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

Dancer, E.; Farina, Alberto

doi:10.1090/S0002-9939-08-09772-4

On the classification of solutions of $-\Delta u= e^u$ on $\mathbb {R}^N$: Stability outside a compact set and applications
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by E. N. Dancer and Alberto Farina PDF

Proc. Amer. Math. Soc. 137 (2009), 1333-1338 Request permission

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

A. Bahri and P.-L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math. 45 (1992), no. 9, 1205–1215. MR 1177482, DOI 10.1002/cpa.3160450908
Marie-Françoise Bidaut-Véron and Laurent Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math. 106 (1991), no. 3, 489–539. MR 1134481, DOI 10.1007/BF01243922
B. Buffoni, E. N. Dancer, and J. F. Toland, The regularity and local bifurcation of steady periodic water waves, Arch. Ration. Mech. Anal. 152 (2000), no. 3, 207–240. MR 1764945, DOI 10.1007/s002050000086
E. N. Dancer, Infinitely many turning points for some supercritical problems, Ann. Mat. Pura Appl. (4) 178 (2000), 225–233. MR 1849387, DOI 10.1007/BF02505896
E. N. Dancer, Stable solutions on $\Bbb R^n$ and the primary branch of some non-self-adjoint convex problems, Differential Integral Equations 17 (2004), no. 9-10, 961–970. MR 2082455
E. N. Dancer, Finite Morse index solutions of exponential problems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 25 (2008), no. 1, 173–179 (English, with English and French summaries). MR 2383085, DOI 10.1016/j.anihpc.2006.12.001
E. N. Dancer, Finite Morse index solutions of supercritical problems, J. Reine Angew. Math. 620 (2008), 213–233. MR 2427982, DOI 10.1515/CRELLE.2008.055
Esposito, P., Linear instability of entire solutions for a class of non-autonomous elliptic equations, preprint (2007).
Alberto Farina, Liouville-type results for solutions of $-\Delta u=|u|^{p-1}u$ on unbounded domains of $\Bbb R^N$, C. R. Math. Acad. Sci. Paris 341 (2005), no. 7, 415–418 (English, with English and French summaries). MR 2168740, DOI 10.1016/j.crma.2005.07.006
Alberto Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\Bbb R^N$, J. Math. Pures Appl. (9) 87 (2007), no. 5, 537–561 (English, with English and French summaries). MR 2322150, DOI 10.1016/j.matpur.2007.03.001
Alberto Farina, Stable solutions of $-\Delta u=e^u$ on $\Bbb R^N$, C. R. Math. Acad. Sci. Paris 345 (2007), no. 2, 63–66 (English, with English and French summaries). MR 2343553, DOI 10.1016/j.crma.2007.05.021
D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1972/73), 241–269. MR 340701, DOI 10.1007/BF00250508
James Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302. MR 170096, DOI 10.1007/BF02391014
Neil S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721–747. MR 226198, DOI 10.1002/cpa.3160200406