Let $$\Omega_\alpha=\{(\xi,\eta)\in \mathbb{R}^2 /\xi>\frac{1}{\alpha-1}a^{1-\alpha},0<\eta<1\}$$ ,a>0

we have $\Delta$ an isomorphism of $\mathbb{ w}^{2,p}\bigcap \mathbb{ w}_0^{1,p}(\Omega_\alpha)$ on $L^p(\Omega_\alpha)$

How we show that $\Delta +\frac{1}{\xi}L$ is an isomorphism of $\mathbb{ w}^{2,p}\bigcap \mathbb{ w}_0^{1,p}(\Omega_\alpha)$ on $L^p(\Omega_\alpha))$?

such that $\frac{1}{\xi}L\omega=\frac{-2\gamma}{\xi}D_\xi\omega+\frac{\gamma(\gamma+1)}{\xi^2}\omega+2\alpha c^{\frac{-1}{\beta}}(\frac{\eta}{\xi}D_\xi D_\eta\omega-\frac{\gamma\eta}{\xi^2}D_\eta \omega)+\alpha^2 c^{\frac{-2}{\beta}}\frac{\eta^2}{\xi^2}D^2_\eta\omega+\alpha c^{\frac{-1}{\beta}}(\frac{1}{\xi}D_\xi\omega-\frac{\gamma}{\xi}\omega)+\alpha(\alpha+1)c^{\frac{-2}{\beta}}\frac{\eta}{\xi^2}D_\eta \omega$

$(\alpha ,\beta , \gamma ,c)\in \mathbb{R}^4$

$\omega\in \mathbb{ w}^{2,p}\bigcap \mathbb{ w}_0^{1,p}(\Omega_\alpha)$

Let $$\Omega_\alpha=\left\{(\xi,\eta)\in \mathbb{R}^2 /\xi>\frac{1}{\alpha-1}a^{1-\alpha},0<\eta<1\right\},$$ $a>0$.

we have $\Delta$ an isomorphism of $\mathbb{ w}^{2,p}\cap \mathbb{ w}_0^{1,p}(\Omega_\alpha)$ on $L^p(\Omega_\alpha)$

How we show that $\Delta +\frac{1}{\xi}L$ is an isomorphism of $\mathbb{ w}^{2,p}\cap \mathbb{w}_0^{1,p}(\Omega_\alpha)$ on $L^p(\Omega_\alpha))$?

such that $$ \begin{align} \frac{1}{\xi}L\omega= {} & \frac{-2\gamma}{\xi} D_\xi\omega+\frac{\gamma(\gamma+1)}{\xi^2}\omega \\[6pt] & {} +2\alpha c^{\frac{-1}{\beta}} \left( \frac{\eta}{\xi}D_\xi D_\eta\omega-\frac{\gamma\eta}{\xi^2}D_\eta \omega \right) + \alpha^2 c^{\frac{-2}{\beta}}\frac{\eta^2}{\xi^2}D^2_\eta\omega \\[6pt] & {} +\alpha c^{\frac{-1}{\beta}} \left( \frac{1}{\xi} D_\xi\omega-\frac{\gamma}{\xi}\omega\right) + \alpha(\alpha+1)c^{\frac{-2}{\beta}}\frac{\eta}{\xi^2}D_\eta \omega \end{align} $$

$(\alpha ,\beta , \gamma ,c)\in \mathbb{R}^4$

$\omega\in \mathbb{w}^{2,p}\cap \mathbb{ w}_0^{1,p}(\Omega_\alpha)$