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).

Fully nontrivial solutions to elliptic systems with mixed couplings / Clapp, M.; Pistoia, A.. - In: NONLINEAR ANALYSIS. - ISSN 0362-546X. - 216:(2022). [10.1016/j.na.2021.112694]

Fully nontrivial solutions to elliptic systems with mixed couplings

Pistoia A.
2022

Abstract

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).
2022
Mixed cooperation and competition; Nehari manifold; Positive and sign-changing solutions; Weakly coupled systems
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Fully nontrivial solutions to elliptic systems with mixed couplings / Clapp, M.; Pistoia, A.. - In: NONLINEAR ANALYSIS. - ISSN 0362-546X. - 216:(2022). [10.1016/j.na.2021.112694]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1680269
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