Correction to: Mathematische Annalen
https://doi.org/10.1007/s00208-019-01803-w
In the Original Publication of the article, few errors have been identified in section 5
and acknowledgements section.Agência financiadora
Ministry of Education and Science, Russian Federation

117198
Portuguese Foundation for Science and Technology (FCT), Portugal 	
SFRH/BSAB/135242/2017info:eu-repo/semantics/publishedVersio

Chipot, M.

de Oliveira, H. B.

English

Sapientia (Univ. do Algarve)

Mathematische Annalen (2019) 375:307–313https://doi.org/10.1007/s00208-019-01859-8 Mathematische AnnalenPUBL ISHER CORRECT IONCorrection to: Some results on the p(u)-Laplacian problemM. Chipot1 · H. B. de Oliveira2,3Published online: 4 July 2019© Springer-Verlag GmbH Germany, part of Springer Nature 2019Correction to: Mathematische Annalenhttps://doi.org/10.1007/s00208-019-01803-wIn the Original Publication of the article, few errors have been identified in section 5and acknowledgements section. The corrected section 5 and acknowledgements aregiven below:5 Nonlocal problemsIn this section we consider a real function p such thatp is continuous, 1 < α < p ≤ β, (5.1)for some constants α, β. We denote by b a mapping from W 1,α0 () into R such thatb is continuous, b is bounded, (5.2)i.e. b sends bounded sets of W 1,α0 () into bounded sets of R.Definition 2 A function u is a weak solution to the problem (1.3) if⎧⎨⎩u ∈ W 1,p(b(u))0 (),∫|∇u|p(b(u))−2∇u · ∇v dx = 〈 f , v〉 ∀ v ∈ W 1,p(b(u))0 (),(5.3)The original article can be found online at https://doi.org/10.1007/s00208-019-01803-w.B H. B. de Oliveiraholivei@ualg.ptM. Chipotm.m.chipot@math.uzh.ch1 IMath, Universität Zürich, Winterthurerstrasse 190, 8057 Zurich, Switzerland2 FCT, Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, Portugal3 CMAFCIO, Universidade de Lisboa, Campo Grande, 1749-016 Lisbon, Portugal123308 M. Chipot, H. B. de Oliveirawhere 〈·, ·〉 denotes the duality pairing between (W 1,p(b(u))0 ())′ and W 1,p(b(u))0 ().One should notice that p(b(u)) is here a real number and not a function so thatthe Sobolev spaces involved are the classical ones. We refer to [5, 7–9] for more onnonlocal problems.Then one has:Theorem 5.1 Let  ⊂ Rd , d ≥ 2, be a bounded domain and assume that (5.1) and(5.2) hold together withf ∈ W−1,α′().Then there exists at least one weak solution to the problem (1.3) in the sense ofDefinition 2.The proof of Theorem 5.1 is based on the following result.Lemma 5.1 For n ∈ N, let un be the solution to the problem⎧⎨⎩un ∈ W 1,pn0 (),∫|∇un|pn−2∇un · ∇v dx = 〈 f , v〉 ∀ v ∈ W 1,pn0 (),(5.4)where 〈·, ·〉 denotes here the duality pairing between (W 1,pn0 ())′ and W 1,pn0 ().Suppose thatpn → p, as n → ∞, where p ∈ (1,∞), (5.5)f ∈ W−1,q ′() for some q < p. (5.6)Thenun → u in W 1,q0 (), as n → ∞, (5.7)where u is the solution to the problem⎧⎨⎩u ∈ W 1,p0 (),∫|∇u|p−2∇u · ∇v dx = 〈 f , v〉 ∀ v ∈ W 1,p0 ().