Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: High-order Kirchhoff problems in bounded and unbounded domains   Consider the following $m-$polyharmonic Kirchhoff problem: \begin{eqnarray}
\label{ea} \begin{cases} M\left(\int_{\O}|D_r u|^{m} +a|u|^m\right)[\Delta^r_m
u +a|u|^{m-2}u]= K(x)f(u) &\mbox{in}\quad \Omega, \\
u=\left(\frac{\partial}{\partial \nu}\right)^k u=0, \quad &\mbox{on}\quad
\partial\Omega, \quad k=1, 2,..... , r-1, \end{cases} \end{eqnarray} where $r
\in \N^*$, $m >1$, $N\geq rm+1$, $a\geq 0$, $K\in L^{\infty}(\O)$ is a positive
weight function, $M \in C([0,+\infty))$ and $f\in C(\mathbb{R})$ which will be
specified later. We will study problem \eqref{ea} in the following different
type of domains: \begin{enumerate}
  \item $a=0$ and $K\in L^{\infty}(\O)$ is a positive weight function if
$\Omega$ is a smooth bounded domain of $\R^N$.
  \item $a>0$ and $K\in L^{\infty}(\O)\cap L^{p}(\O)$, $p \geq 1$ if $\Omega$
is an unbounded smooth domain.
  \item $\O=\R^N$ and $a=0$ (which called the $m\gamma$-zero mass case).
\end{enumerate}
  We prove the existence of infinitely many solutions of \eqref{ea} for some
odd functions $f$ in $u$ satisfying subcritical growth conditions at infinity
which are weaker than the analogue of the Ambrosetti-Rabinowitz condition and
the standard subcritical polynomial growth. The new aspect consists in
employing the Schauder basis of $W_0^{r,m}(\O)$ to verify the geometry of the
symmetric mountain pass theorem without any control on $f$ near $0$ if $\Omega$
is a bounded domain and under a suitable condition at $0$ if $\Omega$ is a
unbounded domain allowing only to derive the variational setting of \eqref{ea}.
Moreover, we introduce a positive quantity $\lambda_M$ similar to the first
eigenvalue of the $m$-polyharmonic operator to find a mountain pass solution.

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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
1
The categories of this paper are: cs
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The categories of this paper are: comp-gas
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat
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The categories of this paper are: stat
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The categories of this paper are: solv-int