On Regions of Existence and Nonexistence of solutions for a System of $p$-$q$-Laplacians

Abstract

We give a new region of existence of solutions to the superhomogeneous Dirichlet problem $$ \quad \begin{array}{l} -\Delta_{p} u= v^\delta\quad v>0\quad {in}\quad B,\cr -\Delta_{q} v = u^{\mu}\quad u>0\quad {in}\quad B, \cr u=v=0 \quad {on}\quad \partial B, \end{array}\leqno{(S_R)} $$ where $B$ is the ball of radius $R>0$ centered at the origin in $\RR^N.$ Here $\delta, \mu >0$ and $ \Delta_{m} u={\rm div}(|\nabla u|^{m-2}\nabla u) $ is the $m-$Laplacian operator for $m>1$.