On certain elliptic systems with nonlinear self-cross diffusions (Q1422589)

In the present study the authors deal with the existence of positive solutions to the following strongly-coupled nonlinear elliptic system \[ \begin{gathered} -\Delta[\varphi(u,v)u]= uf(u,v)\quad\text{in }\Omega,\\ -\Delta[\psi(u,v)v]= vg(u,v)\quad\text{in }\Omega,\\ k_1{\partial u\over\partial\nu}+ \tau_1 u= 0\quad\text{on }\partial\Omega,\\ k_2{\partial v\over\partial\nu}+ \tau_2 v= 0\quad\text{on }\partial\Omega,\end{gathered}\tag{1} \] where \(\Omega\) is a bounded domain of \(\mathbb R^n\) with smooth boundary \(\partial\Omega\) and \(\nu\) is the unit outward normal to \(\partial\Omega\). The authors give sufficient conditions for the existence of positive solution to the (1). To this end, they use the method of decomposing operators and nonlinear fixed point theorem.