{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:GIX4EQYHVOJYPW3MXER3GPM3AN","short_pith_number":"pith:GIX4EQYH","schema_version":"1.0","canonical_sha256":"322fc24307ab9387db6cb923b33d9b0360c3ab241e47c1e87274f210ea1611c9","source":{"kind":"arxiv","id":"1110.2314","version":1},"attestation_state":"computed","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 $n\\geq 3,10$ 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