Home Sublinear elliptic problems under radiality. Harmonic NA groups and Euclidean spaces
Article
Licensed
Unlicensed Requires Authentication

Sublinear elliptic problems under radiality. Harmonic NA groups and Euclidean spaces

  • Ewa Damek ORCID logo and Zeineb Ghardallou ORCID logo EMAIL logo
Published/Copyright: May 7, 2019

Abstract

Let be the Laplace operator on d, d3, or the Laplace–Beltrami operator on the harmonic NA group (in particular, on a rank one noncompact symmetric space). For the equation u-φ(,u)=0 we give necessary and sufficient conditions for the existence of entire bounded or large solutions under the hypothesis of radiality of φ with respect to the first variable. A Harnack-type inequality for positive continuous solutions is also proved.


Communicated by Karl-Hermann Neeb


Funding statement: The first author was supported by the National Science Centre Poland, Research grant Opus UMO-2014/15/B/ST1/00060.

Acknowledgements

The authors want to express their gratitude to Krzysztof Bogdan, Konrad Kolesko and Mohamed Selmi for their helpful and kindly suggestions.

References

[1] J.-P. Anker, E. Damek and C. Yacoub, Spherical analysis on harmonic AN groups, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 23 (1996), no. 4, 643–679. Search in Google Scholar

[2] G. Astarita and G. Marrucci, Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, New York, 1974. Search in Google Scholar

[3] R. Atar, S. Athreya and Z.-Q. Chen, Exit time, Green function and semilinear elliptic equations, Electron. J. Probab. 14 (2009), FPag50–71. 10.1214/EJP.v14-597Search in Google Scholar

[4] A. Baalal and W. Hansen, Nonlinear perturbation of balayage spaces, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 1, 163–172. Search in Google Scholar

[5] C. Bandle and M. Marcus, “Large” solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour, J. Anal. Math. 58 (1992), 9–24. 10.1007/BF02790355Search in Google Scholar

[6] J. Berndt, F. Tricerri and L. Vanhecke, Generalized Heisenberg Groups and Damek–Ricci Harmonic Spaces, Lecture Notes in Math. 1598, Springer, Berlin, 1995. 10.1007/BFb0076902Search in Google Scholar

[7] A. Boukricha, Harnack inequality for nonlinear harmonic spaces, Math. Ann. 317 (2000), no. 3, 567–583. 10.1007/PL00004414Search in Google Scholar

[8] A. Boukricha, W. Hansen and H. Hueber, Continuous solutions of the generalized Schrödinger equation and perturbation of harmonic spaces, Expo. Math. 5 (1987), 97–135. Search in Google Scholar

[9] K.-S. Cheng and W.-M. Ni, On the structure of the conformal scalar curvature equation on 𝐑n, Indiana Univ. Math. J. 41 (1992), no. 1, 261–278. 10.1512/iumj.1992.41.41015Search in Google Scholar

[10] E. Damek and Z. Ghardallou, Large versus bounded solutions to sublinear elliptic problems, Bull. Pol. Acad. Sci. Math. (2019), https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/bulletin-polish-acad-sci-math/online. 10.4064/ba8180-12-2018Search in Google Scholar

[11] E. Damek and F. Ricci, Harmonic analysis on solvable extensions of H-type groups, J. Geom. Anal. 2 (1992), no. 3, 213–248. 10.1007/BF02921294Search in Google Scholar

[12] K. El Mabrouk, Entire bounded solutions for a class of sublinear elliptic equations, Nonlinear Anal. 58 (2004), no. 1–2, 205–218. 10.1016/j.na.2004.01.004Search in Google Scholar

[13] K. El Mabrouk and W. Hansen, Nonradial large solutions of sublinear elliptic problems, J. Math. Anal. Appl. 330 (2007), no. 2, 1025–1041. 10.1016/j.jmaa.2006.08.024Search in Google Scholar

[14] Z. Ghardallou, Positive solutions to some nonlinear elliptic problems, Ph.D. thesis, University Tunis El Manar, Tunis, 2016. Search in Google Scholar

[15] Z. Ghardallou, Positive solutions to sublinear elliptic problems, Colloq. Math. 155 (2019), no. 1, 107–125. 10.4064/cm7340-2-2018Search in Google Scholar

[16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. 224, Springer, Berlin, 1977. 10.1007/978-3-642-96379-7Search in Google Scholar

[17] H. Hueber and M. Sieveking, Uniform bounds for quotients of Green functions on C1,1-domains, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 1, 105–117. 10.5802/aif.861Search in Google Scholar

[18] J. B. Keller, On solutions of Δu=f(u), Comm. Pure Appl. Math. 10 (1957), 503–510. 10.1002/cpa.3160100402Search in Google Scholar

[19] A. V. Lair, Large solutions of semilinear elliptic equations under the Keller–Osserman condition, J. Math. Anal. Appl. 328 (2007), no. 2, 1247–1254. 10.1016/j.jmaa.2006.06.060Search in Google Scholar

[20] A. V. Lair, Large solutions of mixed sublinear/superlinear elliptic equations, J. Math. Anal. Appl. 346 (2008), no. 1, 99–106. 10.1016/j.jmaa.2008.05.047Search in Google Scholar

[21] A. V. Lair and A. Mohammed, Entire large solutions of semilinear elliptic equations of mixed type, Commun. Pure Appl. Anal. 8 (2009), no. 5, 1607–1618. 10.3934/cpaa.2009.8.1607Search in Google Scholar

[22] A. V. Lair and A. W. Wood, Large solutions of sublinear elliptic equations, Nonlinear Anal. 39 (2000), no. 6, 745–753. 10.1016/S0362-546X(98)00233-8Search in Google Scholar

[23] A. C. Lazer and P. J. McKenna, On a problem of Bieberbach and Rademacher, Nonlinear Anal. 21 (1993), no. 5, 327–335. 10.1016/0362-546X(93)90076-5Search in Google Scholar

[24] A. Mohammed, On ground state solutions to mixed type singular semi-linear elliptic equations, Adv. Nonlinear Stud. 10 (2010), no. 1, 231–244. 10.1515/ans-2010-0112Search in Google Scholar

[25] R. Osserman, On the inequality Δuf(u), Pacific J. Math. 7 (1957), 1641–1647. 10.2140/pjm.1957.7.1641Search in Google Scholar

[26] S. I. Pohožaev, The Dirichlet problem for the equation Δu=u2, Soviet Math. Dokl. 1 (1960), 1143–1146. Search in Google Scholar

[27] F. Rouvière, Espaces de Damek-Ricci, géométrie et analyse, Analyse sur les groupes de Lie et théorie des représentations (Kénitra 1999), Sémin. Congr. 7, Société Mathématique de France, Paris (2003), 45–100. Search in Google Scholar

[28] M. Selmi, Perturbation et comparaison des résolvantes et des semi-groupes, Thèse, University Tunis El Manar, Tunis, 1992. Search in Google Scholar

Received: 2019-01-18
Revised: 2019-03-19
Published Online: 2019-05-07
Published in Print: 2019-07-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 14.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2019-0014/html
Scroll to top button