Wiener Criteria for Existence of Large Solutions of Quasilinear Elliptic Equations with Absorption

Abstract

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 $(q-p+1)+b(s-p+1)>0$, $\gamma>0$.