An Orlicz space approach to exponential elliptic problems in higher dimensions

We consider semilinear elliptic problems of the form $$-\mathrm{\Delta }u+\lambda u=f(x,u),u\in H_0^1(A),$$ where $A\subset {\mathbb{R}}^N$, $N\ge 3$, is either a bounded or unbounded annulus, and $\lambda \ge 0$. 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 $H_0^1(A)$.