Existence, Uniqueness and Stability of Positive Solutions to Sublinear Elliptic Systems

Reaction–diffusion systems are used to model many chemical and biological phenomena in the natural world [25, 26], and systems of coupled partial differential equations are also used in other physical models such as nonlinear Schrödinger systems in multi-component Bose–Einstein condensates and nonlinear optics [19, 23]. The steady-state solutions or standing-wave solutions of such systems of nonlinear partial differential equations satisfy a nonlinear elliptic system with more than one equation. Much effort has been devoted to the existence of solutions of such systems (see, for example, [4, 8, 10, 11, 13, 15–17, 24, 30, 33, 34]), but it is usually difficult to determine whether or not the solution is unique. We consider the positive solutions of a semilinear elliptic system of the form