Abstract :
Let (M, g) be a complete noncompact Riemannian n-manifold (n 2). If there exist positive constants
α, τ and β such that
sup
u∈W1,n(M), u 1,τ 1
M
eα|u|
n
n−1 −
n −2
k=0
αk|u| nk
n−1
k!
dvg β,
where 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 such that
Volg(Bx(1)) for all x ∈ M. Also we give a sufficient condition under which the Trudinger–Moser
inequality holds, say the Ricci curvature of (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.
© 2012 Elsevier Inc. All rights reserved.
