This question is one of the old exam questions. I could not solve it. Can anyone help me solve this question ? Thank you for your help.
$\Omega \subset \mathbb{R}^{N} $ is bounded, smooth domain, $N \geq 1, \lambda(x)$ positive function in $C^2(\Omega)$ and $u$ is a $C^2$ solution of the problem \begin{align} \Delta u&=\lambda(x)u, \qquad x \in \Omega \\ u&=1, \text{on} \quad \partial\Omega \end{align} Prove that \begin{align} r^{N-1}\frac{\partial}{\partial r}\left( r^{1-N}\int_{\partial B(x,r)} udS\right)=\int_{B(x,r)} \Delta u(y)dy \qquad \forall B(x,r) \subset \Omega \end{align}


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