Journal of the
Korean Mathematical Society JKMS

In this paper, we give several sufficient conditions ensuring that any positive radial solution $(u,v)$ of the following $\gamma$-Laplacian systems in the whole space ${\mathbb R^n}$ has the components symmetry property $u\equiv v$ $$ \left\{ \begin{array}{ll} &-{\rm div}(|\nabla u|^{\gamma-2}\nabla u) =f(u,v)\quad \text{in} \ \ {\mathbb R^n},\\ &-{\rm div}(|\nabla v|^{\gamma-2}\nabla v) =g(u,v)\quad \text{in} \ \ {\mathbb R^n}. \end{array} \right. $$ Here $n>\gamma$, $\gamma>1$. Thus, the systems will be reduced to a single $\gamma$-Laplacian equation: $$ -{\rm div}(|\nabla u|^{\gamma-2} \nabla u) =f(u) \quad \text{in} \ \ {\mathbb R^n}. $$ Our proofs are based on suitable comparation principle arguments, combined with properties of radial solutions.

Keywords: $\gamma$-Laplacian system, components symmetry property, radial solution

MSC numbers: 35B08, 35J47, 35J6