Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Some results about semilinear elliptic problems on half-spaces

LAMFA, CNRS UMR 7352, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens, France

This contribution is part of the Special Issue: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday
Guest Editor: Gianmaria Verzini
Link: www.aimspress.com/mine/article/5753/special-articles

Special Issues: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday

We prove some new results about the growth, the monotonicity and the symmetry of (possibly) unbounded non-negative solutions of −∆u = f (u) on half-spaces, where f is merely a locally Lipschitz continuous function. Our proofs are based on a comparison principle for solutions of semilinear problems on unbounded slab-type domains and on the moving planes method.
  Article Metrics


1. Berestycki H, Caffarelli LA, Nirenberg L (1990) Uniform estimates for regularization of free boundary problems, In: Analysis and Partial Differential Equations, New York: Dekker, 567-617.

2. Berestycki H, Caffarelli LA, Nirenberg L (1993) Symmetry for elliptic equations in the halfspace, In: Boundary Value Problems for PDEs and Applications, Paris: Masson, 27-42.

3. Berestycki H, Caffarelli LA, Nirenberg L (1996) Inequalities for second order elliptic equations with applications to unbouded domains. Duke Math J 81: 467-494.    

4. Berestycki H, Caffarelli LA, Nirenberg L (1997) Further qualitative properties for elliptic equations in unbouded domains. Ann Scuola Norm Sup Pisa Cl Sci 25: 69-94.

5. Berestycki H, Caffarelli LA, Nirenberg L (1997) Monotonicity for elliptic equations in an unbounded Lipschitz domain. Commun Pure Appl Math 50: 1089-1111.    

6. Caffarelli LA, Salsa S (2005) A Geometric Approach To Free Boundary Problems, AMS.

7. Chen Z, Lin CS, Zou W (2014) Monotonicity and nonexistence results to cooperative systems in the half space. J Funct Anal 266: 1088-1105.    

8. Cortázar C, Elgueta M, García-Melián J (2016) Nonnegative solutions of semilinear elliptic equations in half-spaces. J Math Pure Appl 106: 866-876.    

9. Dancer EN (1992) Some notes on the method of moving planes. B Aust Math Soc 46: 425-434.    

10. Dancer EN (2009) Some remarks on half space problems. Disc Cont Dyn Sist 25: 83-88.    

11. Farina A (2003) Rigidity and one-dimensional symmetry for semilinear elliptic equations in the whole of $\mathbb{R}^N$ and in half spaces. Adv Math Sci Appl 13: 65-82.

12. Farina A (2007) On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^N$. J Math Pure Appl 87: 537-561.    

13. Farina A (2015) Some symmetry results and Liouville-type theorems for solutions to semilinear equations. Nonlinear Anal Theor 121: 223-229.    

14. Farina A, Montoro L, Sciunzi B (2012) Monotonicity and one-dimensional symmetry for solutions of −∆pu = f (u) in half-spaces. Calc Var Partial Dif 43: 123-145.    

15. Farina A, Sciunzi B (2016) Qualitative properties and classification of nonnegative solutions to −∆u = f (u) in unbounded domains when f (0) < 0. Rev Mat Iberoam 32: 1311-1330.    

16. Farina A, Sciunzi B (2017) Monotonicity and symmetry of nonnegative solutions to −∆u = f (u) in half-planes and strips. Adv Nonlinear Stud 17: 297-310.

17. Farina A, Soave N (2013) Symmetry and uniqueness of nonnegative solutions of some problems in the halfspace. J Math Anal Appl 403: 215-233.    

18. Farina A, Valdinoci E (2010) Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems. Arch Ration Mech Anal 195: 1025-1058.    

19. Gidas B, Spruck J (1981) A priori bounds for positive solutions of nonlinear elliptic equations. Commun Part Diff Eq 6: 883-901.    

20. Polácik PP, Quittner P, Souplet P (2007) Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems. Duke Math J 139: 555-579.

21. Quaas A, Sirakov B (2006) Existence results for nonproper elliptic equations involving the Pucci operator. Commun Part Diff Eq 31: 987-1003.    

22. Serrin J, Zou H (2002) Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math 189: 79-142.    

23. Sirakov B (2019) A new method of proving a priori bounds for superlinear elliptic PDE. arXiv:1904.03245.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved