An Orlicz space approach to exponential elliptic problems in higher dimensions

We consider semilinear elliptic problems of the form -Δ u + λ 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 λ ≥ 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 ) .