In this paper, we are concerned with the following (p, q)-Laplacian equation \(\begin{aligned} \left\{ \begin{array}{l} -\Delta _{p}w-\Delta _{q}w=\lambda |w|^{q-2}w+|w|^{s-2}w,~x\in {\mathbb {R}}^N,\\ \int _{{\mathbb {R}}^N}|w|^{q}dx=\rho ^{q}>0,~x\in {\mathbb {R}}^N, \end{array} \right. \end{aligned}\) where \(1<p<q<N\) . \(q<s<q^{*}:=\frac{Nq}{N-q}\) , \(\Delta _{p}w:=div(\left| w\right| ^{p-2}\nabla w)\) , \(\Delta _{q}w\) is similar. \(q+\frac{q^{2}}{N}\) is the mass critical exponent for prescribed \(L^{q}\) -norm problems for the (p, q)-Laplacian. We consider the corresponding minimization problem. When \(s<q+\frac{q^{2}}{N}\) , it admits a minimizer. If \(s=q+\frac{q^{2}}{N}\) , a non-existence result is given. If \(s>q+\frac{q^{2}}{N}\) , local minimizer is obtained. The methods used here contain mountain-pass argument on the prescribed \(L^q\) -norm constraint, a new Moser’s iteration and Pohozaev’s identity. We point out that if \(q<s<p^{*}\) , the Lagrange multiplier \(\lambda <0\) . Here \(p^{*}:=\frac{Np}{N-p}\) and \(p^{*}<q^{*}\) . Some asymptotical behaviours are also given as \(\rho \rightarrow 0^{+}\) .
