Infinitely many positive solutions for p-Laplacian equations with singular and critical growth terms

In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p-Laplacian type involving a singularity and a critical Sobolev exponent $\{\begin{array}{cc}-{\mathrm{\Delta }}_{p}u=u^{p^{\ast }-1}+\frac{\lambda }{|u{|}^{\gamma -1}u},\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0,\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}$ \( \textstyle\begin{cases} -\Delta _{p}u=u^{p^{*}-1}+\frac{\lambda}{|u|^{\gamma -1}u}, & \text{in} ~\Omega , \\ u=0, &\text{on}~\partial \Omega , \end{cases} \) where Ω is a bounded domain, $p^{\ast }=\frac{Np}{N-p}$ $p^{\ast}=\frac{Np}{N-p}$ ( $N\ge 3$ $N \geq 3$ ) is the critical Sobolev exponent and $\lambda >0$ $\lambda > 0$ . Based on the cutoff technique, we prove that the above problem possesses infinitely many positive solutions with $0<\gamma <1,1<p<N$ $0 < \gamma <1, 1< p < N$ , and $\lambda >0$ $\lambda > 0$ small enough.