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

In this paper, we study the problem −∆u =„2+α 2 «2|x|αf(λ,u), in B1, u>0, in B1, u =0 , on ∂B1, (P) where B1 is the unit ball of R2, f is a smooth nonlinearity and α, λ are real numbers with α>0. From a careful study of the linearized operator, we compute the Morse index of some radial solutions to (P). Moreover, using the bifurcation theory, we prove the existence of branches of nonradial solutions for suitable values of the positive parameter λ. The casef(λ,u)=λeu provides more detailed informations.