Let $\Omega\subset \mathbb{R}^n$ be a bounded open set. Given $A \in C^\infty( \overline{\Omega},\mathbb{R}^{n\times n})$ and $f\in C^\infty(\overline{\Omega}, \mathbb{R})$ we can search for a weak solution $u:\Omega\to \mathbb{R}$ of the PDE
$$L(u)(x):= -\text{div} (A(x) \nabla u(x))= f(x) \quad \quad (1)$$
It can be proved that if $A(x)\geq cI \quad \forall x \in \Omega$ and $c>0$ (the so called ellipticity condition, please correct me if I am wrong) , then $(1)$ admits a $unique$ solution $u \in H_0^{1,2}(\Omega)$. This is achieved, for example, exploiting the Lax-Milgram theorem thanks to the ellipticity condition. We also have regularity theorems as Schauder estimates that ensures that $u$ is smooth given $f$ smooth. And the spectrum of $L$ is positive.
What happens if we perturb the operator $L$ with a multiplication operator? How can we tune the proofs used in the unperturbed case in order to work?
i.e. consider the operator $\tilde{L}= L+M_V$ where $$M_V:L^{2}(\Omega)\to L^2(\Omega)=g\mapsto Vg$$ where $V\in C^\infty(\overline{\Omega}, \mathbb{R})$ Under which conditions can we ensure existence and regularity?
How does the spectrum of $L$ change after such a perturbation?
I have tried to prove existence through Lax-Milgram, i.e. I tried to prove that the following bilinear form is coercive and bounded in $H_0^{1,2}$
$$B(u,\varphi) = \int_\Omega A(x)\nabla u(x)\cdot \nabla \varphi(x) + V(x) \varphi(x)dx $$ which is trivial if $V $ is bounded below by a positive constant, but this condition seems too restrictive to me. I hope we can solve this pde in more generality.