On a class of coupled nonlinear elliptic equations

Abstract:

For a class of coupled elliptic equations we discuss the existence and property of positive solutions. We first survey some general results according to the coupling mechanism being attractive or repulsive, about existence, multiplicity and qualitative properties of positive solutions. We then turn to the case of mixed couplings and talk about more details in constructing various types of solutions with simultaneous synchronization and segregation phenomena. Some questions will be raised for all cases for further studies.