Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-13T06:14:03.172Z Has data issue: false hasContentIssue false
  • English
  • Français

Nondegeneracy of the bubble for the critical p-Laplace equation

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  20 February 2020

Angela Pistoia  and
Giusi Vaira
Angela Pistoia
Affiliation:
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, Via Scarpa 16, Roma00161, Italy (angela.pistoia@uniroma1.it)
Giusi Vaira
Affiliation:
Dipartimento di Matematica e Fisica, Università degli studi della Campania ‘Luigi Vanvitelli’’, Viale Lincoln 5, Caserta81100, Italy (giusi.vaira@unicampania.it)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the non-degeneracy of the extremals of the Sobolev inequality

\[ \int_{\mathbb R^N}|\nabla u|^p\,\rd x\ge \mathcal S_p\int_{\open R^N}|u|^\frac{Np}{N-p}\,\rd x,\quad u\in \mathcal D^{1,p}(\open R^N) \]
when 1 < p < N, as solutions of a critical quasilinear equation involving the p-Laplacian.

MSC classification

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B33: Critical exponents 35J20: Variational methods for second-order elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 151 - 168
Check for updates
Copyright
Copyright © The Author(s) 2020. Published by The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Aubin, T.. Problemes isoperimtriques et espaces de Sobolev. J. Differ. Geom. 11 (1976), 573–598.CrossRefGoogle Scholar
2Brézis, H. and Nirenberg, L.. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36 (1983), 437–477.CrossRefGoogle Scholar
3Caffarelli, L., Kohn, R. and Nirenberg, L.. First order interpolation inequalities with weights. Compos. Math. 53 (1984), 259–275.Google Scholar
4Clapp, M. and Lopez Rios, L.. Entire nodal solutions to the pure critical exponent problem for the p-Laplacian. J. Differ. Eq 265 (2018), 891–905.CrossRefGoogle Scholar
5Coron, J.-M.. Topologie et cas limtie des injections de Sobolev. C. R. Acad. Sci. Paris Sr. I Math. 299 (1984), 209–212.Google Scholar
6Damascelli, L., Merchán, S., Montoro, L. and Sciunzi, B.. Radial symmetry and applications for a problem involving the $-\Delta _p(\cdot )$ operator and critical nonlinearity in ℝN. Adv. Math. 256 (2014), 313–335.CrossRefGoogle Scholar
7Damascelli, L. and Ramaswamy, M.. Symmetry of $C^1-$ solutions of p-Laplace equations in ℝN. Adv. Nonlinear Stud. 1 (2001), 40–64.CrossRefGoogle Scholar
8Damascelli, L. and Sciunzi, B.. Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations. J. Differ. Eq. 206 (2004), 483–515.CrossRefGoogle Scholar
9del Pino, M., Musso, M., Pacard, F. and Pistoia, A.. Large energy entire solutions for the Yamabe equation. J. Differ. Eq. 251 (2011), 2568–2597.CrossRefGoogle Scholar
10del Pino, M., Musso, M., Pacard, F. and Pistoia, A.. Torus action on Sn and sign-changing solutions for conformally invariant equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 (2013), 209–237.Google Scholar
11Farina, A., Mercuri, C. and Willem, M.. A Liouville theorem for the p-Laplacian and related questions. Calc. Var. Partial Differ. Eq. 58 (2019), 13.CrossRefGoogle Scholar
12Ge, Y., Musso, M. and Pistoia, A.. Sign changing tower of bubbles for an elliptic problem at the critical exponent in pierced non-symmetric domains. Comm. Partial Differ. Eq. 35 (2010), 1419–1457.CrossRefGoogle Scholar
13Ge, Y., Musso, M., Pistoia, A. and Pollack, D.. A refined result on sign changing solutions for a critical elliptic problem. Commun. Pure Appl. Anal. 12 (2013), 125–155.Google Scholar
14Guedda, M. and Véron, L.. Local and global properties of solutions of quasilinear elliptic equations. J. Differ. Eq. 76 (1988), 159–189.CrossRefGoogle Scholar
15Guedda, M. and Véron, L.. Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 13 (1989), 879–902.CrossRefGoogle Scholar
16Mercuri, C. and Pacella, F.. On the pure critical exponent problem for the p-Laplacian. Calc. Var. Partial Differ. Eq. 49 (2014), 1075–1090.CrossRefGoogle Scholar
17Mercuri, C., Sciunzi, B. and Squassina, M.. On Coron's problem for the p-Laplacian. J. Math. Anal. Appl. 421 (2015), 362–369.CrossRefGoogle Scholar
18Musso, M. and Pistoia, A.. Sign changing solutions to a nonlinear elliptic problem involving the critical Sobolev exponent in pierced domains. J. Math. Pures Appl. 86 (2006), 510–528.CrossRefGoogle Scholar
19Nirenberg, L.. On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115–162.Google Scholar
20Rey, O.. The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent. J. Funct. Anal. 89 (1990), 1–52.CrossRefGoogle Scholar
21Sciunzi, B.. Classification of positive $\mathcal D^{1,p}({\mathbb R }^{N^{N}})$-solutions to the critical p-Laplace equation in ${\mathbb R }^{N^{N}}$. Adv. Math. 291 (2016), 12–23.CrossRefGoogle Scholar
22Talenti, G.. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976), 353–372.CrossRefGoogle Scholar
23Véron, L.. Local and global aspects of quasilinear degenerate elliptic equations. Quasilinear elliptic singular problems (Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2017). xv+457 pp. ISBN: 978-981-4730-32-7.CrossRefGoogle Scholar
24Vétois, J.. A priori estimates and application to the symmetry of solutions for critical p-Laplace equations. J. Differ. Eq. 260 (2016), 149–161.CrossRefGoogle Scholar