Hostname: page-component-7479d7b7d-qlrfm Total loading time: 0 Render date: 2024-07-13T01:23:46.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on Entire Solutions for Two Fourth-Order Elliptic Problems

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  23 November 2015

Baishun Lai  and
Dong Ye
Baishun Lai
Affiliation:
Institute of Contemporary Mathematics, Henan University, Kaifeng 475004, People's Republic of China (laibaishun@henu.edu.cn)
Dong Ye
Affiliation:
IECL, UMR 7502, Département de Mathématiques, Université de Lorraine, Ile de Saulcy, 57045 Metz, France (dong.ye@univ-lorraine.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We are interested in entire solutions for the semilinear biharmonic equation Δ2u = f(u) in ℝN, where f(u) = eu or –u–p (p > 0). For the exponential case, we prove that for the polyharmonic problem Δ2mu = eu with positive integer m, any classical entire solution verifies Δ2m–1u < 0; this completes the results of Dupaigne et al. (Arch. Ration. Mech. Analysis208 (2013), 725–752) and Wei and Xu (Math. Annalen313 (1999), 207–228). We also obtain a refined asymptotic expansion of the radial separatrix solution to Δ2u = eu in ℝ3, which answers a question posed by Berchio et al. (J. Diff. Eqns252 (2012), 2569–2616). For the negative power case, we show the non-existence of the classical entire solution for any 0 < p ⩽ 1.

Keywords

biharmonic equationentire solutionasymptotic behaviournon-existence

MSC classification

Secondary: 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35B08: Entire solutions 35B53: Liouville theorems, Phragmén-Lindelöf theorems 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 777 - 786
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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

1. Arioli, G., Gazzola, F. and Grunau, H. C., Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity, J. Diff. Eqns 230 (2006), 743–770.CrossRefGoogle Scholar
2. Berchio, E., Farina, A., Ferrero, A. and Gazzola, F., Existence and stability of entire solutions to a semilinear fourth order elliptic problem, J. Diff. Eqns 252 (2012), 2569–2616.CrossRefGoogle Scholar
3. Chang, S. Y. A. and Chen, W., A note on a class of higher order conformally covariant equations, Discrete Contin. Dynam. Syst. 7 (2001), 275–281.CrossRefGoogle Scholar
4. Choi, Y. S. and Xu, X., Nonlinear biharmonic equations with negative exponents, J. Diff. Eqns 246 (2009), 216–234.CrossRefGoogle Scholar
5. Dávila, J., Flores, L. and Guerra, I., Multiplicity of solutions for a fourth order problem with power-type nonlinearity, Math. Annalen 348 (2010), 143–193.CrossRefGoogle Scholar
6. Dupaigne, L., Ghergu, M., Goubet, O. and Warnault, G., The Gel’fand problem for the biharmonic operator, Arch. Ration. Mech. Analysis 208 (2013), 725–752.CrossRefGoogle Scholar
7. Farina, A. and Ferrero, A., Existence and stability properties of entire solutions to the polyharmonic equation (–Δ) m u = e u for any m ⩾ 1, Preprint (http://arXiv:1403.0729; 2014).Google Scholar
8. Guo, Z. and Wei, J., Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents, Discrete Contin. Dynam. Syst. 34 (2014), 2561–2580.CrossRefGoogle Scholar
9. Lin, C. S., A classification of solutions of a conformally invariant fourth order equation in ℝ N , Comment. Math. Helv. 73 (1998), 206–231.CrossRefGoogle Scholar
10. McKenna, P. J. and Reichel, W., Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry, Electron. J. Diff. Eqns 37 (2003), 1–13.Google Scholar
11. Mitidieri, E. and Pohozaev, S., A priori estimates and blow-up of solutions to nonlinear partial differential equations and inqualities, Proceedings of the Steklov Institute of Mathematics, Volume 234 (Nauka, Moscow, 2001).Google Scholar
12. Walter, W., Ganze Lösungen der Differentialgleichung Δ m u = f(u), Math. Z. 67 (1957), 32–37.CrossRefGoogle Scholar
13. Walter, W., Zur Existenz ganzer Lösungen der Differentialgleichung Δ m u = e u , Arch. Math. 9 (1958), 308–312.CrossRefGoogle Scholar
14. Wei, J. and Xu, X., Classification of solutions of higher order conformally invariant equations, Math. Annalen 313 (1999), 207–228.CrossRefGoogle Scholar
15. Wei, J. and Ye, D., Nonradial solutions for a conformally invariant fourth order equation in ℝ4 , Calc. Var. PDEs 32 (2008), 373–386.CrossRefGoogle Scholar