{"paper":{"title":"Asymptotic properties of ground states of scalar field equations with a vanishing parameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","nlin.PS"],"primary_cat":"math.AP","authors_text":"Cyrill B. Muratov, Vitaly Moroz","submitted_at":"2012-02-15T20:43:49Z","abstract_excerpt":"We study the leading order behaviour of positive solutions of the equation -\\Delta u +\\varepsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\\qquad x\\in\\R^N, where $N\\ge 3$, $q>p>2$ and when $\\varepsilon>0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$, $q$ and $N$. The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent $p^\\ast=\\frac{2N}{N-2}$. For $p<p^\\ast$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p>p^\\ast$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.3426","kind":"arxiv","version":2},"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"}