Abstract

We investigate the following quasilinear and singular problem, $$ \hbox{t}o 2.7cm{}\left\lbrace \begin{array}{ll} - \Delta _p u = \frac{\lambda }{u^\delta } + u^q \quad & \mbox { in }\,\Omega ;\\ u\vert _{\partial \Omega } = 0,\quad u > 0\quad & \mbox { in }\,\Omega, \end{array} \right.\hbox{t}o 2.7cm{}\hbox{ {\rm }} $$ where $\Omega $ is an open bounded domain with smooth boundary, $1 0$, and $0 N$. We employ variational methods in order to show the existence of at least two distinct solutions of problem in $W_0^{1,p}$. While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions to problem in $C^{1,\beta }$ with some $\beta \in $. Furthermore, we show that $\delta < 1$ is a reasonable sufficient condition to obtain solutions of problem in $C^1$