Journal of Functional Analysis 272 (2017) 1296–1299
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Journal of Functional Analysis www.elsevier.com/locate/jfa
Corrigendum
Corrigendum to “Sobolev spaces with variable exponents on complete manifolds” [J. Funct. Anal. 270 (4) (2016) 1379–1415] Michał Gaczkowski a , Przemysław Górka b,∗ , Daniel J. Pons c,∗ a
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland b Department of Mathematics and Information Sciences, Warsaw University of Technology, Ul. Koszykowa 75, 00-662 Warsaw, Poland c Facultad de Ciencias Exactas, Departamento de Matemáticas, Universidad Andres Bello, República 220, Santiago, Chile
a r t i c l e
i n f o
Article history: Received 11 October 2016 Accepted 28 October 2016 Available online 18 November 2016 Communicated by L. Gross
In this note we correct some misprints in our paper [1]. In particular, we give the correct formulation of Theorem 6.1, and for the reader’s convenience, we provide some elements of the proof. The correct form of Lemma 3.1 is: Lemma 0.1. Let p ∈ P log (M ) and (BR (p), φ) be a chart such that 1 δij ≤ gij ≤ 2δij 2 DOI of original article: http://dx.doi.org/10.1016/j.jfa.2015.09.008.
* Corresponding authors. E-mail addresses:
[email protected] (P. Górka),
[email protected],
[email protected] (D.J. Pons). http://dx.doi.org/10.1016/j.jfa.2016.10.023 0022-1236/© 2016 Elsevier Inc. All rights reserved.
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as bilinear forms, where δij is the delta Kronecker symbol. Then p ◦ φ−1 ∈ P log φ B R (p) . 3
The correct form of Theorem 6.1 is: Theorem 0.1. Let H be a non-trivial compact Lie subgroup of Iso(M, g). Assume that the complete n-manifold (M, g) has property Bvol (λ, ϑ), and lim
inf
T →∞ x∈M \BT (o)
M (x, R) = ∞,
where R > 0 and o is any fixed point in M . Let q and p belong to P(M ) and be H-invariant, with q uniformly continuous, and such that 1 < q − ≤ q + < n, q p q ∗ , and q + < p− . Let f : M × R → R be a continuous function satisfying: (i) For each t ∈ R the function f (·, t) : M → R is H-invariant; (ii) There exists some c1 > 0 such that for each t ∈ R we have the bound |f (x, t)| ≤ c1 |t|p(x)−1 ; (iii) There exist some θ > q + and some sufficiently small A ≥ 0 such that for each |t| > A we have 0 < F (x, t) ≤ f (x, t) t/θ, where F (x, t) :=
t 0
f (x, s) ds.
Then equation (36) in [1] has an H-invariant non-trivial weak solution, in the sense that (37) in [1] holds for every φ in DH (M ). Proof. In the original proof there are two gaps; both gaps are concerned with the estimates to prove that the conditions to use the Mountain Pass theorem are fulfilled. • In Step 1, the left hand side of inequality (41) is not correct. Therefore, in the proof q(·) of the existence of some v ∈ L1,H such that J[v] < 0 there is a gap: Now we fill such a gap. From condition (iii), we have for each |t| > A + 1 the inequality F (x, t) ≥ c2 (x)|t|θ , where c2 is a continuous and positive function. Indeed, from (iii) we obtain ∂ ∂t F (x, t)
F (x, t)
≥
θ t
if
t>A
(1)
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and ∂ ∂t F (x, t)
F (x, t)
≤
θ t
if
t < −A.
Integrating the previous inequalities, we get F (x, t) ≥
F (x, A + 1) θ |t| (A + 1)θ
F (x, t) ≥
F (x, −A − 1) θ |t| (A + 1)θ
if t > A + 1, if t < −A − 1,
hence F (x, t) ≥ min
F (x, A + 1) F (x, −A − 1) , (A + 1)θ (A + 1)θ
|t|θ =: c2 (x)|t|θ
whenever |t| > A + 1, as claimed. Fix u in DH (M ); then using condition (ii) and inequality (1) we have K[u] = F (x, u(x)) dVg (x) + F (x, u(x)) dVg (x) |u|≤A+1
|u|>A+1
≥ −c1
|u(x)|
p(x)
|u|≤A+1
|u|>A+1
+
≥ −c1 (A + 1)p
c2 (x)|u(x)|θ dVg (x)
dVg (x) +
|u(x)|θ dVg (x).
dVg (x) + inf c2 (x) x∈spt u
|u|>A+1
spt u
Therefore, we get K[u] ≥ −˜ c1
|u(x)|θ dVg (x).
dVg (x) + c˜2
spt u
|u|>A+1
Next, gathering the above inequality with (40) from [1], we obtain tq J[tu] ≤ − ρq(·) (|∇u|) + ρq(·) (u) − tθ c˜2 q +
|u(x)| dVg (x) + c˜1 θ
|tu|>A+1
dVg (x),
spt u
where t > 1. Since θ > q + , we deduce that J[tu] → −∞ as t → ∞. The existence of q(·) some v ∈ L1,H (M ) with v Lq(·) > r and with J[v] < 0 follows. 1 • In Step 2, there is a gap in the proof of the boundedness of the sequence un: Now we fill such a gap. We know that the sequence {un } satisfies: i) J[un ] ≤ L for some L > 0,
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ii) | un , DJ[un ] | ≤ θ un Lq(·) for n large enough. 1 To simplify the arguments, we use the estimate 1 1
u Lq(·) (M ) ≤ ρq(·) (u) + 1 q− + ρq(·) (|∇u|) + 1 q− 1
≤2
q − −1 q−
1 ρq(·) (u) + ρq(·) (|∇u|) + 2 q− .
Making the same calculations as on page 1411 of [1], we have 1 L + un Lq(·) ≥ J[un ] − un , DJ[un ] 1 θ
1 1 ≥ ρq(·) (|∇un |) + ρq(·) (un ) − + q θ
1 F (x, un (x)) − f (x, un (x))un (x) dVg (x) − θ |un |≤A
1 1 ≥ ρq(·) (|∇un |) + ρq(·) (un ) − q+ θ
1 +1 |un (x)|p(x)−q(x) |un (x)|q(x) dVg (x) − c1 θ
≥
|un |≤A
1 1 − − c1 + q θ
1 p− −q + ρq(·) (|∇un |) + ρq(·) (un ) . +1 A θ
Abbreviating ρq(·) (|∇un |) + ρq(·) (un ) by an , we get L+2
q − −1 q−
1
(an + 2) q− ≥
1 1 − − c1 q+ θ
− + 1 + 1 Ap −q an . θ
Hence, since A is small, we conclude that an is bounded. 2 References [1] M. Gaczkowski, P. Górka, D.J. Pons, Sobolev spaces with variable exponents on complete manifolds, J. Funct. Anal. 270 (4) (2016) 1379–1415.
