{"paper":{"title":"Positive solutions to an elliptic equation in $\\mathbb{R}^N$ of the Kirchhoff type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Yisheng Huang, Yuanze Wu, Zeng Liu","submitted_at":"2016-03-24T04:07:32Z","abstract_excerpt":"In this paper, we consider the following Kirchhoff type problem $$\\left\\{\\aligned&-\\biggl(a + b\\int_{\\mathbb{R}^N} |\\nabla u|^2 dx \\biggr) \\Delta u + V(x) u = |u|^{p-2}u &\\text{ in } \\mathbb{R}^N,\\cr &u\\in H^1(\\mathbb{R}^N), \\endaligned\\right. \\eqno{(\\mathcal{P}_{a,b})} $$ where $N\\geq3$, $2<p<2^*=\\frac{2N}{N-2}$, $a,b>0$ are parameters and $V(x)$ is a potential function. Under some mild conditions on $V(x)$, we prove that $(\\mathcal{P}_{a,b})$ has a positive solution for $b$ small enough by the variational method, a non-existence result is also established in the cases $N\\geq4$. Our results i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.07428","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"}