S}chrödinger equation: $$\{\begin{array}{l}-\mathrm{\Delta }u-\frac{\kappa }{2}\mathrm{\Delta }\left(u^2\right)u=h\left(u\right)\text{ in }{\mathbb{R}}^N,\\ u\in H^1\left({\mathbb{R}}^N\right),\end{array}$$ where $N\ge 3,\kappa >0$ is a parameter, and $h$ satisfies Berestycki-Lions condition. By using a critical point theory on a topological manifold, we obtain the existence of a ground state for $N\ge 3$, a nonradial ground state for $N\ge 4$, and infinitely many nonradial solutions for $N=4$ or $N\ge 6$. Our results generalize several classical works into quasilinear equations.