Liouville-type results for solutions of $ -\mathrm{\Delta }u={|u|}^{p-1}u$ on unbounded domains of $ {\mathbb{R}}^{N}$

Farina, Alberto

doi:10.1016/j.crma.2005.07.006

Partial Differential Equations

Liouville-type results for solutions of

- Δ u = {| u |}^{p - 1} u

on unbounded domains of

R^{N}

[Résultats de type Liouville pour des solutions de

- Δ u = {| u |}^{p - 1} u

dans des domaines non-bornés de

R^{N}

]

Présenté par : Philippe G. Ciarlet

Farina, Alberto ¹

¹ LAMFA, CNRS UMR 6140, université de Picardie Jules Verne, faculté de mathématiques et d'informatique, 33, rue Saint-Leu, 80039 Amiens, France

Comptes Rendus. Mathématique, Tome 341 (2005) no. 7, pp. 415-418.

Résumé
Abstract

Cette Note porte sur l'étude des solutions, éventuellement non-bornées et de signe quelconque, de l'équation $- Δ u = {| u |}^{p - 1} u$ dans des domaines non-bornés de $R^{N}$ avec $N ⩾ 2$ et $p > 1$ . Nous démontrons des résultats de type Liouville ainsi que des théorèmes de classification pour les solutions régulières appartenant à une des classes suivantes : solutions stables, solutions d'indice de Morse fini et solutions stables à l'extérieur d'un compact. Nous étendons aussi, au cas d'un épigraphe coercif régulier, les célèbres résultats de Gidas et Spruck concernant les solutions positives de l'équation considérée.

In this Note we study solutions, possibly unbounded and sign-changing, of the equation $- Δ u = {| u |}^{p - 1} u$ on unbounded domains of $R^{N}$ with $N ⩾ 2$ and $p > 1$ . We prove some Liouville-type results and a classification theorem for $C^{2}$ solutions belonging to one of the following classes: stable solutions, finite Morse index solutions and solutions which are stable outside a compact set. We also extend, to smooth coercive epigraphs, the well-known results of Gidas and Spruck concerning non-negative solutions of the considered equation.

Reçu le : 2005-06-27
Accepté le : 2005-06-29
Publié le : 2005-09-19

DOI : 10.1016/j.crma.2005.07.006

Affiliations des auteurs :

Farina, Alberto ¹

¹ LAMFA, CNRS UMR 6140, université de Picardie Jules Verne, faculté de mathématiques et d'informatique, 33, rue Saint-Leu, 80039 Amiens, France

@article{CRMATH_2005__341_7_415_0,
     author = {Farina, Alberto},
     title = {Liouville-type results for solutions of $ -\mathrm{\Delta }u={|u|}^{p-1}u$ on unbounded domains of $ {\mathbb{R}}^{N}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {415--418},
     publisher = {Elsevier},
     volume = {341},
     number = {7},
     year = {2005},
     doi = {10.1016/j.crma.2005.07.006},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2005.07.006/}
}

TY  - JOUR
AU  - Farina, Alberto
TI  - Liouville-type results for solutions of $ -\mathrm{\Delta }u={|u|}^{p-1}u$ on unbounded domains of $ {\mathbb{R}}^{N}$
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 415
EP  - 418
VL  - 341
IS  - 7
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2005.07.006/
DO  - 10.1016/j.crma.2005.07.006
LA  - en
ID  - CRMATH_2005__341_7_415_0
ER  -

%0 Journal Article
%A Farina, Alberto
%T Liouville-type results for solutions of $ -\mathrm{\Delta }u={|u|}^{p-1}u$ on unbounded domains of $ {\mathbb{R}}^{N}$
%J Comptes Rendus. Mathématique
%D 2005
%P 415-418
%V 341
%N 7
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2005.07.006/
%R 10.1016/j.crma.2005.07.006
%G en
%F CRMATH_2005__341_7_415_0

Farina, Alberto. Liouville-type results for solutions of $ -\mathrm{\Delta }u={|u|}^{p-1}u$ on unbounded domains of $ {\mathbb{R}}^{N}$. Comptes Rendus. Mathématique, Tome 341 (2005) no. 7, pp. 415-418. doi : 10.1016/j.crma.2005.07.006. http://archive.numdam.org/articles/10.1016/j.crma.2005.07.006/

Bibliographie
Cité par

[1] Bahri, A.; Coron, J.-M. On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., Volume 41 (1988) no. 3, pp. 253-294

[2] Bahri, A.; Lions, P.-L. Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math., Volume 45 (1992) no. 9, pp. 1205-1215

[3] Brézis, H. Elliptic equations with limiting Sobolev exponents – the impact of topology, Frontiers of the Mathematical Sciences: 1985 (New York, 1985), Comm. Pure Appl. Math., Volume 39 (1986) no. Suppl. S, p. S17-S39

[4] Cabré, X.; Capella, A. On the stability of radial solutions of semilinear elliptic equations in all of $R^{N}$ , C. R. Math. Acad. Sci. Paris, Ser. I, Volume 338 (2004) no. 10, pp. 769-774

[5] Dancer, E.N. Some notes on the method of moving planes, Bull. Austral. Math. Soc., Volume 46 (1992) no. 3, pp. 425-434

[6] Esteban, M.J.; Lions, P.-L. Existence and nonexistence results for semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, Volume 93 (1982/83) no. 1–2, pp. 1-14

[7] A. Farina, forthcoming

[8] Fowler, R.H. Further studies of Emden's and similar differential equations, Quart. J. Math. Oxford Ser. (2) (1931), pp. 259-288

[9] Gidas, B.; Spruck, J. A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, Volume 6 (1981) no. 8, pp. 883-901

[10] Gidas, B.; Spruck, J. Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., Volume 34 (1981) no. 4, pp. 525-598

[11] Joseph, D.D.; Lundgren, T.S. Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., Volume 49 (1972/73), pp. 241-269

[12] Passaseo, D. Nontrivial solutions of elliptic equations with supercritical exponent in contractible domains, Duke Math. J., Volume 92 (1998) no. 2, pp. 429-457

[13] Pohozaev, S.I. On the eigenfunctions of the equation $Δ u + λ f (u) = 0$ , Dokl. Akad. Nauk SSSR, Volume 165 (1965), pp. 36-39 (English translation in Soviet Math. Dokl., 6, 1965, pp. 1408-1411)

[14] Wang, X. On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., Volume 337 (1993) no. 2, pp. 549-590

[15] Ambrosio, L.; Cabré, X. Entire solutions of semilinear elliptic equations in $R^{3}$ and a conjecture of De Giorgi, J. Amer. Math. Soc., Volume 13 (2000) no. 4, pp. 725-739

[16] Caffarelli, L.A.; Gidas, B.; Spruck, J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., Volume 42 (1989) no. 3, pp. 271-297

Cité par Sources :