{"paper":{"title":"Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Florica C. C\\^irstea, Fr\\'ed\\'eric Robert","submitted_at":"2016-01-20T19:36:34Z","abstract_excerpt":"Given $B_1(0)$ the unit ball of $\\mathbb{R}^n$ ($n\\geq 3$), we study smooth positive singular solutions $u\\in C^2(B_1(0)\\setminus \\{0\\})$ to $-\\Delta u=\\frac{u^{2^\\star(s)-1}}{|x|^s}-\\mu u^q$. Here $0< s<2$, $2^\\star(s):=2(n-s)/(n-2)$ is critical for Sobolev embeddings, $q>1$ and $\\mu> 0$. When $\\mu=0$ and $s=0$, the profile at the singularity $0$ was fully described by Caffarelli-Gidas-Spruck. We prove that when $\\mu>0$ and $s>0$, besides this profile, two new profiles might occur. We provide a full description of all the singular profiles. Special attention is accorded to solutions such that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.05382","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}