Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Positive solutions to an elliptic equation in $\mathbb{R}^N$ of the
  Kirchhoff type   In this paper, we consider the following Kirchhoff type problem
$$\left\{\aligned&-\biggl(a + b\int_{\mathbb{R}^N} |\nabla u|^2 dx \biggr)
\Delta u + V(x) u = |u|^{p-2}u &\text{ in } \mathbb{R}^N,\cr &u\in
H^1(\mathbb{R}^N), \endaligned\right. \eqno{(\mathcal{P}_{a,b})} $$ where
$N\geq3$, $2<p<2^*=\frac{2N}{N-2}$, $a,b>0$ are parameters and $V(x)$ is a
potential function. Under some mild conditions on $V(x)$, we prove that
$(\mathcal{P}_{a,b})$ has a positive solution for $b$ small enough by the
variational method, a non-existence result is also established in the cases
$N\geq4$. Our results in the case $N=3$ partial improve the results in
\cite{G15,LY14} and our results in the cases $N\geq4$ are totally new to the
best of our knowledge. By combining the scaling technique, we also give a
global description on the structure of the positive solutions to the autonomous
form of $(\mathcal{P}_{a,b})$, that is $V(x)\equiv\lambda>0$. This result can
be seen as a partial complement of the studies in \cite{A12,A13}.

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The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
28
The categories of this paper are: alg-geom
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
6
The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol
29
The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
24
The categories of this paper are: dg-ga
18
The categories of this paper are: econ