Cette Note porte sur l'étude des solutions, éventuellement non-bornées et de signe quelconque, de l'équation dans des domaines non-bornés de avec et . 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 on unbounded domains of with and . We prove some Liouville-type results and a classification theorem for 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.
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@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/
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