Trudinger–Moser inequalities on complete noncompact Riemannian manifolds

Let (M,g)(M,g) be a complete noncompact Riemannian n  -manifold (n⩾2n⩾2). If there exist positive constants α, τ and β such thatsupu∈W1,n(M),‖u‖1,τ⩽1∫M(eα|u|nn−1−∑k=0n−2αk|u|nkn−1k!)dvg⩽β, where ‖u‖1,τ=‖∇gu‖Ln(M)+τ‖u‖Ln(M)‖u‖1,τ=‖∇gu‖Ln(M)+τ‖u‖Ln(M), then we say that the Trudinger–Moser inequality holds. Suppose the Trudinger–Moser inequality holds, we prove that there exists some positive constant ϵ   such that Volg(Bx(1))⩾ϵVolg(Bx(1))⩾ϵ for all x∈Mx∈M. Also we give a sufficient condition under which the Trudinger–Moser inequality holds, say the Ricci curvature of (M,g)(M,g) has lower bound and its injectivity radius is positive. Moreover, the Adams inequality is discussed in this paper. For application of the Trudinger–Moser inequality, we obtain existence results for some quasilinear equations with nonlinearity of exponential growth.