# Series and solution of $-\Delta u + \lambda u = f(x)$

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?

• 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.
– Ali
Feb 17, 2021 at 14:01
• @Ali Thank you. Could you expand this comment into an answer by adding some details?
– Hiro
Feb 17, 2021 at 14:11
• Since $\lambda > - \lambda_1$ you should be able to apply Riesz directly to get the existence of a solution on $H_0^1$. Feb 18, 2021 at 23:26