Symmetry of components for radial solutions of \(\gamma\)-Laplacian systems (Q2813352)

Under consideration is the \(\gamma\)-Laplacian system NEWLINE\[NEWLINE -\operatorname{div}(|\nabla u|^{\gamma-2}\nabla u)=f(u,v),\;-\operatorname{div}(|\nabla v|^{\gamma-2}\nabla v)=g(u,v),\;x\in{\mathbb R}^{n}.\eqno{(1)} NEWLINE\]NEWLINE It is assumed that \((u-v)(f(u,v)-g(u,v))\leq 0\) for all \(u,v\geq 0\) and, for every \(\eta>0\), there exists a constant \(c>0\) such that NEWLINE\[NEWLINE f(u,v)\geq cu^{p},\;g(u,v)\geq cv^{p} NEWLINE\]NEWLINE for all \(v\geq \eta, u\geq 0\) and \(u\geq \eta, v\geq 0\), respectively, where \(0\leq p,t\leq (n-1)\gamma/(n-\gamma) -1\). Under these conditions it is proven that every nonnegative radial solution \((u,v)\) to the system (1) is semitrivial or \(u=v\). The former means that either \(u\equiv 0\) or \(v\equiv 0\).