Blow-up solutions for a Blow-up solutions for a Blow-up solutions for a $p$-Laplacian elliptic equation of logistic type with singular nonlinearity

TL;DR

This paper investigates the existence, uniqueness, and boundary behavior of blow-up solutions for a $p$-Laplacian elliptic equation with singular nonlinearity, using variational methods and comparison principles.

Contribution

It introduces a novel approach combining sub and super solutions with variational techniques to analyze blow-up solutions for singular $p$-Laplacian problems.

Findings

Established existence and uniqueness of blow-up solutions.

Determined the exact boundary rate of solutions.

Developed a comparison principle for the problem.

Abstract

In this paper, we deal with existence, uniqueness and exact rate of boundary behavior of blow-up solutions for a class of logistic type quasilinear problem in a smooth bounded domain involving the $p$-Laplacian operator, where the nonlinearity can have a singular behavior. In the proof of the existence of solution, we have used the sub and super solution method in conjunction with variational techniques and comparison principles. Related to the rate on boundary and uniqueness, we combine a comparison principle proved in the present paper together with the our result of existence of solution.