Positive steady-states for two interacting species models with linear self-cross diffusions
The authors consider the following coupled system of elliptic partial differential equations: \[ -\Delta[(d_1+ \beta_{11} u+\beta_{12}\upsilon)u]=(a_1-b_{11}u-b_{12}\upsilon)u \quad\text{in }\Omega, \] \[ -\Delta[(d_2+\beta_{21}u+ \beta_{22}\upsilon)\upsilon]=(a_2\pm b_{21}u-b_{22}\upsilon)\upsilon \quad\text{in }\Omega, \] \[ \kappa_1\frac{\partial u}{\partial n}+ \tau_1u=0\quad\text{on }\partial \Omega, \] \[ \kappa_2\frac{\partial\upsilon}{\partial n} + \tau_2\upsilon=0\quad\text{on }\partial \Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb R^n\) with smooth boundary \(\partial \Omega\) and \(d_i, \beta_{ij}, a_i, b_{ij}, \kappa_i, \tau_i\) for \(i,j=1,2\) are nonnegative constants. They use the methods of fixed point index on cones in Banach spaces and that of decomposing operators to obtain sufficient conditions for the existence of positive solutions. Furthermore, a priori bounds for the positive solutions are obtained.
nonlinear system of elliptic partial differential equations
scientific article; zbMATH DE number 1941564