Existence and non-existence of Blow-up solutions for a non-autonomous problem with indefinite and gradient terms

TL;DR

This paper investigates conditions under which solutions to a non-autonomous quasilinear PDE either exist or do not exist, especially solutions that blow up at infinity, using advanced analytical methods.

Contribution

It provides more general criteria for existence and non-existence of solutions to a complex PDE with indefinite and gradient terms, extending previous results.

Findings

Established new existence conditions for blow-up solutions.

Derived non-existence results using Mitidieri-Pohozaev method.

Analyzed an associated convective ground state problem.

Abstract

We deal with existence and non-existence of non-negative entire solutions that blow-up at infinity for a quasilinear problem depending on a non-negative real parameter. Our main objectives in this paper are to provide far more general conditions for existence and non-existence of solutions. To this end, we explore an associated $\mu$-parameter convective ground state problem, sub and super solutions method combined and an approximation arguments to show existence of solutions. To show the result of non-existence of solutions, we follow an idea due to Mitidieri-Pohozaev.