American Institute of Mathematical Sciences

September  2008, 7(5): 1269-1273. doi: 10.3934/cpaa.2008.7.1269

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

 1 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China, China

Received  June 2007 Revised  January 2008 Published  June 2008

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].

Citation: Guangyue Huang, Wenyi Chen. Uniqueness for the solution of semi-linear elliptic Neumann problems in $\mathbb R^3$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1269-1273. doi: 10.3934/cpaa.2008.7.1269
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