Fully nontrivial solutions to elliptic systems with mixed couplings

We study the existence of fully nontrivial solutions to the system −∆ui + λiui = l ∑ j=1 βij |uj | |ui| p−2 ui in Ω, i = 1, . . . , l, in a bounded or unbounded domain Ω in R , N ≥ 3. The λi’s are real numbers, and the nonlinear term may have subcritical (1 < p < N N−2 ), critical (p = N N−2 ), or supercritical growth (p > N N−2 ). The matrix (βij) is symmetric and admits a block decomposition such that the diagonal entries βii are positive, the interaction forces within each block are attractive (i.e., all entries βij in each block are non-negative) and the interaction forces between different blocks are repulsive (i.e., all other entries are non-positive). We obtain new existence and multiplicity results of fully nontrivial solutions, i.e., solutions where every component ui is nontrivial. We also find fully synchronized solutions (i.e., ui = ciu1 for all i = 2, . . . , l) in the purely cooperative case whenever p ∈ (1, 2).