Abstract We consider semilinear elliptic problems of the form - u + u = f (x, u), u H¹₀ (A), - Δ u + λ u = f (x, u), u ∈ H 0 1 (A), where A RN A ⊂ R N, N 3 N ≥ 3, is either a bounded or unbounded annulus, and 0 λ ≥ 0. We study a broad class of nonlinearities f with superlinear growth at infinity, including exponential- and power-type ones. Under suitable assumptions, we establish the existence of a positive nonradial solution via techniques in the spirit of Szulkin’s nonsmooth critical point theory, applied within a convex cone in Orlicz spaces. Notably, the Trudinger-Moser inequality fails in the whole Sobolev space H¹₀ (A) H 0 1 (A).
Boscaggin et al. (Tue,) studied this question.