Fully nontrivial solutions to elliptic systems with mixed couplings

We study the existence of fully nontrivial solutions to the system −Δui+λiui=∑j=1ℓβij|uj|p|ui|p−2uiinΩ,i=1,…,ℓ,in a bounded or unbounded domain Ω in RN,N≥3. The λi’s are real numbers, and the nonlinear term may have subcritical (1<[Formula presented]), critical (p=[Formula presented]), or supercritical growth (p>[Formula presented]). 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).