Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A characterization related to Schr\"odinger equations on Riemannian
  manifolds   In this paper we consider the following problem $$\begin{cases}
-\Delta_{g}u+V(x)u=\lambda\alpha(x)f(u), & \mbox{in }M\\ u\geq0, & \mbox{in
}M\\ u\to0, & \mbox{as }d_{g}(x_{0},x)\to\infty \end{cases}$$where $(M,g)$ is a
$N$-dimensional ($N\geq3)$, non-compact Riemannian manifold with asymptotically
non-negative Ricci curvature, $\lambda$ is a real parameter, $V$ is a positive
coercive potential, $\alpha$ is a bounded function and $f$ is a suitable
nonlinearity. By using variational methods we prove a characterization result
for existence of solutions for our problem.

4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
26
The categories of this paper are: bayes-an
34
The categories of this paper are: chem-ph
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg