where is an open bounded domain, are real parameters, in the variables with . and are called cooperative and noncooperative, respectively. In addition, is assumed to be positive for the noncooperative case. Note that systems are closely related to reaction-diffusion systems arising in various chemical/physical and biological phenomena.

In study of cooperative elliptic systems with degeneracy, we have developed a local Min-Orthogonal Method (LMOM, refer to the Paper for detail) aimed at finding multiple co-existing solutions based on a local Min-Orthogonal Characterization on saddle points. Such degenerate elliptic systems appear extensively in the study of spatial vector solitons in nonlinear optics and the study of multicomponent Bose-Einstein condensates in condensed matter physics, etc. For noncooperative elliptic systems, the big challenge we encountered is that under some standard assumptions their (weak) solutions correspond to critical (saddle) points (of certain strongly indefinite functionals) with infinite Morse index and co-index. To overcome this difficulty, we have recently proposed a local Min-Max-Orthogonal Characterization as well as a local Min-Max-Orthogonal Algorithm (LMMOA) to capture saddle points of such kind. Our numerical experiments and investigations show that the algorithm is very robust and reliable for computing saddle points with infinite Morse index. Some interesting numerical results have been obtained for noncooperative systems, Hamiltonian elliptic systems  as well as semilinear biharmonic problems.