Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^N$ $(N\geq 3)$, $\phi\in H_0^1(\Omega)$ is a solution of $$ \begin{cases}\Delta \phi+ \phi=h & \text { in } \Omega, \\ \phi=0 & \text { on } \partial \Omega,\end{cases} $$where $h\in C^{\alpha}(\Omega)$ for some $0<\alpha<1$. Wirte the above equation in a weak form: finding a $\phi\in H_0^1(\Omega)$ such that $$ \langle\phi, \psi\rangle_{H_0^1(\Omega)}=\int_{\Omega}(\phi-h)\psi,\quad \forall \psi\, \in H_0^1(\Omega). $$My question is: How to use Riesz’s representation theorem to rewrite the above equation in $H_0^1(\Omega)$ in the operational form $\phi=T(\phi)+\tilde{h}$ with certain $\tilde{h}\in H_0^1(\Omega)$ which depends linearly in $h$ and where $T$ is a compact operator in $H_0^1(\Omega)$. This problem is motivated by using Fredholm’s alternative theorem to study the uniqueness of the existence of solutions.