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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 [email protected]
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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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