Volume 297, Issue 5 pp. 1818-1830

ORIGINAL ARTICLE

Global integrability for solutions to quasilinear elliptic systems with degenerate coercivity

Ya Li,

Ya Li

College of Mathematics and Information Science, Hebei University, Baoding, China

Search for more papers by this author

Gaoyang Liu,

Gaoyang Liu

College of Mathematics and Information Science, Hebei University, Baoding, China

Search for more papers by this author

Hongya Gao,

Corresponding Author

Hongya Gao

[email protected]

College of Mathematics and Information Science, Hebei University, Baoding, China

Correspondence

Hongya Gao, College of Mathematics and Information Science, Hebei University, Baoding, 071002, China.

Email: [email protected]

Search for more papers by this author

Ya Li,

Ya Li

College of Mathematics and Information Science, Hebei University, Baoding, China

Search for more papers by this author

Gaoyang Liu,

Gaoyang Liu

College of Mathematics and Information Science, Hebei University, Baoding, China

Search for more papers by this author

Hongya Gao,

Corresponding Author

Hongya Gao

[email protected]

College of Mathematics and Information Science, Hebei University, Baoding, China

Correspondence

Hongya Gao, College of Mathematics and Information Science, Hebei University, Baoding, 071002, China.

Email: [email protected]

Search for more papers by this author

First published: 11 January 2024

https://doi.org/10.1002/mana.202200550

Share a link

Email
Wechat
Bluesky

Abstract

This paper deals with global integrability for solutions to quasilinear elliptic systems involving $N$ equations of the form

\begin{equation*} {\begin{cases} \displaystyle -\sum _{i=1}^n D_i {\left(\sum _{\beta =1}^N \sum _{j=1}^n a^{\alpha, \beta } _{i,j} (x,u(x)) D_j u^\beta (x) \right)} =f^\alpha (x), & \mbox{ in } \Omega, \\[10pt] \displaystyle u(x)=0, &\displaystyle \mbox{ on } \partial \Omega, \end{cases}} \end{equation*}

where

\Omega

is an open bounded subset of

\mathbb {R}^n

n&gt;2

u=(u^1,u^2,\ldots,u^N):\Omega \subset \mathbb {R}^n \rightarrow \mathbb {R}^N

N\ge 2

. Under degenerate coercivity condition of the diagonal coefficients and proportional condition of the off-diagonal coefficients, we obtain some global integrability results.

