The aim of this paper is to study the quasilinear elliptic equation \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u-\Delta (u^2)u-\lambda u=Q(x)h(u),\,\, x\in \mathbb {R}^N,\\ \int _{\mathbb {R}^N}|u|^2\text {d}x=a, \end{array} \right. \end{aligned}\) where \(N\ge 2\) , \(\lambda \in \mathbb {R}\) , \(a>0\) is a given mass and \(Q\in \mathcal {C}(\mathbb {R}^N,[0,+\infty ))\) is a nonlinear function. By a suitable change of variables, the existence of ground state normalized solutions with the non-autonomous nonlinearity are established via a dual method, which has rarely been considered for quasilinear elliptic problems. Our results are the supplement of quasilinear elliptic equations with prescribed mass.
