Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Symmetry breaking and Morse index of solutions of nonlinear elliptic
  problems in the plane   In this paper we study the problem
  -\Delta u =\left(\frac{2+\alpha}{2}\right)^2\abs{x}^{\alpha}f(\lambda,u), &
\hbox{in}B_1 \\ u > 0, & \hbox{in}B_1 u = 0, & \hbox{on} \partial B_1 where
$B_1$ is the unit ball of $\R^2$, $f$ is a smooth nonlinearity and $\a$, $\l$
are real numbers with $\a>0$. From a careful study of the linearized operator
we compute the Morse index of some radial solutions to \eqref{i0}. Moreover,
using the bifurcation theory, we prove the existence of branches of nonradial
solutions for suitable values of the positive parameter $\l$. The case
$f(\lambda,u)=\l e^u$ provides more detailed information.

6
The categories of this paper are: astro-ph
7
The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
4
The categories of this paper are: cond-mat
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th