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We mainly investigate the effects of the potentials and the nonlinear coupling terms on the structure of solutions.\n  Applying the Lyapunov-Schmidt reduction method, we prove the ex"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Applying the Lyapunov-Schmidt reduction method, we prove the existence of infinitely many solutions to the system. Specifically, the solutions we obtain satisfy that some components are synchronized with each other but segregated from the others, and that some components are positive while others are sign-changing.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The potentials V_i(x) are continuous bounded radial functions and the coupling constants β_ij allow the reduced functional to have the required critical points for the mixed synchronization-segregation and sign patterns; the abstract does not specify the precise range of β_ij or decay conditions on V_i.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Infinitely many solutions exist for the 3-component Hartree system with some positive and some sign-changing components, constructed via Lyapunov-Schmidt reduction as the first such mixed-sign application.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A three-component nonlinear Hartree system possesses infinitely many multi-peak solutions with mixed synchronization, segregation, and sign patterns.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"2207a817a91306c880037f569518c0284e1905d8275034715216213fa2706007"},"source":{"id":"2605.13531","kind":"arxiv","version":1},"verdict":{"id":"fbea96cc-c5df-4606-a5d4-64cbba73fe8c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:57:04.959517Z","strongest_claim":"Applying the Lyapunov-Schmidt reduction method, we prove the existence of infinitely many solutions to the system. Specifically, the solutions we obtain satisfy that some components are synchronized with each other but segregated from the others, and that some components are positive while others are sign-changing.","one_line_summary":"Infinitely many solutions exist for the 3-component Hartree system with some positive and some sign-changing components, constructed via Lyapunov-Schmidt reduction as the first such mixed-sign application.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The potentials V_i(x) are continuous bounded radial functions and the coupling constants β_ij allow the reduced functional to have the required critical points for the mixed synchronization-segregation and sign patterns; the abstract does not specify the precise range of β_ij or decay conditions on V_i.","pith_extraction_headline":"A three-component nonlinear Hartree system possesses infinitely many multi-peak solutions with mixed synchronization, segregation, and sign patterns."},"references":{"count":54,"sample":[{"doi":"","year":2004,"title":"Ackermann, On a periodic Schr¨ odinger equation with nonlocal superlinear part, Math","work_id":"51b1a414-09b0-43cc-b0a8-79eb7ee7fa3c","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"C. 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