In this talk, we present an existence result for semilinear elliptic problems of the form -Delta u + u = f(u), u > 0, u in H^1_0(A), where A denotes either an annulus or the exterior of a ball in R^N, with N>=3. We consider a broad class of nonlinearities f exhibiting superlinear growth at infinity, including both exponential-type and supercritical power-type behaviors. Under suitable assumptions on f and A, we prove the existence of a positive nonradial solution via techniques in the spirit of Szulkin's nonsmooth critical point theory, applied within invariant convex cones. The results presented are based on joint papers with Alberto Boscaggin (Università di Torino), Benedetta Noris (Politecnico di Milano), Federica Sani (Università di Modena e Reggio Emilia), and Tobias Weth (University of Frankfurt).