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In the critical regime p ≥ (N+2)/(N-2) on a ball and 0 < β < 2/(p-1) a radial positive solution exists."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The proofs assume that positive solutions exist for each fixed β > 0 (or that the problem is well-posed in the appropriate function space) and that the domain is smooth; the existence statement for the critical case further assumes the domain is a ball and restricts β to a specific interval."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"As beta goes to zero, solutions to -Δu = u^p with Robin conditions behave like constants that blow up for p<1, stay finite for p=1, and go to zero for p>1; existence of radial solutions is shown for balls in the critical regime when beta is small."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Positive solutions to the Robin problem -Δu = u^p approach a constant as β tends to zero, with uniform blow-up for p < 1, convergence to a fixed constant for p = 1, and uniform decay to zero for p > 1."}],"snapshot_sha256":"70d26819962dca7bd86716fa128d679f7f7f01a190ab9818f7e02b3726de3691"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2604.10139/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we investigate the asymptotic behavior, as $\\beta \\to 0$, of positive solutions to the semilinear elliptic Robin problem \\begin{equation*} \\begin{cases} -\\Delta u = u^p, & \\text{in } \\Omega,\\\\ u > 0, & \\text{in } \\Omega,\\\\ \\frac{\\partial u}{\\partial \\nu} + \\beta u = 0, & \\text{on } \\partial \\Omega, \\end{cases} \\end{equation*} where $p \\ge 0$, $\\beta > 0$, and $\\Omega$ is a bounded smooth domain.\n  We will prove that, for all $p\\ge0$, the solution $u_\\beta$ behaves like a constant as $\\beta\\to0$. However, the value of this constant is strongly influenced by the value of $p$. Inde","authors_text":"Massimo Grossi, Mengyao Chen, Qi Li","cross_cats":[],"headline":"Positive solutions to the Robin problem -Δu = u^p approach a constant as β tends to zero, with uniform blow-up for p < 1, convergence to a fixed constant for p = 1, and uniform decay to zero for p > 1.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-04-11T10:08:31Z","title":"Asymptotic behavior of solutions to elliptic problems with Robin boundary conditions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.10139","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-10T16:20:06.439632Z","id":"1994ec6d-3fd8-44d8-9419-d3bc3975b801","model_set":{"reader":"grok-4.3"},"one_line_summary":"As beta goes to zero, solutions to -Δu = u^p with Robin conditions behave like constants that blow up for p<1, stay finite for p=1, and go to zero for p>1; existence of radial solutions is shown for balls in the critical regime when beta is small.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Positive solutions to the Robin problem -Δu = u^p approach a constant as β tends to zero, with uniform blow-up for p < 1, convergence to a fixed constant for p = 1, and uniform decay to zero for p > 1.","strongest_claim":"For all p ≥ 0 the solution u_β behaves like a constant as β → 0; specifically, u_β blows up uniformly if 0 ≤ p < 1, converges to a constant if p = 1, and converges uniformly to zero if p > 1. 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