{"paper":{"title":"Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David G. Costa, Pedro M. Gir\\~ao","submitted_at":"2014-07-23T14:25:41Z","abstract_excerpt":"Let $\\Omega$ be a smooth bounded domain in $\\mathbb{R}^{N}$, with $N\\geq 5$, $a>0$, $\\alpha\\geq 0$ and $2^*=\\frac{2N}{N-2}$. We show that the the exponent $q=\\frac{2(N-1)}{N-2}$ plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem $$ \\left\\{\\begin{array}{ll} -\\Delta u+au=u^{2^*-1}-\\alpha u^{q-1}&\\mbox{in}\\ \\Omega,\\\\ u>0&\\mbox{in}\\ \\Omega,\\\\ \\frac{\\partial u}{\\partial\\nu}=0&\\mbox{on}\\ \\partial\\Omega. \\end{array}\\right. $$ Namely, we prove that when $q=\\frac{2(N-1)}{N-2}$ there exists an $\\alpha_{0}>0$ such that the problem has a least "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6232","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"}