Summary: Consider the problem \[ -\Delta u= u^{{n+2\over n-2}}-\lambda u\quad \text{in }B(1),\;u>0,\;u\text{ is radial},\quad {\partial u\over\partial\nu}=0\quad \text{on }\partial B(1), \] where \(B(1)\) is the unit ball in \(\mathbb{R}^n\). Here, we prove that for \(n\geq 7\), there exists a \(\lambda_0>0\) such that for \(0<\lambda<\lambda_0\), the above problem does not admit a nonconstant solution.