TY - JOUR
T1 - Corrigendum to “Sobolev spaces with variable exponents on complete manifolds” (Journal of Functional Analysis (2016) 270(4) (1379–1415) (S0022123615003675) (10.1016/j.jfa.2015.09.008))
AU - Gaczkowski, Michał
AU - Górka, Przemysław
AU - Pons, Daniel J.
N1 - Publisher Copyright:
© 2016 Elsevier Inc.
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.
PY - 2017/2/1
Y1 - 2017/2/1
N2 - 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 [formula presented] and [formula presented] be a chart such that [formula presented] as bilinear forms, where [formula presented] is the delta Kronecker symbol. Then [formula presented] The correct form of Theorem 6.1 is: Theorem 0.1 Let H be a non-trivial compact Lie subgroup of [formula presented]. Assume that the complete n-manifold [formula presented] has property [formula presented], and [formula presented] where [formula presented] and o is any fixed point in M. Let q and p belong to [formula presented] and be H-invariant, with q uniformly continuous, and such that [formula presented] Let [formula presented] be a continuous function satisfying: (i) For each [formula presented] the function [formula presented] is H-invariant;(ii) There exists some [formula presented] such that for each [formula presented] we have the bound [formula presented];(iii) There exist some [formula presented] and some sufficiently small [formula presented] such that for each [formula presented] we have [formula presented] where [formula presented].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 [formula presented]. 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 of the existence of some [formula presented] such that [formula presented] there is a gap: Now we fill such a gap. From condition (iii), we have for each [formula presented] the inequality [formula presented] where [formula presented] is a continuous and positive function. Indeed, from (iii) we obtain [formula presented] and [formula presented] Integrating the previous inequalities, we get [formula presented] hence [formula presented] whenever [formula presented], as claimed. Fix u in [formula presented]; then using condition (ii) and inequality (1) we have [formula presented] Therefore, we get [formula presented] Next, gathering the above inequality with (40) from [1], we obtain [formula presented] where [formula presented]. Since [formula presented], we deduce that [formula presented] as [formula presented]. The existence of some [formula presented] with [formula presented] and with [formula presented] follows.• In Step 2, there is a gap in the proof of the boundedness of the sequence [formula presented]: Now we fill such a gap. We know that the sequence [formula presented] satisfies: i) [formula presented] for some [formula presented],ii) [formula presented] for n large enough.To simplify the arguments, we use the estimate [formula presented] Making the same calculations as on page 1411 of [1], we have [formula presented] Abbreviating [formula presented] by [formula presented], we get [formula presented] Hence, since A is small, we conclude that [formula presented] is bounded. □
AB - 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 [formula presented] and [formula presented] be a chart such that [formula presented] as bilinear forms, where [formula presented] is the delta Kronecker symbol. Then [formula presented] The correct form of Theorem 6.1 is: Theorem 0.1 Let H be a non-trivial compact Lie subgroup of [formula presented]. Assume that the complete n-manifold [formula presented] has property [formula presented], and [formula presented] where [formula presented] and o is any fixed point in M. Let q and p belong to [formula presented] and be H-invariant, with q uniformly continuous, and such that [formula presented] Let [formula presented] be a continuous function satisfying: (i) For each [formula presented] the function [formula presented] is H-invariant;(ii) There exists some [formula presented] such that for each [formula presented] we have the bound [formula presented];(iii) There exist some [formula presented] and some sufficiently small [formula presented] such that for each [formula presented] we have [formula presented] where [formula presented].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 [formula presented]. 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 of the existence of some [formula presented] such that [formula presented] there is a gap: Now we fill such a gap. From condition (iii), we have for each [formula presented] the inequality [formula presented] where [formula presented] is a continuous and positive function. Indeed, from (iii) we obtain [formula presented] and [formula presented] Integrating the previous inequalities, we get [formula presented] hence [formula presented] whenever [formula presented], as claimed. Fix u in [formula presented]; then using condition (ii) and inequality (1) we have [formula presented] Therefore, we get [formula presented] Next, gathering the above inequality with (40) from [1], we obtain [formula presented] where [formula presented]. Since [formula presented], we deduce that [formula presented] as [formula presented]. The existence of some [formula presented] with [formula presented] and with [formula presented] follows.• In Step 2, there is a gap in the proof of the boundedness of the sequence [formula presented]: Now we fill such a gap. We know that the sequence [formula presented] satisfies: i) [formula presented] for some [formula presented],ii) [formula presented] for n large enough.To simplify the arguments, we use the estimate [formula presented] Making the same calculations as on page 1411 of [1], we have [formula presented] Abbreviating [formula presented] by [formula presented], we get [formula presented] Hence, since A is small, we conclude that [formula presented] is bounded. □
UR - http://www.scopus.com/inward/record.url?scp=85006001574&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2016.10.023
DO - 10.1016/j.jfa.2016.10.023
M3 - Comment/debate
AN - SCOPUS:85006001574
VL - 272
SP - 1296
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 3
ER -