Symmetry and monotonicity results for positive solutions of p-Laplace systems

Abstract

In this paper, we extend to a system of the type \[ \begin{cases} \begin{array}{c} -\Delta_{p_{1}}u=f\left(v\right)\quad in\,\Omega,\quad u>0\quad in\,\Omega\quad u=0\quad on\,\partial\Omega,\\ -\Delta_{p_{2}}v=g\left(u\right)\quad in\,\Omega,\quad v>0\quad in\,\Omega\quad v=0\quad on\,\partial\Omega, \end{array}\end{cases} \] where $\Omega\subset\mathbb{R}^{N}$ is bounded, the monotonicity and simmetry results of Damascelli and Pacella obtained in $\left[5\right]$ in the case of a scalar p-Laplace equation with 1 < p < 2. For this purpose, we use the moving hyperplanes method and we suppose that $f,g\::\:\mathbb{R}\rightarrow\mathbb{R}^{+}$ are increasing on $\mathbb{R}^{+}$ and locally Lipschitz continuous on $\mathbb{R}$ and p$_{1},$ p$_{2}$ $\epsilon$ (1, 2) or p$_{1}\:\epsilon\left(1,\infty\right),$ p$_{2}$=2