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

Abstract

In this paper, we study the problem [Formula: see text] where [Formula: see text] is the unit ball of [Formula: see text], [Formula: see text] is a smooth nonlinearity and [Formula: see text], [Formula: see text] are real numbers with [Formula: see text]. From a careful study of the linearized operator, we compute the Morse index of some radial solutions to ([Formula: see text]). Moreover, using the bifurcation theory, we prove the existence of branches of nonradial solutions for suitable values of the positive parameter [Formula: see text]. The case [Formula: see text] provides more detailed informations.