Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the classification of solutions of $ -\Delta u= e^u$ on $ \mathbb{R}^N$: Stability outside a compact set and applications

Authors: E. N. Dancer and Alberto Farina
Journal: Proc. Amer. Math. Soc. 137 (2009), 1333-1338
MSC (2000): Primary 35J60, 35B05, 35J25, 35B32
Published electronically: December 4, 2008
MathSciNet review: 2465656
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this short paper we prove that, for $ 3 \le N \le 9$, the problem $ -\Delta u = e^u$ on the entire Euclidean space $ \mathbb{R}^N$ does not admit any solution stable outside a compact set of $ \mathbb{R}^N$. This result is obtained without making any assumption about the boundedness of solutions. Furthermore, as a consequence of our analysis, we also prove the non-existence of finite Morse Index solutions for the considered problem. We then use our results to give some applications to bounded domain problems.

References [Enhancements On Off] (What's this?)

  • 1. Bahri, A., and Lions, P.-L., Solutions of superlinear elliptic equations and their Morse indices. Comm. Pure Appl. Math. 45 (1992), no. 9, 1205-1215. MR 1177482 (93m:35077)
  • 2. Bidaut-Véron, M., and Véron, L., Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math. 106 (1991), 489-539. MR 1134481 (93a:35045)
  • 3. Buffoni, B., Dancer, E.N., and Toland, J.F., The sub-harmonic bifurcation of Stokes waves, Arch. Rat. Mech. Anal. 152 (2000), no. 3, 241-271. MR 1764946 (2002e:76010b)
  • 4. Dancer, E.N., Infinitely many turning points for some supercritical problems, Ann. Mat. Pura Appl. (4) 178 (2000), 225-233. MR 1849387 (2002g:35077)
  • 5. Dancer, E.N., Stable solutions on $ \mathbb{R}^n$ and the primary branch of some non-self-adjoint convex problems, Differential and Integral Equations 17 (2004), 961-970. MR 2082455 (2006b:35093)
  • 6. Dancer, E.N., Finite Morse index solutions of exponential problems, Ann. Inst. H. Poincaré Analyse Non Linéaire 25 (2008), 173-179. MR 2383085
  • 7. Dancer, E.N., Finite Morse index solutions of supercritical problems, J. Reine Angewandte Math. 620 (2008), 213-233. MR 2427982
  • 8. Esposito, P., Linear instability of entire solutions for a class of non-autonomous elliptic equations, preprint (2007).
  • 9. Farina, A., Liouville-type results for solutions of $ - \Delta u = \vert u \vert^{p-1} u$ on unbounded domains of $ \mathbb{R}^N$, C. R. Math. Acad. Sci. Paris 341 (2005), 415-418. MR 2168740 (2006d:35074)
  • 10. Farina, A., On the classification of solutions of the Lane-Emden equation on unbounded domains of $ \mathbb{R}^N$, J. Math. Pures Appl. 87 (2007), 537-561. MR 2322150 (2008c:35070)
  • 11. Farina, A., Stable solutions of $ -\Delta u = e^u$ on $ \mathbb{R}^N$, C. R. Math. Acad. Sci. Paris 345 (2007), 63-66. MR 2343553 (2008e:35063)
  • 12. Joseph, D.D., and Lundgren, T.S., Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal. 49 (1972/73), 241-269. MR 0340701 (49:5452)
  • 13. Serrin, J., Local behavior of solutions of quasi-linear equations. Acta Math. 111 (1964), 247-302. MR 0170096 (30:337)
  • 14. Trudinger, Neil S., On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. 20 (1967), 721-747. MR 0226198 (37:1788)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35J60, 35B05, 35J25, 35B32

Retrieve articles in all journals with MSC (2000): 35J60, 35B05, 35J25, 35B32

Additional Information

E. N. Dancer
Affiliation: School of Mathematics and Statistics, The University of Sydney, New South Wales 2006, Australia

Alberto Farina
Affiliation: LAMFA, CNRS UMR 6140, Université de Picardie Jules Verne, Faculté de Mathématiques et d’Informatique, 33, rue Saint-Leu, 80039 Amiens, France

Received by editor(s): November 8, 2007
Published electronically: December 4, 2008
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society