Symmetry breaking and Morse index of solutions of nonlinear elliptic problems in the plane

Abstract: 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.