| Abstract: |
| We are concerned with the study of the solutions of any sign of the system
\begin{equation*}
\left\{
\begin{array}{c}
{-\Delta u}_{1}=\left\vert \nabla u_{2}\right\vert ^{p}, \
{-\Delta u}_{2}=\left\vert \nabla u_{1}\right\vert ^{q},%
\end{array}%
\right.
\end{equation*}%
in a domain of $\mathbb{R}^{N},$ $N\geqq 3$ and $p,q>0,$ $pq>1..$ We show
their relation with Lane-Emden Hardy-H\`{e}non equations
\begin{equation*}
-{\Delta }_{\mathbf{p}}^{\mathbf{N}}w{=\varepsilon r}^{\sigma }w^{\mathbf{q}%
},\qquad \varepsilon =\pm 1,
\end{equation*}%
where $u\mapsto {\Delta }_{\mathbf{p}}^{\mathbf{N}}{u}$ $(\mathbf{p}>1)$ is
the $\mathbf{p}$-Laplacian in dimension $\mathbf{N,}$ $\mathbf{q}>\mathbf{p}%
-1$ and $\sigma \in \mathbb{R}.$ We make a
complete description of the radial solutions of the system and of the
Hardy-Henon equations and give nonradial a priori estimates and Liouville
type results. |
|