$m$ such that $\nu \notin {\mathbb Z}$ to avoid complications with logarithms. Extension to $\Delta \in [\frac{d}{2}-1, \frac{d}{2}]$ with the lower end of the interval saturating the unitary bound is possible with the added complication of taking proper account of the necessary boundary terms.

We will begin by rewriting this solution in terms of the quantities on the Dirichlet surface which we denote with a hat to distinguish them from the boundary data: $$\hat{J}_\phi \equiv \left[r^{d-\Delta}\,\Phi(r)\right]_{r\to r_D}=J_\phi+ \frac{\phi}{(2\nu)\,r_D^{2\nu}} \ ,
\label{j0}$$ $$\hat{\phi} \equiv \left[-r^{2\nu}\times r\partial_r \left( r^{d-\Delta}\Phi \right)\right]_{r\to r_D} =\phi \ .
\label{ph0}$$

Since the transformation between the data on the boundary $\{J_\phi,\phi\}$ and that on the hypersurface $\{\hat{J}_\phi, \hat{\phi}\}$ is linear it is a simple matter to write the bulk solution in terms of the hypersurface variables. One simply has $$\Phi(r) =\frac{\hat{\phi}}{(2\nu)\, r^\Delta} +  r^{\Delta-d}  \left(\hat{J}_\phi- \frac{\hat{\phi}}{(2\nu)r_D^{2\nu}}\right) . 
\label{pDsol}$$ This is the answer we seek and all that remains is to interpret this result.

It is now easy to notice that the imposition of the Dirichlet condition on a hypersurface inside the bulk is equivalent to making the boundary source a specific function of the vev. From we can read off the specific deformation of the boundary CFT action 