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

Texto

(1)Mathematische Annalen (2019) 375:307–313 https://doi.org/10.1007/s00208-019-01859-8. Mathematische Annalen. PUBLISHER CORRECTION. 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 W01,α () into R such that b is continuous,. b is bounded,. (5.2). i.e. b sends bounded sets of W01,α () into bounded sets of R. Definition 2 A function u is a weak solution to the problem (1.3) if ⎧ ⎨ u ∈ W01, p(b(u)) (), ⎩ |∇u| p(b(u))−2 ∇u · ∇v d x = f , v ∀ v ∈ W01, p(b(u)) (),. (5.3). . The original article can be found online at https://doi.org/10.1007/s00208-019-01803-w.. B. H. B. de Oliveira [email protected] M. Chipot [email protected]. 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.

(2) 308. M. Chipot, H. B. de Oliveira 1, p(b(u)). where ·, · denotes the duality pairing between (W0. ()) and W0. 1, p(b(u)). ().. 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 u n be the solution to the problem ⎧ 1, p ⎨ u n ∈ W0 n (), ⎩. . |∇u n | pn −2 ∇u n · ∇v d x = f , v ∀ v ∈ W0. 1, pn. 1, pn. where ·, · denotes here the duality pairing between (W0 Suppose that. ()) and W0. pn → p, as n → ∞, where p ∈ (1, ∞), f ∈W. −1,q . () for some q < p.. Then. (5.4). (),. 1, pn. ().. (5.5) (5.6). 1,q. u n → u in W0 (), as n → ∞,. (5.7). where u is the solution to the problem ⎧ 1, p ⎨ u ∈ W0 (), ⎩. . 1, p. |∇u| p−2 ∇u · ∇v d x = f , v ∀ v ∈ W0 ().. (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 = u n in the equation of (5.4) we get . 123. |∇u n | pn d x ≤

(3) f

(4) −1,q

(5) ∇u n

(6) q .. (5.10).

(7) Correction to: Some results on the p(u)-Laplacian problem. 309. Recall that

(8) f

(9) −1,q denotes the strong dual norm of f associated to the norm

(10) ∇ ·

(11) q . On the other hand, by using Hölder’s inequality and (5.9), we have

(12) ∇u n

(13) q ≤ C

(14) ∇u n

(15) pn ,. (5.11). for some positive constant C = C( p, q, ). Plugging (5.11) into (5.10) it comes

(16) ∇u n

(17) pn ≤ C,. (5.12). for some other positive constant C = C( p, q, , f ). Combining (5.11) with (5.12), it follows that

(18) ∇u n

(19) q ≤ C, (5.13) for some positive constant C independent of n. From (5.13) we deduce then that for 1,q some subsequence still labelled by n and for some u ∈ W0 () ∇u n ∇u. in L q (), 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 |∇u n | pn d x ≥ |∇u| p d x. n→∞. As a consequence we have. . . 1, p. u ∈ W0 ().. (5.15). Clearly the equation in (5.4) is equivalent to 1, p |∇u n | pn −2 ∇u n · ∇(v − u n ) d x ≥ f , v − u n ∀ v ∈ W0 n (). . and by the Minty lemma to 1, p |∇v| pn −2 ∇v · ∇(v − u n ) d x ≥ f , v − u n ∀ v ∈ W0 n (). . (5.16). Taking v ∈ C0∞ (), 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) d x ≥ f , v − u ∀ v ∈ C0∞ (). (5.17) . Using the density of C0∞ () in W0 (), we see that (5.17) also holds for all v ∈ 1, p 1, p W0 (). In this case, taking v = u ± δz, with z ∈ W0 () and δ > 0, and letting δ → 0 after simplifying the resulting inequality, one obtains 1, p |∇u| p−2 ∇u · ∇z d x = f , z ∀ z ∈ W0 (). 1, p. . Thus u is the solution to the problem (5.8).. 123.

