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THE ROBIN PROBLEM FOR THE HÉNON EQUATION

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  27 June 2013

HAIYANG HE
HAIYANG HE*
Affiliation:
College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), Hunan Normal University, Changsha, Hunan 410081, PR China email hehy917@hotmail.com
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Abstract

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In this paper, we consider the following Robin problem:

$$\begin{eqnarray*}\displaystyle \left\{ \begin{array}{ @{}ll@{}} \displaystyle - \Delta u= \mid x{\mathop{\mid }\nolimits }^{\alpha } {u}^{p} , \quad & \displaystyle x\in \Omega , \\ \displaystyle u\gt 0, \quad & \displaystyle x\in \Omega , \\ \displaystyle \displaystyle \frac{\partial u}{\partial \nu } + \beta u= 0, \quad & \displaystyle x\in \partial \Omega , \end{array} \right.&&\displaystyle\end{eqnarray*}$$
where $\Omega $ is the unit ball in ${ \mathbb{R} }^{N} $ centred at the origin, with $N\geq 3$, $p\gt 1$, $\alpha \gt 0$, $\beta \gt 0$, and $\nu $ is the unit outward vector normal to $\partial \Omega $. We prove that the above problem has no solution when $\beta $ is small enough. We also obtain existence results and we analyse the symmetry breaking of the ground state solutions.

Keywords

Hénon equationsymmetry breakingRobin problemground state solutions

MSC classification

Secondary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 1 , August 2013 , pp. 1 - 11
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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