scholarly journals Asymptotic behaviour of ground state solutions for the fractional Hénon equation

2021 ◽  
pp. 125456
Author(s):  
Haixia Chen ◽  
Xiaolin Xu ◽  
Xiaolong Yang
Keyword(s):  
Asymptotic Behaviour ◽  
Ground State ◽  
Ground State Solutions ◽  
Hénon Equation

scholarly journals THE ROBIN PROBLEM FOR THE HÉNON EQUATION

2013 ◽  
Vol 88 (1) ◽  
pp. 1-11
Author(s):  
HAIYANG HE
Keyword(s):  
Symmetry Breaking ◽  
Unit Ball ◽  
Ground State ◽  
Existence Results ◽  
Robin Problem ◽  
Ground State Solutions ◽  
Hénon Equation ◽  
Image Position

AbstractIn 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.

scholarly journals The Brezis–Nirenberg Problem for the Hénon Equation: Ground State Solutions

2012 ◽  
Vol 12 (2) ◽  
Author(s):  
Simone Secchi
Keyword(s):  
Dirichlet Problem ◽  
Ground State ◽  
Ground State Solutions ◽  
Hénon Equation ◽  
Nirenberg Problem

AbstractThis work is devoted to the Dirichlet problem for the equation −Δu = λu + |x|α|u|

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