Fully nontrivial solutions to elliptic systems with mixed couplings

We study the existence of fully nontrivial solutions to system ??ui+?iui=?j=1??ij|uj|p|ui|p?2uiin?,i=1,…,?,in a bounded or unbounded domain ? in RN, N?3. The ?i’s are real numbers, and nonlinear term may have subcritical (1NN?2). matrix (?ij) is symmetric admits block decomposition such that diagonal entries ?ii positive, interaction forces within each attractive (i.e., all ?ij non-negative) between different blocks repulsive other non-positive). obtain new multiplicity results solutions, i.e., where every component ui nontrivial. also find synchronized ui=ciu1 for i=2,…,?) purely cooperative case whenever p?(1,2).