Given the following boundary value problem \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \Delta v + m(x)v &=& g,~\text{in}~ \Omega, \\ v &=& 0, ~\text{on}~\partial\Omega, \end{eqnarray*}
(a) Suppose that $m(x)$ is small enough. Prove that a classical solution exists and that it is unique.
(b) Also prove the existence of the solution in the Sobolev space $H^1(\Omega)$ assuming that $g\in L^2(\Omega)$.
In trying to get the solution to the above, I read Sections 9.5 and 9.6 of the book Functional Analysis, Sobolev Spaces and Partial Differential Equations of Haim Brezis. I feel that the solutions rely heavily on using some theorems: For instance, by Theorem 9.32, for all $g\in L^p(\Omega)$, there exists a unique solution $u\in W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega)$ of the equation $-\Delta v + v = g,~\text{in}~ \Omega$. I believe this may provide the answer for (b). Unfortunately, in the problem ''$\Delta v$'' has no negative sign. Further, I do not make sense of the condition ''$m(x)$ is small''. Note that in the cited Theorem, $m(x)=1$ in $-\Delta v + v = g,~\text{in}~ \Omega$
My questions are the following:
(a) Is there a direct way of computing the solutions without necessarily citing theorems throughout?
(b) Is there any handout or book with easy (elementary) worked examples of similar nature to help me do more revision.
I will assume that $\|\tilde {m}\|_{L^\infty(\Omega)} \ll 1$ ($\tilde m := -m$). Then, we have the following bound from below on the bilinear form associated to uniformly elliptic operator $-\Delta v + \tilde{m} v$: \begin{align} B[v,v] \geq& \int_{\Omega}|\nabla v|^2 \, \mathrm{d}x - \|m \|_{L^\infty(\Omega)} \| v\|_{L^2(\Omega)}^2 \\ \geq & \frac{1}{2}\int_{\Omega}|\nabla v|^2\, \mathrm{d} x +(\frac{c_p}{2} - \|m \|_{L^\infty(\Omega)}) \| v\|_{L^2(\Omega)}^2 \\ \geq & \frac{1}{2}\lVert v\rVert_{H^1_0(\Omega)}^2 \, . \end{align} for any $v \in H^1_0(\Omega)$, where in the penultimate step I have simply applied Poincaré's inequality. The form $B$ is thus coercive and by the Lax--Milgram theorem (after proving the required upper bound) I have a unique solution.
For a general $m$, I do not think there is a direct way of computing the solution.