Wiener criteria for existence of large solutions of quasilinear elliptic equations with absorption

TL;DR

This paper establishes Wiener-type criteria involving capacities to determine the existence of large solutions for certain quasilinear elliptic equations with absorption in bounded domains.

Contribution

It introduces new capacity-based Wiener criteria for large solutions of quasilinear elliptic equations with absorption terms, extending previous existence results.

Findings

Derived sufficient conditions for large solutions using Wiener tests.

Applied criteria to specific equations with exponential and power nonlinearities.

Provided a framework connecting capacities with solution existence in bounded domains.

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