Iam working on the following problem:
Let $\Omega \subseteq \mathbb{R}^n$ and $v$ be the solution of $(-\Delta+q-\lambda)v=F$ on $\Omega$ such that $q$ is bounded and $v=exp(-ax)u$ where $u\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$
For a large $X\in \mathbb{R}$, we define a cut-off function $\chi_{X}$ on $\Omega_{X}$: $\chi_{X}=\begin{cases} 1 & \text{on }[0,X-1]\\ 1\leq\chi_{X}\leq0 & \text{on} (X-1,X)\\ 0 & \text{on} [X,\infty)\end{cases}$
Iam trying to find $L^2(\Omega_{X})$-norm of $(\chi_{X}-1)(-\Delta v)+(\chi_{X}-1)(q-\lambda)v$ ..
I startred by the following: $\|(\chi_{X}-1)(-\Delta v)+(\chi_{X}-1)(q-\lambda)v\|^2_{L^2(\Omega_{X})}=\\ \|(\chi_{X}-1)(-\Delta+q-\lambda)v\|^2_{L^2(\Omega_{X})}=\\ \int_{\Omega_{X}}(\chi_{X}-1)^2((-\Delta+q-\lambda)v)^2$
Now Iam not sure how I can finish the calculation, I know $(\chi_{X}-1)$ is bounded on $\Omega_{X}$ hence $ \int_{\Omega_{X}}(\chi_{X}-1)^2((-\Delta+q-\lambda)v)^2= \int_{\Omega_{X}}|\chi_{X}-1|^2((-\Delta+q-\lambda)v)^2\leq C \int_{\Omega_{X}}((-\Delta+q-\lambda)v)^2$ Are these steps correct? and how I can find the $L^2$ norm on $\Omega_{X}$ for the laplacian term?