(5.8)Proof of Lemma 5.1 We shall split this proof into two steps.1. Weak convergence: We first observe that, in view of pn → p, as n → ∞, andq < p, we may assume thatp + 1 > pn > q ∀ n ∈ N. (5.9)Taking v = un in the equation of (5.4) we get∫|∇un|pn dx ≤ ‖ f ‖−1,q ′ ‖∇un‖q . (5.10)123Correction to: Some results on the p(u)-Laplacian problem 309Recall that ‖ f ‖−1,q ′ denotes the strong dual norm of f associated to the norm ‖∇ · ‖q .On the other hand, by using Hölder’s inequality and (5.9), we have‖∇un‖q ≤ C‖∇un‖pn , (5.11)for some positive constant C = C(p, q,). Plugging (5.11) into (5.10) it comes‖∇un‖pn ≤ C, (5.12)for some other positive constant C = C(p, q,, f ). Combining (5.11) with (5.12),it follows that‖∇un‖q ≤ C, (5.13)for some positive constant C independent of n. From (5.13) we deduce then that forsome subsequence still labelled by n and for some u ∈ W 1,q0 ()∇un⇀∇u in Lq(), as n → ∞. (5.14)Due to (5.5), (5.9), (5.12) and (5.14), we can also apply Lemma 3.1 so thatlim infn→∞∫|∇un|pn dx ≥∫|∇u|pdx .As a consequence we haveu ∈ W 1,p0 (). (5.15)Clearly the equation in (5.4) is equivalent to∫|∇un|pn−2∇un · ∇(v − un) dx ≥ 〈 f , v − un〉 ∀ v ∈ W 1,pn0 ().and by the Minty lemma to∫|∇v|pn−2∇v · ∇(v − un) dx ≥ 〈 f , v − un〉 ∀ v ∈ W 1,pn0 (). (5.16)Taking v ∈ C∞0 (), one can use (5.5) and (5.14) to pass to the limit in (5.16), asn → ∞, so that∫|∇v|p−2∇v · ∇(v − u) dx ≥ 〈 f , v − u〉 ∀ v ∈ C∞0 (). (5.17)Using the density of C∞0 () in W1,p0 (), we see that (5.17) also holds for all v ∈W 1,p0 (). In this case, taking v = u ± δz, with z ∈ W 1,p0 () and δ > 0, and lettingδ → 0 after simplifying the resulting inequality, one obtains∫|∇u|p−2∇u · ∇z dx = 〈 f , z〉 ∀ z ∈ W 1,p0 ().Thus u is the solution to the problem (5.8).123310 M. Chipot, H. B. de Oliveira2. Strong convergence: We want to show that the convergence (5.14) is in fact strong.To prove this, we first note that, taking v = un in the equation of (5.4) and using (5.14)to pass to the limit, we obtain∫|∇un|pn dx = 〈 f , un〉 → 〈 f , u〉 =∫|∇u|pdx, as n → ∞. (5.18)Consider the case of the pn’s such thatpn ≥ p ∀ n ∈ N.One has by Hölder’s inequality∫|∇un|pdx ≤(∫|∇un|pn dx) ppn ||1− ppn ,where || denotes the d-Lebesgue measure of . Thus by (5.18) for such a sequencelim supn→∞∫|∇un|pdx ≤∫|∇u|pdx ≤ lim infn→∞∫|∇un|pdx,which shows (since ‖∇un‖p → ‖∇u‖p, as n → ∞)un → u strongly in W 1,p0 (), as n → ∞. (5.19)Since W 1,p0 () ⊂ W 1,q0 (), (5.19) implies (5.7).Next, consider the pn’s such thatq < pn < p ∀ n ∈ N (5.20)and setAn :=∫(|∇un|pn−2∇un − |∇u|pn−2∇u)· (∇un − ∇u) dx . (5.21)Due to the monotonicity, An ≥ 0 and, because of (5.4), one hasAn = 〈 f , un − u〉 −∫|∇u|pn−2∇u · ∇(un − u) dx .From (5.6) and (5.14), we have〈 f , un − u〉 → 0, as n → ∞. (5.22)Moreover, from (5.15) one easily gets∣∣∣|∇u|pn−2∇u∣∣∣ ≤ max{1, |∇u|}p−1 ∈ L p′(). (5.23)123Correction to: Some results on the p(u)-Laplacian problem 311Hence, (5.20), (5.22) and (5.23) ensure thatAn → 0, as n → ∞. (5.24)Assume first thatpn ≥ 2.This allows us to use property (3.10) of Lemma 3.2 in (5.21) so thatAn ≥ 12pn−1∫|∇(un − u)|pn dx . (5.25)Since, by (5.20), pn > q, we have by Hölder’s inequality, (5.20), (5.24) and (5.25)∫|∇(un − u)|q dx ≤( ∫|∇(un − u)|pn dx) qpn ||1− qpn → 0,when n → ∞. This proves (5.7) in this case.Consider now the case whenpn < 2.Here, we use Hölder’s inequality as follows∫|∇(un − u)|pn dx=∫|∇(un − u)|pn (|∇un| + |∇u|) (pn−2)pn2 (|∇un| + |∇u|) (2−pn )pn2 dx≤[∫|∇(un − u)|2 (|∇un|+|∇u|)pn−2 dx] pn2[∫(|∇un|+|∇u|)pn dx]1− pn2.