{"paper":{"title":"Multiplicity of positive solutions of nonlinear Schr\\\"odinger \\'equations concentrating at a potential well","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kazunaga Tanaka, Louis Jeanjean, Silvia Cingolani","submitted_at":"2013-05-16T05:29:13Z","abstract_excerpt":"We consider singularly perturbed nonlinear Schr\\\"odinger equations \\be \\label{eq:0.1} - \\varepsilon^2 \\Delta u + V(x)u = f(u), \\ \\ u > 0, \\ \\ v \\in H^1(\\R^N) \\ee where $V \\in C(\\R^N, \\R)$ and $f$ is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain $\\Omega \\subset \\R^N$ such that \\[m_0 \\equiv \\inf_{x \\in \\Omega} V(x) < \\inf_{x \\in \\partial \\Omega} V(x) \\] and we set $K = \\{x \\in \\Omega \\ | \\ V(x) = m_0\\}$. For $\\e >0$ small we prove the existence of at least ${\\cuplength}(K) + 1$ solutions to (\\ref{eq:0.1}) concentrating, a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.3685","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"}