Fully nontrivial solutions to elliptic systems with mixed couplings. (March 2022) - Clapp, Mónica
Abstract: We study the existence of fully nontrivial solutions to the system − Δ u i + λ i u i = ∑ j = 1 ℓ β i j | u j | p | u i | p − 2 u i i n Ω, i = 1, …, ℓ, in a bounded or unbounded domain Ω in R N, N ≥ 3 . The λ i '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 ( β i j ) is symmetric and admits a block decomposition such that the diagonal entries β i i are positive, the interaction forces within each block are attractive (i.e., all entries β i j 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 u i is nontrivial. We also find fully synchronized solutions (i.e., u i = c i u 1 for all i = 2, …, ℓ ) in the purely cooperative case whenever p ∈ ( 1, 2 ) .