(5.26)Using property (3.11) of Lemma 3.2 we haveAn ≥ C∫|∇(un − u)|2 (|∇un| + |∇u|)pn−2 dx, (5.27)for some positive constant C = C(pn). Now, by using (5.26), (5.27) together with(5.12) we deduce that∫|∇(un − u)|pn dx → 0, as n → ∞.Thus, as above, (5.7) holds true also in this case. unionsqLet us now show how Lemma 5.1 can be applied to prove the existence of weaksolutions to the nonlocal problem (1.3).123312 M. Chipot, H. B. de OliveiraProof of Theorem 5.1 Note that f ∈ (W 1,α0 ())′ ⊂ (W 1,δ0 ())′ for any δ > α. Thusfor each λ ∈ R, there exists a unique solution u = uλ to the p(λ)-Laplacian problem⎧⎨⎩u ∈ W 1,p(λ)0 (),∫|∇u|p(λ)−2∇u · ∇v dx = 〈 f , v〉 ∀ v ∈ W 1,p(λ)0 ().(5.28)Taking v = u = uλ in (5.28) one derives∫|∇uλ|p(λ) dx ≤ ‖ f ‖−1,α′ ‖∇uλ‖α. (5.29)By Hölder’s inequality one has‖∇uλ‖α ≤ ‖∇uλ‖p(λ)||1α− 1p(λ) . (5.30)Thus by (5.29) it comes‖∇uλ‖p(λ)−1p(λ) ≤ ‖ f ‖−1,α′ ||1α− 1p(λ) . (5.31)Gathering (5.30) and (5.31), and using (5.1) we obtain‖∇uλ‖α ≤ ‖ f ‖1p(λ)−1−1,α′ ||(1α− 1p(λ))p(λ)p(λ)−1 ≤ maxp∈[α,β] ‖ f ‖1p−1−1,α′ ||(1α− 1p)pp−1 = C,(5.32)for some positive constant C = C(α, β,, f ). Due to the boundedness of b, see (5.2),and to (5.32), there exists L ∈ R such thatb(uλ) ∈ [−L, L] ∀ λ ∈ R.Let us now consider the mapλ → b(uλ), (5.33)from [−L, L] into itself. This map is continuous. Indeed, if λn → λ as n → ∞, due to(5.1), we have p(λn) → p(λ). Applying now Lemma 5.1 with pn = p(λn), it followsthatuλn → uλ in W 1,α0 (), as n → ∞.Now, b being continuous (see (5.2)), it follows that b(uλn ) −→ b(uλ), as n → ∞,and thus the map (5.33) is also continuous. It has then a fixed point λ0 and uλ0 is thensolution to (5.3). unionsqThe original article has been corrected.123Correction to: Some results on the p(u)-Laplacian problem 313Acknowledgements We are very grateful to the referees for their constructive remarks. This work wasperformed when the first author was visiting the USTC in Hefei and during a part time employment atthe S. M. Nikolskii Mathematical Institute of RUDN University, 6 Miklukho-Maklay St, Moscow, 117198,supported by the Ministry of Education and Science of the Russian Federation. He is grateful to theseinstitutions for their support. Main part of this work was carried out also during the visit of the second authorto the University of Zurich during the first quarter of 2018. Besides the Grant SFRH/BSAB/135242/2017of the Portuguese Foundation for Science and Technology (FCT), Portugal, which made this visit possible,the second author also wishes to thank to Prof. Michel Chipot who kindly welcomed him at the Universityof Zurich.Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published mapsand institutional affiliations.123

Correction to: Some results on the p(u)-Laplacian problem

Universidade do Algarve

Mathematische Annalen (2019) 375:307–313
https://doi.org/10.1007/s00208-019-01859-8 Mathematische Annalen
PUBL ISHER CORRECT ION
Correction to: Some results on the p(u)-Laplacian problem
M. Chipot1 · H. B. de Oliveira2,3
Published online: 4 July 2019
© Springer-Verlag GmbH Germany, part of Springer Nature 2019
Correction to: Mathematische Annalen
https://doi.org/10.1007/s00208-019-01803-w
In the Original Publication of the article, few errors have been identified in section 5
and acknowledgements section. The corrected section 5 and acknowledgements are
given below:
5 Nonlocal problems
In this section we consider a real function p such that
p is continuous, 1 < α < p ≤ β, (5.1)
for some constants α, β. We denote by b a mapping from W 1,α0 () into R such that
b is continuous, b is bounded, (5.2)
i.e. b sends bounded sets of W 1,α0 () into bounded sets of R.
Definition 2 A function u is a weak solution to the problem (1.3) if
⎧
⎨
⎩
u ∈ W 1,p(b(u))0 (),∫

|∇u|p(b(u))−2∇u · ∇v dx = 〈 f , v〉 ∀ v ∈ W 1,p(b(u))0 (),
(5.3)
The original article can be found online at https://doi.org/10.1007/s00208-019-01803-w.
B H. B. de Oliveira
holivei@ualg.pt
M. Chipot
m.m.chipot@math.uzh.ch
1 IMath, Universität Zürich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
2 FCT, Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, Portugal
3 CMAFCIO, Universidade de Lisboa, Campo Grande, 1749-016 Lisbon, Portugal
123
308 M. Chipot, H. B. de Oliveira
where 〈·, ·〉 denotes the duality pairing between (W 1,p(b(u))0 ())′ and W 1,p(b(u))0 ().
One should notice that p(b(u)) is here a real number and not a function so that
the Sobolev spaces involved are the classical ones. We refer to [5, 7–9] for more on
nonlocal problems.
Then one has:
Theorem 5.1 Let  ⊂ Rd , d ≥ 2, be a bounded domain and assume that (5.1) and
(5.2) hold together with
f ∈ W−1,α′().
Then there exists at least one weak solution to the problem (1.3) in the sense of
Definition 2.
The proof of Theorem 5.1 is based on the following result.
Lemma 5.1 For n ∈ N, let un be the solution to the problem
⎧
⎨
⎩
un ∈ W 1,pn0 (),∫

|∇un|pn−2∇un · ∇v dx = 〈 f , v〉 ∀ v ∈ W 1,pn0 (),
(5.4)
where 〈·, ·〉 denotes here the duality pairing between (W 1,pn0 ())′ and W 1,pn0 ().
Suppose that
pn → p, as n → ∞, where p ∈ (1,∞), (5.5)
f ∈ W−1,q ′() for some q < p. (5.6)
Then
un → u in W 1,q0 (), as n → ∞, (5.7)
where u is the solution to the problem
⎧
⎨
⎩
u ∈ W 1,p0 (),∫

|∇u|p−2∇u · ∇v dx = 〈 f , v〉 ∀ v ∈ W 1,p0 ().
(5.8)
Proof of Lemma 5.1 We shall split this proof into two steps.
1. Weak convergence: We first observe that, in view of pn → p, as n → ∞, and
q < p, we may assume that
p + 1 > pn > q ∀ n ∈ N. (5.9)
Taking v = un in the equation of (5.4) we get
∫

|∇un|pn dx ≤ ‖ f ‖−1,q ′ ‖∇un‖q . (5.10)
123
Correction to: Some results on the p(u)-Laplacian problem 309
Recall that ‖ f ‖−1,q ′ denotes the strong dual norm of f associated to the norm ‖∇ · ‖q .
On the other hand, by using Hölder’s inequality and (5.9), we have
‖∇un‖q ≤ C‖∇un‖pn , (5.11)
for some positive constant C = C(p, q,). Plugging (5.11) into (5.10) it comes
‖∇un‖pn ≤ C, (5.12)
for some other positive constant C = C(p, q,, f ). Combining (5.11) with (5.12),
it follows that
‖∇un‖q ≤ C, (5.13)
for some positive constant C independent of n. From (5.13) we deduce then that for
some subsequence still labelled by n and for some u ∈ W 1,q0 ()
∇un⇀∇u in Lq(), as n → ∞. (5.14)
Due to (5.5), (5.9), (5.12) and (5.14), we can also apply Lemma 3.1 so that
lim inf
n→∞
∫

|∇un|pn dx ≥
∫

|∇u|pdx .
As a consequence we have
u ∈ W 1,p0 (). (5.15)
Clearly the equation in (5.4) is equivalent to
∫

|∇un|pn−2∇un · ∇(v − un) dx ≥ 〈 f , v − un〉 ∀ v ∈ W 1,pn0 ().
and by the Minty lemma to
∫

|∇v|pn−2∇v · ∇(v − un) dx ≥ 〈 f , v − un〉 ∀ v ∈ W 1,pn0 (). (5.16)
Taking v ∈ C∞0 (), one can use (5.5) and (5.14) to pass to the limit in (5.16), as
n → ∞, so that
∫

|∇v|p−2∇v · ∇(v − u) dx ≥ 〈 f , v − u〉 ∀ v ∈ C∞0 (). (5.17)
Using the density of C∞0 () in W
1,p
0 (), we see that (5.17) also holds for all v ∈
W 1,p0 (). In this case, taking v = u ± δz, with z ∈ W 1,p0 () and δ > 0, and letting
δ → 0 after simplifying the resulting inequality, one obtains
∫

|∇u|p−2∇u · ∇z dx = 〈 f , z〉 ∀ z ∈ W 1,p0 ().
Thus u is the solution to the problem (5.8).
123
310 M. Chipot, H. B. de Oliveira
2. Strong convergence: We want to show that the convergence (5.14) is in fact strong.
To prove this, we first note that, taking v = un in the equation of (5.4) and using (5.14)
to pass to the limit, we obtain
∫

|∇un|pn dx = 〈 f , un〉 → 〈 f , u〉 =
∫

|∇u|pdx, as n → ∞. (5.18)
Consider the case of the pn’s such that
pn ≥ p ∀ n ∈ N.
One has by Hölder’s inequality
∫

|∇un|pdx ≤
(∫

|∇un|pn dx
) p
pn ||1− ppn ,
where || denotes the d-Lebesgue measure of . Thus by (5.18) for such a sequence
lim sup
n→∞
∫

|∇un|pdx ≤
∫

|∇u|pdx ≤ lim inf
n→∞
∫

|∇un|pdx,
which shows (since ‖∇un‖p → ‖∇u‖p, as n → ∞)
un → u strongly in W 1,p0 (), as n → ∞. (5.19)
Since W 1,p0 () ⊂ W 1,q0 (), (5.19) implies (5.7).
Next, consider the pn’s such that
q < pn < p ∀ n ∈ N (5.20)
and set
An :=
∫

(
|∇un|pn−2∇un − |∇u|pn−2∇u
)
· (∇un − ∇u) dx . (5.21)
Due to the monotonicity, An ≥ 0 and, because of (5.4), one has
An = 〈 f , un − u〉 −
∫

|∇u|pn−2∇u · ∇(un − u) dx .
From (5.6) and (5.14), we have
〈 f , un − u〉 → 0, as n → ∞. (5.22)
Moreover, from (5.15) one easily gets
∣
∣
∣|∇u|pn−2∇u
∣
∣
∣ ≤ max{1, |∇u|}p−1 ∈ L p′(). (5.23)
123
Correction to: Some results on the p(u)-Laplacian problem 311
Hence, (5.20), (5.22) and (5.23) ensure that
An → 0, as n → ∞. (5.24)
Assume first that
pn ≥ 2.
This allows us to use property (3.10) of Lemma 3.2 in (5.21) so that
An ≥ 12pn−1
∫

|∇(un − u)|pn dx . (5.25)
Since, by (5.20), pn > q, we have by Hölder’s inequality, (5.20), (5.24) and (5.25)
∫

|∇(un − u)|q dx ≤
( ∫

|∇(un − u)|pn dx
) q
pn ||1− qpn → 0,
when n → ∞. This proves (5.7) in this case.
Consider now the case when
pn < 2.
Here, we use Hölder’s inequality as follows
∫

