Existence of radial solution for a quasilinear equation with singular nonlinearity

TL;DR

This paper proves the existence of radially symmetric solutions for a quasilinear PDE with singular nonlinearity in a bounded domain, using blow-up arguments, Liouville theorems, and a regularization approach.

Contribution

It introduces a novel combination of blow-up and Liouville techniques to establish solutions for a class of singular quasilinear equations.

Findings

Existence of solutions for small positive mbda

Solutions are radially symmetric and positive

Method applicable to equations with singular nonlinearities

Abstract

We prove that the equation \begin{eqnarray*} -\Delta_p u =\lambda\Big( \frac{1} {u^\delta} + u^q + f(u)\Big)\;\text{ in } \, B_R(0) u =0 \,\text{ on} \; \partial B_R(0), \quad u>0 \text{ in } \, B_R(0) \end{eqnarray*} admits a weak radially symmetric solution for $\lambda>0$ sufficiently small, $0<\delta<1$ and $p-1<q<p^{*}-1$. We achieve this by combining a blow-up argument and a Liouville type theorem to obtain a priori estimates for the regularized problem. Using a variant of a theorem due to Rabinowitz we derive the solution for the regularized problem and then pass to the limit.