Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence and convergence of solutions to fractional pure critical
  exponent problems   We study existence and convergence properties of least-energy symmetric
solutions (l.e.s.s.) to the pure critical problem \begin{equation*}
(-\Delta)^su_s=|u_s|^{2^\star_s-2}u_s, \quad u_s\in D^s_0(\Omega),\quad
2^\star_s:=\frac{2N}{N-2s}, \end{equation*} where $s$ is any positive number,
$\Omega$ is either $\mathbb{R}^N$ or a smooth symmetric bounded domain, and
$D^s_0(\Omega)$ is the homogeneous Sobolev space. Depending on the kind of
symmetry considered, solutions can be sign changing. We show that, up to a
subsequence, a l.e.s.s. $u_s$ converges to a l.e.s.s. $u_{t}$ as $s$ goes to
any $t>0$. In bounded domains, this convergence can be characterized in terms
of an homogeneous fractional norm of order $t-\varepsilon$. A similar
characterization is no longer possible in unbounded domains due to scaling
invariance and an incompatibility with the functional spaces; to circumvent
these difficulties, we use a suitable rescaling and characterize the
convergence via cut-off functions. If $t$ is an integer, these results describe
in a precise way the nonlocal-to-local transition. Finally, we also include a
nonexistence result of nontrivial nonnegative solutions in a ball for any
$s>1$.

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The categories of this paper are: adap-org
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The categories of this paper are: q-bio
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The categories of this paper are: patt-sol
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The categories of this paper are: funct-an
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The categories of this paper are: stat
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The categories of this paper are: math-ph
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The categories of this paper are: physics
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The categories of this paper are: eess
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The categories of this paper are: supr-con
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The categories of this paper are: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th