In this paper, we study the following degenerate elliptic problem with prescribed mass \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta _\gamma u+\lambda u=f(u)\ \text {in}\ {\mathbb {R}^{N}}, \\ \int _{\mathbb {R}^{N}}|u|^2\mathrm dx=a^2, \end{array} \right. \end{aligned}\) where \(\Delta _\gamma :=\Delta _x+(1+\gamma )^2|x|^{2\gamma }\Delta _y\) is the Grushin operator, \(\gamma >0\) , \(z=(x,y)\in {\mathbb {R}}^{N}\) , \(N=k+l\) , \(k,l\ge 1\) , \(f\in C({\mathbb {R}},{\mathbb {R}})\) and \(\lambda \in {\mathbb {R}}\) appears as an unknown Lagrange multiplier. We establish the existence of normalized solutions for general mass subcritical, general mass supercritical and mixed nonlinearities. In particular, for the mass subcritical case, we give the least action characterization, which states that any normalized ground state is a least action solution of the associated unconstrained problem and vice versa. For the mixed case, we consider \(f(u)=|u|^{p-2}u+|u|^{q-2}u\) with \(2<p<2+\frac{4}{N_\gamma }<q\le 2_\gamma ^*\) , which involves the critical Sobolev exponent \(2_\gamma ^*=\frac{2 N_\gamma }{N_\gamma -2}\) . The number \(2+\frac{4}{N_\gamma }\) is the mass critical exponent and \(N_\gamma =k+(1+\gamma )l\) is the so-called homogeneous dimension attached to the Grushin operator. Our results are not only the first contribution to the above degenerate elliptic problem with prescribed mass, but also present the new existence for the unconstrained problem.
