In this paper we consider harmonic maps u(r, θ) from an annular domain Ωρ = B1¯Bρ to S2 with the boundary conditions: u(ρ, θ) = (cosθ, sin θ, 0), and u(1, θ) = (cos (θ + θ0) , sin (θ + θ0) , 0), where θ0 ∈ [0, π[ is a fixed angle. This problem arises from the theory of liquid crystals. We prove, with elementary time map arguments, a bifurcation result, namely the existence of a not trivial (that is not planar) harmonic map of minimum energy uθ0 , for suitable combination of value of ρ and θ0. This result improves the one in Greco (Proc Am Math Soc 129(4):1199–1206, 2000). In the case θ0 = π, so that u(1, θ) = (−cos θ,−sin θ, 0), no bifurcation occurs, since the minimum of the energy is not trivial, and we study the behavior of the harmonic maps uθ0 as θ0 → π.

In this paper we illustrate the lineguides of our research group. We describe some recent results concerning the study of some nonlinear differential equations and systems having a variational nature and arising from physics, geometry and applied sciences. In particular we report existence, multiplicity and regularity results for the solutions of these nonlinear problems. We point out that, in treating the above problems, the used methods for finding solutions are variational and topological, indeed the existence of solutions of the considered equations is obtained searching for critical points of suitable functionals defined on manifolds embedded into infinite dimensional functional spaces, while the regularity of the solutions is studied by means of geometric and harmonic analysis tools.