Necessary and sufficient conditions for existence of Blow-up solutions for elliptic problems in Orlicz-Sobolev spaces

TL;DR

This paper generalizes classical results on blow-up solutions for quasilinear elliptic problems to Orlicz-Sobolev spaces, encompassing models in physics and engineering, using comparison, variational, and topological methods.

Contribution

It extends Keller-Osserman type conditions for existence of blow-up solutions to a broader class of elliptic problems in Orlicz-Sobolev spaces, including various physical models.

Findings

Established necessary and sufficient conditions for blow-up solutions.

Generalized classical results to Orlicz-Sobolev space setting.

Applied comparison, variational, and topological methods.

Abstract

This paper is principally devoted to revisit the remarkable works of Keller and Osserman and generalize some previous results related to the those for the class of quasilinear elliptic problem $$ \left\{ \begin{array}{l} {\rm{div}} \left( \phi(|\nabla u|)\nabla u\right) = a(x)f(u)\quad \mbox{in } \Omega,\\ u\geq0\ \ \mbox{in}\ \Omega,\ \ u=\infty\ \mbox{on}\ \partial\Omega, \end{array} \right. $$ where either $\Omega \subset \mathbb{R}^N$ with $N \geq 1$ is a smooth bounded domain or $\Omega = \mathbb{R}^N$. The function $\phi$ includes special cases appearing in mathematical models in nonlinear elasticity, plasticity, generalized Newtonian fluids, and in quantum physics. The proofs are based on comparison principle, variational methods and topological arguments on the Orlicz-Sobolev spaces.