Fully nontrivial solutions to elliptic systems with mixed couplings

We study the existence of fully nontrivial solutions to the system -Delta u(i) +lambda(i)u(i) = Sigma(l)beta(ij)|u(j)|p|u(i)|p(-2)u(i) in Omega, i = 1,.. .,l, in a bounded or unbounded domain Omega in R-N, N >= 3. The lambda(i)'s are real numbers, and the nonlinear term may have subcritical (1 < p N/N-2). The matrix (beta(ij)) is symmetric and admits a block decomposition such that the diagonal entries beta ii are positive, the interaction forces within each block are attractive (i.e., all entries beta(ij) 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 is an element of(1, 2). (C) 2021 Elsevier Ltd. All rights reserved.