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

Abstract

In this paper we study the problem -Δu =\left(\frac{2+α}{2}\right)^2\abs{x}^αf(λ,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 $\a$, $ł$ 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 $ł$. The case $f(λ,u)=łe^u$ provides more detailed information.