Qualitative properties of a nonlinear system involving the $p$-Laplacian operator

Abstract

In this article we consider the nonlinear system involving the $p$-Laplacian $$\{\begin{array}{lccc}|u^{\prime }{|}^{p-2}{u}^{\prime \prime }=u^{p-1}{v}^p& |v^{\prime }{|}^{p-2}{v}^{\prime \prime }=v^{p-1}{u}^p& \mathrm{on}\mathbb{R},u\ge 0,v\ge 0& \end{array}$$ for which we prove symmetry, asymptotic behavior and non degeneracy properties. This can help to a better understanding to what happens in the $N$ dimensional case, for which several authors prove a De Giorgi Type result under some additional growth and monotonicity assumptions.

arXiv comment29 pages, no figure