Elliptic Hamilton-Jacobi systems and Lane-Emden Hardy-H{\'e}non equations

TL;DR

This paper investigates solutions to elliptic Hamilton-Jacobi systems and their connection to Lane-Emden Hardy-H{é}non equations, providing a comprehensive analysis of radial solutions, a priori estimates, and Liouville theorems in various parameter ranges.

Contribution

It establishes a novel link between Hamilton-Jacobi systems and Hardy-H{é}non equations, analyzing solutions in less-studied parameter regimes and offering new classification results.

Findings

Complete description of radial solutions

Nonradial a priori estimates

Liouville type theorems for various parameters

Abstract

Here we study the solutions of any sign of the system --$\Delta$u 1 = |$\nabla$u 2 | p , --$\Delta$u 2 = |$\nabla$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{\'e}non equations --$\Delta$ N p w= $\epsilon$r $\sigma$ w q , $\epsilon$ = $\pm$1, where u $\rightarrow$ $\Delta$ 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.