Existence of positive solutions for some Dirichlet problems with an asymptotically homogeneous operator

Abstract

Dirichlet problem of the form $${ m div,} (A(|Du|)Du)=f(u) { m in } Omega $$ $$ u = 0 { m on } partialOmega $$ is studied by using blow-up techniques. It is proven here that by choosing the functions $sA(s)$ and $f(s)$ among a certain class called {em asymptotically homogeneous}, the blow-up method still provides the a-priori bounds for positive solutions. Existence is proved then by using degree theory.

Keywords

• Dirichlet Problem
• Positive Solution
• Blow up.