Laplacian instability, as well as the conditions of stability under these perturbations. We found that these conditions are similar and reduce to very simple conditions $$\begin{aligned}
{\cal F}(r)>0,\quad {\cal G}(r)>0,\quad {\cal H}(r)>0\,,\nonumber\end{aligned}$$ where $r$ runs from the event horizon to infinity or cosmological horizon and $$\begin{aligned}
{\cal F}&=2\Bigl[
G_4+\frac{1}{2}B\phi'X'G_{5X}-XG_{5\phi}
\Bigr]\,,\nonumber
\\
{\cal G}&=2\left[ G_4-2XG_{4X}
+X\left(\frac{A'}{2A}B\phi'G_{5X}+G_{5\phi}\right)
\right]\,,\nonumber
\\
{\cal H}&=2\left[G_4-2X G_{4X}
+X\left(\frac{  C' }{2 C} B \phi' G_{5X}+ G_{5\phi}\right)\right]\,.\nonumber\end{aligned}$$ We have applied the results to various solutions encountered in the literature. We found that for some spacetimes, the S-deformation is not possible because of a boundary term which makes the argument impossible. The condition to use the S-deformation and therefore the previous results is $S(r=r_{\text{Horizon}})\geq 0$ and $S(r=+\infty)\leq 0$ where $+\infty$ can be the cosmological horizon in the case of an asymptotically de Sitter solution. $$\begin{aligned}
S=\frac{\sqrt{AB}}{2}\Bigl(\frac{C'}{C}+\frac{{\cal F}'}{{\cal F}}\Bigr)\end{aligned}$$ The condition will often be violated in asymptotically AdS solution, and therefore this S-deformation will not be used for these spacetimes. In these particular cases, we can study directly the potential and show its positivity as we have done in this paper or use an other S-deformation.

Acknowledgments {#acknowledgments .unnumbered}
===============

A. Ganguly wants to thank Claude Leon Foundation for financial support. 