{"paper":{"title":"Distributional solutions of the stationary nonlinear Schr\\\"odinger equation: singularities, regularity and exponential decay","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Rainer Mandel, Wolfgang Reichel","submitted_at":"2011-10-11T09:38:39Z","abstract_excerpt":"We consider the nonlinear Schr\\\"{o}dinger equation $-\\Delta u + V(x) u = \\Gamma(x) |u|^{p-1}u$ in $\\R^n$ where the spectrum of $-\\Delta+V(x)$ is positive. In the case $n\\geq 3$ we use variational methods to prove that for all $p\\in (\\frac{n}{n-2},\\frac{n}{n-2}+\\eps)$ there exist distributional solutions with a point singularity at the origin provided $\\eps>0$ is sufficiently small and $V,\\Gamma$ are bounded on $\\R^n\\setminus B_1(0)$ and satisfy suitable H\\\"{o}lder-type conditions at the origin. In the case $n=1,2$ or $n\\geq 3,1<p<\\frac{n}{n-2}$, however, we show that every distributional solut"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.2314","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"}