We obtain sufficient conditions expressed in terms of Wiener type tests involving Hausdorff or Bessel capacities for the existence of large solutions to equations (1) $-\Gd_pu+e^{\lambda u}+\beta=0$ or (2) $-\Gd_pu+\lambda |u|^{q-1}u+\beta=0$ in a bounded domain $\Gw$ when $q>p-1>0, \lambda>0$ and $\beta\in\mathbb{R}$. We apply our results to equations (3) $-\Gd_pu+a\abs{\nabla u}^{q}+bu^{s}=0$, (4) $\Gd_p u+u^{-\gamma}=0$ with $10, b\geq 0$ and $(q-p+1)+b(s-p+1)>0$, $\gamma>0$.