Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence results for superlinear elliptic equations with nonlinear
  boundary value conditions   In this paper, we study the existence of solutions for the following
superlinear elliptic equation with nonlinear boundary value condition $$
  \left\{ \begin{array}{ll} -\Delta u+u=|u|^{r-2}u &\text{in} \; \Omega,\\ \\
\frac{\partial u}{\partial \nu}=|u|^{q-2}u &\text{on}\;\partial\Omega,
\end{array} \right. $$ where $\Omega\subset \mathbb R^N, N\geq 3$ is a bounded
domain with smooth boundary. We will prove the existence results for the above
equation under four different cases: (i) Both $q$ and $r$ are subcritical; (ii)
$r$ is critical and $q$ is subcritical; (iii) $r$ is subcritical and $q$ is
critical; (iv) Both $q$ and $r$ are critical.

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The categories of this paper are: cond-mat
1
The categories of this paper are: cs
10
The categories of this paper are: q-bio
37
The categories of this paper are: supr-con
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
18
The categories of this paper are: econ
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph