Avoid common mistakes on your manuscript.
1 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:
2 Nonlocal problems
In this section we consider a real function p such that
for some constants \(\alpha , \beta \). We denote by b a mapping from \(W_0^{1,\alpha }(\Omega )\) into \(\mathbb {R}\) such that
i.e. b sends bounded sets of \(W_0^{1,\alpha }(\Omega )\) into bounded sets of \(\mathbb {R}\).
Definition 2
A function u is a weak solution to the problem (1.3) if
where \(\langle \cdot ,\cdot \rangle \) denotes the duality pairing between \((W_0^{1,p(b(u))}(\Omega ))'\) and \(W_0^{1,p(b(u))}(\Omega )\).
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 \(\Omega \subset \mathbb {R}^d\), \(d\ge 2\), be a bounded domain and assume that (5.1) and (5.2) hold together with
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 \(n\in \mathbb {N}\), let \(u_n\) be the solution to the problem
where \(\langle \cdot ,\cdot \rangle \) denotes here the duality pairing between \((W_0^{1,p_n}(\Omega ))'\) and \(W_0^{1,p_n}(\Omega )\). Suppose that
Then
where u is the solution to the problem
Proof of Lemma 5.1
We shall split this proof into two steps.
1. Weak convergence: We first observe that, in view of \(p_n\rightarrow p\), as \(n\rightarrow \infty \), and \(q<p\), we may assume that
Taking \(v=u_n\) in the equation of (5.4) we get
Recall that \( \Vert f\Vert _{-1,q'}\) denotes the strong dual norm of f associated to the norm \(\Vert \varvec{\nabla }\cdot \Vert _{q}\). On the other hand, by using Hölder’s inequality and (5.9), we have
for some positive constant \(C=C(p,q,\Omega )\). Plugging (5.11) into (5.10) it comes
for some other positive constant \(C=C(p,q,\Omega , f)\). Combining (5.11) with (5.12), it follows that
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 \(u \in W^{1,q}_0(\Omega )\)
Due to (5.5), (5.9), (5.12) and (5.14), we can also apply Lemma 3.1 so that
As a consequence we have
Clearly the equation in (5.4) is equivalent to
and by the Minty lemma to
Taking \(v\in C_0^\infty (\Omega )\), one can use (5.5) and (5.14) to pass to the limit in (5.16), as \(n\rightarrow \infty \), so that
Using the density of \(C_0^\infty (\Omega )\) in \(W_0^{1,p}(\Omega )\), we see that (5.17) also holds for all \(v\in W_0^{1,p}(\Omega )\). In this case, taking \(v=u\pm \delta z\), with \(z\in W_0^{1,p}(\Omega )\) and \(\delta >0\), and letting \(\delta \rightarrow 0\) after simplifying the resulting inequality, one obtains
Thus u is the solution to the problem (5.8).
2. Strong convergence: We want to show that the convergence (5.14) is in fact strong. To prove this, we first note that, taking \(v=u_n\) in the equation of (5.4) and using (5.14) to pass to the limit, we obtain
Consider the case of the \(p_n\)’s such that
One has by Hölder’s inequality
where \(|\Omega |\) denotes the d-Lebesgue measure of \(\Omega \). Thus by (5.18) for such a sequence
which shows (since \( \Vert \varvec{\nabla }u_n\Vert _{p} \rightarrow \Vert \varvec{\nabla }u\Vert _{p}\), as \(n\rightarrow \infty \))
Since \(W^{1,p}_0(\Omega )\subset W^{1,q}_0(\Omega )\), (5.19) implies (5.7).
Next, consider the \(p_n\)’s such that
and set
Due to the monotonicity, \(A_n\ge 0\) and, because of (5.4), one has
From (5.6) and (5.14), we have
Moreover, from (5.15) one easily gets
Hence, (5.20), (5.22) and (5.23) ensure that
Assume first that
This allows us to use property (3.10) of Lemma 3.2 in (5.21) so that
Since, by (5.20), \(p_n>q\), we have by Hölder’s inequality, (5.20), (5.24) and (5.25)
when \(n \rightarrow \infty \). This proves (5.7) in this case.
Consider now the case when
Here, we use Hölder’s inequality as follows
Using property (3.11) of Lemma 3.2 we have
for some positive constant \(C=C(p_n)\). Now, by using (5.26), (5.27) together with (5.12) we deduce that
Thus, as above, (5.7) holds true also in this case. \(\square \)
Let us now show how Lemma 5.1 can be applied to prove the existence of weak solutions to the nonlocal problem (1.3).
Proof of Theorem 5.1
Note that \(f \in (W_0^{1,\alpha }(\Omega ))' \subset (W_0^{1,\delta }(\Omega ))'\) for any \(\delta > \alpha \). Thus for each \(\lambda \in \mathbb {R}\), there exists a unique solution \(u=u_\lambda \) to the \(p(\lambda )\)-Laplacian problem
Taking \(v=u=u_\lambda \) in (5.28) one derives
By Hölder’s inequality one has
Thus by (5.29) it comes
Gathering (5.30) and (5.31), and using (5.1) we obtain
for some positive constant \(C=C(\alpha ,\beta ,\Omega ,f)\). Due to the boundedness of b, see (5.2), and to (5.32), there exists \(L\in \mathbb {R}\) such that
Let us now consider the map
from \([-L,L]\) into itself. This map is continuous. Indeed, if \(\lambda _n\rightarrow \lambda \) as \(n\rightarrow \infty \), due to (5.1), we have \(p(\lambda _n)\rightarrow p(\lambda )\). Applying now Lemma 5.1 with \(p_n=p(\lambda _n)\), it follows that
Now, b being continuous (see (5.2)), it follows that \(b(u_{\lambda _n})\longrightarrow b(u_\lambda )\), as \(n\rightarrow \infty \), and thus the map (5.33) is also continuous. It has then a fixed point \(\lambda _0\) and \(u_{\lambda _0}\) is then solution to (5.3). \(\square \)
The original article has been corrected.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chipot, M., de Oliveira, H.B. Correction to: Some results on the p(u)-Laplacian problem. Math. Ann. 375, 307–313 (2019). https://doi.org/10.1007/s00208-019-01859-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-019-01859-8