Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$ $(N\geq2)$ and $\lambda>0$ is a small parameter. I wonder if there exists a Green function such that $$(\Delta+\lambda) G(x,y)=\delta_x\quad \text{ in }\, \Omega, \quad G(x,y)=0\quad \text{ on }\,\partial \Omega,$$ where $\delta_x$ is the Dirac measure at $x\in\Omega$. If such function exists, what is the expansion profie around $x$?
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