REFERENCES

1P. Baroni, Riesz potential estimates for a general class of quasilinear equations, Calc. Var. Partial Differential Equations 53 (2015), 803–846.
10.1007/s00526-014-0768-z
Web of Science® Google Scholar
2M. F. Betta, T. Del Vecchio, and M. R. Posteraro, Existence and regularity results for nonlinear degenerate elliptic equations with measure data, Ric. Mat. 47 (1998), 277–295.
Google Scholar
3L. Boccardo, Problemi differenziali ellittici e parabolici con dati misure, Boll. Unione Mat. Ital. Sez. A 11 (1997), 439–461.
Google Scholar
4L. Boccardo and G. Croce, Esistenza e regolarità di soluzioni di alcuni problemi ellittici, Quaderni dell $^{\prime }$ UMI, vol. 51, Pitagora Editrice, Bologna, 2010.
Google Scholar
5L. Boccardo and G. Croce, Elliptic partial differential equations, De Gruyter Stud. Math. 55 (2014), 50–51.
Google Scholar
6L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1998), 149–169.
10.1016/0022-1236(89)90005-0
Google Scholar
7L. Boccardo, T. Gallouët, and P. Marcellini, Anisotropic equations in $L^1$ , Differ. Integral Equ. 9 (1996), 209–212.
Google Scholar
8G. R. Cirmi, On the existence of solutions to non-linear degenerate elliptic equations with measure data, Ric. Mat. 42 (1993), 315–329.
Google Scholar
9G. R. Cirmi and S. Leonardi, Regularity results for the gradient of solutions linear elliptic equations with $L^{1,\lambda }$ data, Ann. Mat. Pura Appl. 185 (2006), 537–553.
10.1007/s10231-005-0167-3
Web of Science® Google Scholar
10M. Fuchs and J. Reuling, Non-linear elliptic systems involving measure data, Rend. Mat. Appl. 15 (1995), 101–109.
Google Scholar
11H. Gao, H. Deng, M. Huang, and W. Ren, Generalizations of Stampacchia Lemma and applications to quasilinear elliptic systems, Nonlinear Anal. 208 (2021), 112297.
10.1016/j.na.2021.112297
Web of Science® Google Scholar
12H. Gao, M. Huang, H. Deng, and W. Ren, Global integrability for solutions to quasilinear elliptic systems, Manuscripta Math. 164 (2021), 23–37.
10.1007/s00229-020-01183-5
Web of Science® Google Scholar
13H. Gao, S. Liang, and Y. Cui, Regularity for anisotropic solutions to some nonlinear elliptic system, Front. Math. China 11 (2016), 77–87.
10.1007/s11464-015-0443-5
Web of Science® Google Scholar
14H. Gao, J. Zhang, and H. Ma, A generalization of Stampacchia Lemma and applications, https://arxiv.org/abs/2212.04237.
Google Scholar
15P. Di Geronimo, F. Leonetti, M. Macrí, and P. V. Petricca, Existence of bounded solutions for some quasilinear degenerate elliptic systems, Minimax Theory its Appl. 6 (2021), 321–340.
Web of Science® Google Scholar
16M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Princeton University Press, Princeton, NJJ, 1980.
Google Scholar
17M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31–46.
10.1007/BF02392725
Web of Science® Google Scholar
18E. De Giorgi, Un esempio estremali discontinue per un problema variazionale di tipo ellittico, Boll. Unione Mat. Ital. 4 (1968), 135–137.
Google Scholar
19A. I. Koshelev, Regularity problem for quasilinear elliptic and parabolic systems, Springer, Berlin, 1995.
10.1007/BFb0094482
Google Scholar
20J. Kottas, Interior regularity for the quasilinear elliptic systems with nonsmooth coefficients, Comm. Math. Univ. Carolinae 28 (1997), 95–102.
Google Scholar
21T. Kuusi and G. Mingione, Universal potential estimates, J. Funct. Anal. 262 (2012), 4205–4269.
10.1016/j.jfa.2012.02.018
Web of Science® Google Scholar
22S. Leonardi, F. Leonetti, C. Pignotti, E. Rocha, and V. Staicu, Maximum principles for some quasilinear elliptic systems, Nonlinear Anal. 194 (2020), 111377.
10.1016/j.na.2018.11.004
Web of Science® Google Scholar
23S. Leonardi, F. Leonetti, C. Pignotti, E. Rocha, and V. Staicu, Local boundedness for weak solutions to some quasilinear elliptic systems, Minimax Theory its Appl. 6 (2021), 365–372.
Web of Science® Google Scholar
24S. Leonardi, J. Kottas, and J. Stara, Hölder regularity of the solutions of some classes of elliptic systems in convex nonsmooth domains, Nonlinear Anal. 60 (2005), 925–944.
10.1016/j.na.2004.09.056
Web of Science® Google Scholar
25F. Leonetti and P. V. Petricca, Existence of bounded solutions to some nonlinear degenerate elliptic systems, Discrete Contin. Dyn. Syst. Ser. B 11 (2009), 191–203.
Web of Science® Google Scholar
26F. Leonetti, E. Rocha, and V. Staicu, Quasilinear elliptic systems with measure data, Nonlinear Anal. 154 (2017), 210–224.
10.1016/j.na.2016.04.002
Web of Science® Google Scholar
27F. Leonetti, E. Rocha, and V. Staicu, Smallness and cancellation in some elliptic systems with measure data, J. Math. Anal. Appl. 465 (2018), 885–902.
10.1016/j.jmaa.2018.05.047
Web of Science® Google Scholar
28M. Meier, Boundedness and integrability properties of weak solutions of quasilinear elliptic systems, J. Reine Angew. Math. 333 (1982), 191–220.
Web of Science® Google Scholar
29G. Mingione, The Calderón-Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa 6 (2007), 195–261.
Web of Science® Google Scholar
30G. Mingione, Gradient estimates below the duality exponent, Math. Ann. 346 (2010), 571–627.
10.1007/s00208-009-0411-z
Web of Science® Google Scholar
31G. Mingione, La teoria di Calderon-Zygmund dal caso lineare a quello non lineare, Boll. Unione Mat. Ital. 9 (2013), 269–297.
Google Scholar
32G. Mingione, Nonlinear measure data problems, Milan J. Math. 79 (2011), 429–496.
10.1007/s00032-011-0168-1
Web of Science® Google Scholar
33J. Necas, On the regularity of weak solutions to nonlinear elliptic systems of partial differential equations, Lectures at Scuola Normale Superiore, Pisa, 1979.
Google Scholar
34P. Oppezzi and A. M. Rossi, Esistenza di soluzioni per problemi unilateri con dato misura o in $L^1$ , Ric. Mat. 45 (1996), 491–513.
Google Scholar
35K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems, Acta Math. 138 (1977), 219–240.
10.1007/BF02392316
Web of Science® Google Scholar
36Z. Q. Yan, Everywhere regularity for solutions to quasilinear elliptic systems of triangular form, Partial differential equations (Tianjin, 1986), Lecture Notes in Math., vol. 1306, Springer, Berlin, 1988, pp. 255–261.
Google Scholar
37S. Zhou, A note on nonlinear elliptic systems involving measure, Electron J. Differential Equations 8 (2000), 1–6.
Google Scholar

Volume297, Issue5

May 2024

Pages 1818-1830

Global integrability for solutions to quasilinear elliptic systems with degenerate coercivity

Abstract

REFERENCES

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Global integrability for solutions to quasilinear elliptic systems with degenerate coercivity

Abstract

REFERENCES

References

Related

Information