Uniqueness for the solution of semi-linear elliptic Neumann problems in $\mathbb R^3$
Guangyue Huang, Wenyi Chen · Communications on Pure & Applied Analysis · 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 u}=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].