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In this talk, I will present some recent results on a quasilinear Schrödinger equation with a power nonlinearity. After showing the uniqueness and the non-degeneracy of the positive radial solution $u_\omega$ for all $\omega>0$, I will describe its asymptotic behavior in the limit $\omega\to 0$. This gives some important information about the orbital stability of $u_\omega$ and the uniqueness of normalized ground states. Joint work with François Genoud.