Existence of Green functions and some properties

Let $\Omega$ be a smooth domain in $\mathbb{R}^N$， $N\geq 3$, $p\in \Omega$ is a fixed point, $\lambda$ is a parameter (can be 0,>0,<0), if there exisits a Green function $G_{\lambda}(x,p)$ such that $$(-\Delta_{x} -\lambda)G_{\lambda}(x,p)=\delta_p, \quad \text{ in }\, \Omega, \quad G_{\lambda}(x,p)=0\quad \text{ on }\, \partial \Omega,$$ where $\delta_p$ is a Dirac function at $p$. If so,what are the properties of such function $G_{\lambda}(x,p)$, like positivity and bounds estimate, and asymptotic behavior like $G_{\lambda}(x,p)\to G_{0}(x,p)$ when $\lambda\to 0$? Any comments and references are welcome!