  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.
