Exponential problem on a compact Riemannian manifold without boundary

Let (M,g)(M,g) be an nn-dimensional compact Riemannian manifold without boundary. A Trudinger–Moser-type inequality says that sup‖u‖W1,n≤1∫Meαn|u|nn−1dvg<∞, where ‖u‖W1,n‖u‖W1,n is the usual Sobolev norm of u∈W1,n(M)u∈W1,n(M), αn=nωn−11n−1, and ωn−1ωn−1 is the area of the unit sphere Sn−1Sn−1. Using this inequality, when ε>0ε>0 is small enough, we establish sufficient conditions under which the quasilinear equation −Δnu+|u|n−2u=f(x,u)+εh(x)−Δnu+|u|n−2u=f(x,u)+εh(x) has at least two nontrivial weak solutions in W1,n(M)W1,n(M), where −Δnu=−divg(|∇u|n−2∇u), f(x,u)f(x,u) behaves like eγ|u|nn−1 as |u|→∞|u|→∞ for some γ>0γ>0, and h≢0h≢0 belongs to the dual space of W1,n(M)W1,n(M).