Elliptic Hamilton-Jacobi systems and Lane-Emden Hardy-H{é}non equations

Here we study the solutions of any sign of the system –$\mathrm{\Delta }$u 1 = |$\mathrm{\nabla }$u 2 | p , –$\mathrm{\Delta }$u 2 = |$\mathrm{\nabla }$u 1 | q , in a domain of R N , N 3 and p, q > 0, pq > 1… We show their relation with Lane-Emden Hardy-H{é}non equations –$\mathrm{\Delta }$ N p w= $\epsilon$r $\sigma$ w q , $\epsilon$ = $\pm$1, where u $\to$ $\mathrm{\Delta }$ N p u (p > 1) is the p-Laplacian in dimension N, q > p – 1 and $\sigma$ $\in$ R. This leads us to explore these equations in not often tackled ranges of the parameters N, p, $\sigma$. 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.