(20) 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 = u n in the equation of (5.4) and using (5.14) to pass to the limit, we obtain |∇u n | pn d x = f , u n → f , u = |∇u| p d x, as n → ∞. (5.18) . . Consider the case of the pn ’s such that pn ≥ p ∀ n ∈ N. One has by Hölder’s inequality . |∇u n | d x ≤. . p. . |∇u n | d x pn. . p pn. ||. 1− ppn. ,. where || denotes the d-Lebesgue measure of . Thus by (5.18) for such a sequence p p lim sup |∇u n | d x ≤ |∇u| d x ≤ lim inf |∇u n | p d x, n→∞. . n→∞. . . which shows (since

(21) ∇u n

(22) p →

(23) ∇u

(24) p , as n → ∞) 1, p. u n → u strongly in W0 (), as n → ∞. 1, p. (5.19). 1,q. Since W0 () ⊂ W0 (), (5.19) implies (5.7). Next, consider the pn ’s such that q < pn < p ∀ n ∈ N. (5.20). and set . An := |∇u n | pn −2 ∇u n − |∇u| pn −2 ∇u · (∇u n − ∇u) d x. . (5.21). Due to the monotonicity, An ≥ 0 and, because of (5.4), one has An = f , u n − u − |∇u| pn −2 ∇u · ∇(u n − u) d x. . From (5.6) and (5.14), we have f , u n − u → 0, as n → ∞. Moreover, from (5.15) one easily gets. . |∇u| pn −2 ∇u ≤ max{1, |∇u|} p−1 ∈ L p ().. 123. (5.22). (5.23).

(25) 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 ≥. . 1 2 pn −1. . |∇(u n − u)| pn d x.. (5.25). Since, by (5.20), pn > q, we have by Hölder’s inequality, (5.20), (5.24) and (5.25) |∇(u n − u)| d x ≤ q. . . |∇(u n − u)| pn d x. q pn. ||. 1− pqn. → 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 |∇(u n − u)| pn d x ( pn −2) pn (2− pn ) pn = |∇(u n − u)| pn (|∇u n | + |∇u|) 2 dx (|∇u n | + |∇u|) 2. . . ≤. . |∇(u n − u)|2 (|∇u n |+|∇u|) pn −2 d x.

(26) pn 2. .

(27) 1− pn (|∇u n |+|∇u|) pn d x. 2. .. (5.26) Using property (3.11) of Lemma 3.2 we have An ≥ C. . |∇(u n − u)|2 (|∇u n | + |∇u|) pn −2 d x,. (5.27). for some positive constant C = C( pn ). Now, by using (5.26), (5.27) together with (5.12) we deduce that . |∇(u n − u)| pn d x → 0, as n → ∞.. Thus, as above, (5.7) holds true also in this case.. . 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.

(28) 312. M. Chipot, H. B. de Oliveira. Proof of Theorem 5.1 Note that f ∈ (W01,α ()) ⊂ (W01,δ ()) for any δ > α. Thus for each λ ∈ R, there exists a unique solution u = u λ to the p(λ)-Laplacian problem ⎧ 1, p(λ) ⎨ u ∈ W0 (), ⎩. . 1, p(λ). |∇u| p(λ)−2 ∇u · ∇v d x = f , v ∀ v ∈ W0. (5.28). ().. Taking v = u = u λ in (5.28) one derives . |∇u λ | p(λ) d x ≤

(29) f

(30) −1,α

(31) ∇u λ

(32) α .. (5.29). By Hölder’s inequality one has 1.

(33) ∇u λ

(34) α ≤

(35) ∇u λ

(36) p(λ) || α. 1 − p(λ). .. (5.30). Thus by (5.29) it comes p(λ)−1.

(37) ∇u λ

(38) p(λ). 1. ≤

(39) f

(40) −1,α || α. 1 − p(λ). .. (5.31). Gathering (5.30) and (5.31), and using (5.1) we obtain 1. . p(λ)−1

(41) ∇u λ

(42) α ≤

(43) f

(44) −1,α ||. 1 1 α − p(λ). p(λ) p(λ)−1. 1. p−1 ≤ max

(45) f

(46) −1,α ||. p∈[α,β]. . 1 1 α− p. 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(u λ ) ∈ [−L, L] ∀ λ ∈ R. Let us now consider the map λ → b(u λ ),. (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 u λn → u λ in W01,α (), as n → ∞. Now, b being continuous (see (5.2)), it follows that b(u λn ) −→ b(u λ ), as n → ∞, and thus the map (5.33) is also continuous. It has then a fixed point λ0 and u λ0 is then solution to (5.3). . The original article has been corrected.. 123.

(47) 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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