Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Singular Liouville Equations on $S^2$: Sharp Inequalities and Existence
  Results   We prove a sharp Onofri-type inequality and non-existence of extremals for a
Moser-Tudinger functional on the sphere in the presence of potentials having
positive order singularities. We also investigate the existence of critical
points and give some sufficient conditions under symmetry or nondegeneracy
assumptions.

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The categories of this paper are: patt-sol
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The categories of this paper are: chem-ph
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The categories of this paper are: alg-geom
3
The categories of this paper are: physics
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-th
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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat