Existence of positive solutions for nonlinear fractional systems in bounded domains

Abstract

We prove the existence of positive continuous solutions to the nonlinear fractional system $$displaylines{ (-Delta|_D) ^{alpha/2}u+lambda g(.,v) =0, cr (-Delta|_D) ^{alpha/2}v+mu f(.,u) =0, }$$ in a bounded $C^{1,1}$-domain $D$ in $mathbb{R}^n$ $(ngeq 3)$, subject to Dirichlet conditions, where $0<alpha leq 2$, $lambda $ and $mu $ are nonnegative parameters. The functions f and g are nonnegative continuous monotone with respect to the second variable and satisfying certain hypotheses related to the Kato class.

Keywords

• Fractional nonlinear systems
• Green function
• positive solutions
• blow-up solutions