Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Blow-up of solutions of critical elliptic equation in three dimensions   We describe the asymptotic behavior of positive solutions $u_\epsilon$ of the
equation $-\Delta u + au = 3\,u^{5-\epsilon}$ in $\Omega\subset\mathbb{R}^3$
with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to
be critical in the sense of Hebey and Vaugon and the functions $u_\epsilon$ are
assumed to be an optimizing sequence for the Sobolev inequality. Under a
natural nondegeneracy assumption we derive the exact rate of the blow-up and
the location of the concentration point, thereby proving a conjecture of
Br\'ezis and Peletier (1989). Similar results are also obtained for solutions
of the equation $-\Delta u + (a+\epsilon V) u = 3\,u^5$ in $\Omega$.

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The categories of this paper are: physics
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The categories of this paper are: acc-phys
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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: supr-con
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The categories of this paper are: chao-dyn
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The categories of this paper are: mtrl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an