Differential and Integral Equations

Symmetry and nonexistence results for Emden-Fowler equations in cones

Jérôme Busca

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Abstract

The purpose of this paper is to state some qualitative properties of the solutions to the Emden-Fowler equation $\Delta u + r^\sigma u^p = 0$ in a cone with Dirichlet boundary conditions. Namely one can show that every solution has the same symmetry as the cone in some sense; furthermore it is possible to extend the nonexistence results for regular solutions to this equation already stated by C. Bandle and M. Essen in [2]. For this one needs to establish some asymptotics for the solutions as $r\rightarrow 0$ or $r\rightarrow \infty$, relying on methods used by Veron in [25] for similar equations, but in different geometries, and then use a special form of the moving-planes method on a sphere in the spirit of [22].

Article information

Source
Differential Integral Equations, Volume 14, Number 8 (2001), 897-912.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356123171

Mathematical Reviews number (MathSciNet)
MR1827094

Zentralblatt MATH identifier
1056.35062

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B40: Asymptotic behavior of solutions 35B50: Maximum principles

Citation

Busca, Jérôme. Symmetry and nonexistence results for Emden-Fowler equations in cones. Differential Integral Equations 14 (2001), no. 8, 897--912. https://projecteuclid.org/euclid.die/1356123171


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