In this paper, we study the following quasilinear (p, q)-equation \(\begin{aligned} -\Delta _p u-u\Delta _q u^2+\lambda |u|^{p-2}u=\mu |u|^{l-2}u+|u|^{m-2}u,\quad \text {in}\;{\mathbb {R}}^N, \end{aligned}\) with prescribed mass \(\begin{aligned} \int _{{\mathbb {R}}^N}|u|^p=c^p, \end{aligned}\) where \(c>0\) , \(\mu \ge 0\) , \(2\le p<q<N\) , \(\Delta _p u=\text {div}(|\nabla u|^{p-2}\nabla u)\) , \(\begin{aligned} \Delta _q u^2=2^{q-1}(|u|^{q-2}u\text {div}(|\nabla u|^{q-2}\nabla u)+(q-1)|u|^{q-3}u|\nabla u|^q), \end{aligned}\) \(\lambda \) is a Lagrange multiplier and \(p<l<m<p^*:=\frac{Np}{N-p}\) . We first consider the case \(\frac{pq}{N}+2q<m<p^*\) , \(\mu =0\) and we prove the existence of normalized solutions in the purely supercritical case by using the perturbation method. Then we obtain multiplicity of normalized solutions in the case \(\begin{aligned} p<l<\frac{p^2}{N}+p,\quad \frac{pq}{N}+2q<m<p^*,\quad \mu >0, \end{aligned}\) namely when the two nonlinearities have a different character with respect to the \(L^p\) -critical exponent. This case presents substantial differences concerning the purely supercritical case.
