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THE ROBIN PROBLEM FOR THE HÉNON EQUATION
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 88 / Issue 1 / August 2013
- Published online by Cambridge University Press:
- 27 June 2013, pp. 1-11
- Print publication:
- August 2013
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- Article
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In this paper, we consider the following Robin problem:
where
$$\begin{eqnarray*}\displaystyle \left\{ \begin{array}{ @{}ll@{}} \displaystyle - \Delta u= \mid x{\mathop{\mid }\nolimits }^{\alpha } {u}^{p} , \quad & \displaystyle x\in \Omega , \\ \displaystyle u\gt 0, \quad & \displaystyle x\in \Omega , \\ \displaystyle \displaystyle \frac{\partial u}{\partial \nu } + \beta u= 0, \quad & \displaystyle x\in \partial \Omega , \end{array} \right.&&\displaystyle\end{eqnarray*}$$
$\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.
