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Consider the equation \begin{equation*} -\Delta_p u =\lambda|u|^{p-2}u+\mu|u|^{q-2}u+|u|^{p^\ast-2}u\ \ {\rm in}\ \R^N \end{equation*}under the normalized constraint $$\int_{ \R^N}|u|^p=c^p,$$ where$-\Delta_pu={\rm div} (|\nabla u|^{p-2}\nabla u)$, $1<p<N$,$p<q<p^\ast=\frac{Np}{N-p}$, $c,\mu>0$ and $\lambda\in\R$. In the purely$L^p$-subcritical case, we obtain the existence of ground state solution byvirtue of truncation technique, and obtain multiplicity of normalizedsolutions. In the purely $L^p$-critical and supercritical case, we drive theexistence of positive ground state solution, respectively.  Finally, we investigate the asymptotic behavior of ground state solutionsobtained above as $\mu\to0^+$.

Normalized solutions for some quasilinear elliptic equation with critical Sobolev exponent