Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Normalized solutions to a class of Kirchhoff equations with Sobolev
  critical exponent   In this paper, we consider the existence and asymptotic properties of
solutions to the following Kirchhoff equation
\begin{equation}\label{1}\nonumber - \Bigl(a+b\int_{{\R^3}} {{{\left| {\nabla
u} \right|}^2}}\Bigl) \Delta u
  =\lambda u+ {| u |^{p - 2}}u+\mu {| u |^{q - 2}}u \text { in } \mathbb{R}^{3}
\end{equation} under the normalized constraint $\int_{{\mathbb{R}^3}}
{{u}^2}=c^2$, where $a\!>\!0$, $b\!>\!0$, $c\!>\!0$,
$2\!<\!q\!<\!\frac{14}{3}\!<\! p\!\leq\!6$ or $\frac{14}{3}\!<\!q\!<\!
p\!\leq\! 6$, $\mu\!>\!0$ and $\lambda\!\in\!\R$ appears as a Lagrange
multiplier. In both cases for the range of $p$ and $q$, the Sobolev critical
exponent $p\!=\!6$ is involved and the corresponding energy functional is
unbounded from below on $S_c=\Big\{ u \in H^{1}({\mathbb{R}^3}):
\int_{{\mathbb{R}^3}} {{u}^2}=c^2 \Big\}$. If $2\!<\!q\!<\!\frac{10}{3}$ and
$\frac{14}{3}\!<\! p\!<\!6$, we obtain a multiplicity result to the equation.
If $2\!<\!q\!<\!\frac{10}{3}\!<\! p\!=\!6$ or $\frac{14}{3}\!<\!q\!<\!
p\!\leq\! 6$, we get a ground state solution to the equation. Furthermore, we
derive several asymptotic results on the obtained normalized solutions.
  Our results extend the results of N. Soave (J. Differential Equations 2020
$\&$ J. Funct. Anal. 2020), which studied the nonlinear Schr\"{o}dinger
equations with combined nonlinearities, to the Kirchhoff equations. To deal
with the special difficulties created by the nonlocal term $({\int_{{\R^3}}
{\left| {\nabla u} \right|} ^2}) \Delta u$ appearing in Kirchhoff type
equations, we develop a perturbed Pohozaev constraint approach and we find a
way to get a clear picture of the profile of the fiber map via careful
analysis. In the meantime, we need some subtle energy estimates under the
$L^2$-constraint to recover compactness in the Sobolev critical case.

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The categories of this paper are: eess
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: funct-an
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The categories of this paper are: gr-qc
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The categories of this paper are: econ
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ph
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The categories of this paper are: patt-sol
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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: plasm-ph