Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Branch continuation inside the essential spectrum for the nonlinear
  Schr\"odinger equation   We consider the nonlinear stationary Schr\"odinger equation \begin{equation*}
  -\Delta u -\lambda u= Q(x)|u|^{p-2}u, \qquad \text{in }\mathbb{R}^N
\end{equation*} in the case where $N \geq 3$, $p$ is a superlinear, subcritical
exponent, $Q$ is a bounded, nonnegative and nontrivial weight function with
compact support in $\mathbb{R}^N$ and $\lambda \in \mathbb{R}$ is a parameter.
Under further restrictions either on the exponent $p$ or on the shape of $Q$,
we establish the existence of a continuous branch $\mathcal{C}$ of nontrivial
solutions to this equation which intersects $\{\lambda \} \times
L^{s}(\mathbb{R}^N)$ for every $\lambda \in (-\infty, \lambda_Q)$ and $s>
\frac{2N}{N-1}$. Here $\lambda_Q>0$ is an explicit positive constant which only
depends on $N$ and $\text{diam}(\text{supp }Q)$. In particular, the set of
values $\lambda$ along the branch enters the essential spectrum of the operator
$-\Delta$.

31
The categories of this paper are: ao-sci
36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
12
The categories of this paper are: nucl-th
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
19
The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex