In joint works with F.Pacella, we are interested in the number and symmetry properties of solutions of semilinear or quasilinear equations. The tools rely on the moving plane method and various forms of the maximum principle.

One direction of work consists in proving the uniqueness and nondegeneracy of positive solutions of nonlinear problems of the p laplacian type in the unit ball with Dirichlet boundary conditions. The main ideas rely on the Maximum Principle and an  implicit function theorem that we derive in a suitable weighted space. We are also able to get results on the spectrum of the linearized operator in a suitable weighted space of radial functions and derive information on the Morse index.

More recently, we are dealing with sign changing solutions, which are thus of index 2 and try to derive symmetry properties about the nodal line.