|∇(un − u)|pn dx
=
∫

|∇(un − u)|pn (|∇un| + |∇u|) (pn−2)pn2 (|∇un| + |∇u|) (2−pn )pn2 dx
≤
[∫

|∇(un − u)|2 (|∇un|+|∇u|)pn−2 dx
] pn
2
[∫

(|∇un|+|∇u|)pn dx
]1− pn2
.
(5.26)
Using property (3.11) of Lemma 3.2 we have
An ≥ C
∫

|∇(un − u)|2 (|∇un| + |∇u|)pn−2 dx, (5.27)
for some positive constant C = C(pn). Now, by using (5.26), (5.27) together with
(5.12) we deduce that
∫

|∇(un − u)|pn dx → 0, as n → ∞.
Thus, as above, (5.7) holds true also in this case. unionsq
Let us now show how Lemma 5.1 can be applied to prove the existence of weak
solutions to the nonlocal problem (1.3).
123
312 M. Chipot, H. B. de Oliveira
Proof of Theorem 5.1 Note that f ∈ (W 1,α0 ())′ ⊂ (W 1,δ0 ())′ for any δ > α. Thus
for each λ ∈ R, there exists a unique solution u = uλ to the p(λ)-Laplacian problem
⎧
⎨
⎩
u ∈ W 1,p(λ)0 (),∫

|∇u|p(λ)−2∇u · ∇v dx = 〈 f , v〉 ∀ v ∈ W 1,p(λ)0 ().
(5.28)
Taking v = u = uλ in (5.28) one derives
∫

|∇uλ|p(λ) dx ≤ ‖ f ‖−1,α′ ‖∇uλ‖α. (5.29)
By Hölder’s inequality one has
‖∇uλ‖α ≤ ‖∇uλ‖p(λ)||
1
α
− 1p(λ) . (5.30)
Thus by (5.29) it comes
‖∇uλ‖p(λ)−1p(λ) ≤ ‖ f ‖−1,α′ ||
1
α
− 1p(λ) . (5.31)
Gathering (5.30) and (5.31), and using (5.1) we obtain
‖∇uλ‖α ≤ ‖ f ‖
1
p(λ)−1
−1,α′ ||
(
1
α
− 1p(λ)
)
p(λ)
p(λ)−1 ≤ max
p∈[α,β] ‖ f ‖
1
p−1
−1,α′ ||
(
1
α
− 1p
)
p
p−1 = C,
(5.32)
for some positive constant C = C(α, β,, f ). Due to the boundedness of b, see (5.2),
and to (5.32), there exists L ∈ R such that
b(uλ) ∈ [−L, L] ∀ λ ∈ R.
Let us now consider the map
λ → b(uλ), (5.33)
from [−L, L] into itself. This map is continuous. Indeed, if λn → λ as n → ∞, due to
(5.1), we have p(λn) → p(λ). Applying now Lemma 5.1 with pn = p(λn), it follows
that
uλn → uλ in W 1,α0 (), as n → ∞.
Now, b being continuous (see (5.2)), it follows that b(uλn ) −→ b(uλ), as n → ∞,
and thus the map (5.33) is also continuous. It has then a fixed point λ0 and uλ0 is then
solution to (5.3). unionsq
The original article has been corrected.
123
Correction to: Some results on the p(u)-Laplacian problem 313
Acknowledgements We are very grateful to the referees for their constructive remarks. This work was
performed when the first author was visiting the USTC in Hefei and during a part time employment at
the S. M. Nikolskii Mathematical Institute of RUDN University, 6 Miklukho-Maklay St, Moscow, 117198,
supported by the Ministry of Education and Science of the Russian Federation. He is grateful to these
institutions for their support. Main part of this work was carried out also during the visit of the second author
to the University of Zurich during the first quarter of 2018. Besides the Grant SFRH/BSAB/135242/2017
of the Portuguese Foundation for Science and Technology (FCT), Portugal, which made this visit possible,
the second author also wishes to thank to Prof. Michel Chipot who kindly welcomed him at the University
of Zurich.
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps
and institutional affiliations.
123


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