Is the following and isomorphism $H_0^1(\Omega) \to H^{-1}(\Omega)$?

Question

While working with differentiation with respect to the domain I stumbled upon the following question. Let $\Omega \subset \mathbb R^N$ be bounded, $\alpha \in L^\infty (\Omega)$ be non-negative and let $$ \varphi \in H_0^1(\Omega) \mapsto -\Delta \varphi + \alpha \varphi \in H^{-1}(\Omega)$$ be a linear application. I would like to show that this application is an isomorphism. I have no troubles as to proof that it is injective, but I can go no further.

I would wholeheartedly thank any suggestion.

Regards, DG

Answer 1

Hint: Apply Browder–Minty theorem.

Answer 2

I would like to add a final remark. This can much more be proven with Lax-Milgram's Theorem. The above mentioned operator can be seen as a bilinear form.