Consider a bounded smooth set $\Omega \subset \mathbb R^n$ (for example, we can take a ball). Can we write down the solution of \begin{align*} -\Delta u(x) + \lambda u(x) &= f(x), & x \in \Omega \\ u(x)&= 0, & x \in \partial \Omega, \end{align*} with $\lambda >0$ and $f$ smooth as, $$u(x) = \sum_{i=1}^\infty(-\lambda)^i G^{i+1} f(x),$$ where $G f(x)$ denotes the convolution of $f$ with the Green function of the Laplacian? In other words, is it true that $(I+\lambda G)$ is invertible? What is a good reference for this result?
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2$\begingroup$ This is to show that $I+\lambda G$ is invertible: By compactness and Fredholm theory it suffices to show that the kernel is trivial. But that easily follows from energy estimates. $\endgroup$– AliCommented Feb 17, 2021 at 14:01
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$\begingroup$ @Ali Thank you. Could you expand this comment into an answer by adding some details? $\endgroup$– HiroCommented Feb 17, 2021 at 14:11
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$\begingroup$ Since $ \lambda > - \lambda_1$ you should be able to apply Riesz directly to get the existence of a solution on $H_0^1$. $\endgroup$– Math604Commented Feb 18, 2021 at 23:26
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