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 $\begin{equation}\begin{array}{l}-\Delta_{p}u=v^{\delta},\quad v>0\ \mbox{in}\ B,\\-\Delta_{q}v=u^{\mu},\quad u>0\ \mbox{in}\ B,\\u=v=0\quad\mbox{on}\ \curpartial B,\end{array}\label{(S_R)}\end{equation}$ where B is the ball of radius R>0 centered at the origin in $\mathbb {R}^{N}$ . Here δ,μ>0 and Δ_mu=div(|∇u|^m−2∇u) is the m-Laplacian operator for m>1.