In this talk, I will present an existence result for the Dirichlet problem associated with the elliptic equation -\Delta u + u = a(x)|u|^{p-2}u set in an annulus of R^N. Here p>2 is allowed to be supercritical in the sense of Sobolev embeddings, and a(x) is a positive weight with additional symmetry and monotonicity properties, which are shared by the solution that we construct. For this problem, we find a new type of positive, axially symmetric solutions. Moreover, in the case where the weight a(x) is constant, we detect a condition, depending only on the exponent p and on the inner radius of the annulus, that ensures that the solution is nonradial. In this setting, the major difficulty to overcome is the lack of compactness in a nonradial framework. The proofs rely on a combination of variational methods and dynamical system techniques. This is joint work with Alberto Boscaggin (Università di Torino), Benedetta Noris (Politecnico di Milano), and Tobias Weth (Goethe-Universitat Frankfurt).