Prove that the solution operator is continuous

Let $\Omega\subset\mathbb R^n$ be open and bounded and let $f,g\in L^2(\Omega)$. Let $\phi\in H^1_0(\Omega)$. Let $\psi\in H^1_0(\Omega)$ be the weak solution of$$\int_\Omega \nabla \psi\cdot\nabla v dx + \int_\Omega a\psi v dx = \int_{\Omega} f v dx \quad\forall v\in H^1_0(\Omega).$$

I want to show that the solution operator $S:H^1_0(\Omega)\to H^1_0(\Omega)$ such that $\phi\in H^1_0(\Omega)\mapsto S(\phi)=\psi\in H^1_0(\Omega)$ is continuous.

I need to show that if $\{\phi_n\}\subset H^1_0(\Omega)$ is such that $\phi_n\to\phi$ strongly in $H^1_0(\Omega)$, then $S(\phi_n)\to S(\phi)$ in $H^1_0(\Omega)$.

I am confused because I used to test continuity for operators in integral form, like$$\varphi\mapsto\int_{\Omega} f\varphi dx.$$

I have no idea about the strategy to employ in this case. Does anyone know how to prove it?