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Communications on Pure and Applied Analysis (CPAA)
 

Uniqueness for the solution of semi-linear elliptic Neumann problems in $\mathbb R^3$

Pages: 1269 - 1273, Volume 7, Issue 5, September 2008      doi:10.3934/cpaa.2008.7.1269

 
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Guangyue Huang - School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China (email)
Wenyi Chen - School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China (email)

Abstract: Considering the positive solution of the following nonlinear elliptic Neumann problem

$\Delta_0 u-\lambda u+f(u)=0, u>0,\ $ in $\Omega,\quad \frac{\partial u}{\partial\nu}=0\ $ on $\partial\Omega$

where $\Omega$ is convex and $f(u)$ defined by (2). We prove that for $1< p_i < 5$, $i=1,\cdots, K$ and $\lambda$ small, the only solution to the above problem is constant. This can be seen as a generalization of Theorem 1 in [7].

Keywords:  Uniqueness, Neumann problem, nonlinear elliptic equation.
Mathematics Subject Classification:  Primary: 35J60; Secondary: 35C21.

Received: June 2007;      Revised: January 2008;      Available Online: June 2008.