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Symmetry and nonexistence of low Morse index solutions in unbounded domains

Gladiali, Francesca Maria and Pacella, Filomena and Weth, Tobias (2010) Symmetry and nonexistence of low Morse index solutions in unbounded domains. Journal de mathématiques pures et appliqués, Vol. 93 (5), p. 536-558. ISSN 0021-7824. Article.

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DOI: 10.1016/j.matpur.2009.08.003

Abstract

In this paper we prove symmetry results for classical solutions of semilinear elliptic equations in the whole RN or in the exterior of a ball, N≥2, in the case when the nonlinearity is either convex or has a convex first derivative. More precisely we prove that solutions having Morse index j≤N are foliated Schwarz symmetric, i.e. they are axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis. From this we deduce some nonexistence results for positive or sign changing solutions in the case when the nonlinearity does not depend explicitly on the space variable.

Nous démontrons des résultats de symétrie de solutions classiques de problèmes elliptiques semilinéaires dans RN ou à l'extérieur d'une boule dans les cas N≥2 où la non-linéarité est convexe et la dérivée est aussi convexe. Plus précisément, nous démontrons que toute solution dont l'indice de Morse est inférieur ou égal à N est à symétrie axiale et monotone relativement à l'angle polaire. Partant de ce résultat nous déduisons des théorèmes de non-existence de solutions positives ou de solutions qui changent de signe.

Item Type:Article
ID Code:4721
Status:Published
Refereed:Yes
Uncontrolled Keywords:Semilinear elliptic equations, symmetry results, unbounded domains, nonexistence of solutions
Subjects:Area 01 - Scienze matematiche e informatiche > MAT/05 Analisi matematica
Divisions:001 Università di Sassari > 03 Istituti > Matematica e fisica
ISSN:0021-7824
Deposited On:22 Oct 2010 12:18

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