String in arbitrary dilaton background I have only seen literature string theory with linear dilaton i.e. $\Phi(X) = V.X $. What if I consider something like $\Phi(X) = X^2$ ?
 A: As Wakabaloola point out, you must check if the beta functions are zero in order to discover if a "quadratic dilaton background" is a consistent background for Weyl-invariant strings. 
Although the direct computation would be interesting, it also evident that it may be not very illuminating. Several intuitive arguments of why such backgrounds cannot be allowed in string theory can be elaborated. The key idea about all of them is to recognize the fundamental origin of the linear dilaton background as the near-horizon geometry of a stack of NS-5 branes. You can check this "Brief review of little string theory" for details.
The existence of a dual holographic description for the quadratic dilaton background is doubious:  The linear dilaton background is a consistent string background with an emergent direction $\phi$ over which the string coupling vary as $$g_{s}(\phi) = exp({ \frac{Q\phi}{2}}) $$ For some constant  $Q$.
A (non-gravitational) holographic description for the dilaton background exist. Such construction is possible because the string interactions vanish at $\phi \rightarrow - 
\infty $ and are indeed confined to the compact region at which $\phi \rightarrow + 
\infty $. The region outside the compact locii of the geometry where the theory is strongly coupled is called "the bulk theory" and the strongly coupled regime is identified as the holographic dual of the bulk theory, namely little string theory. 
Why is all of the above relevant for the question? Well, an expectation for holography is the existence of a weakly coupled regime at whith computations can be performed. Relevant S-matrix computations can be done in the locci where $\phi \rightarrow - 
\infty $ for the linear dilaton theory case. The latter is an ubiquitous quality of holography that is not satisfied in the quadratic dilaton background at which we cannot find a region where $g_{s}(\phi) = exp({ \frac{Q\phi^{2}}{2}}) \rightarrow 0 $ to perform actual computations using perturbative string theory.
The latter argument of course cannot rule the possibility of the non-perturbative consistency of the quadratic dilaton theory. But at least shows the difficulty to obtain it from a brane configuration; and in any case,its consistency can not be checked in perturbative string theory because the theory is necessary at finite coupling at any point over the $\phi$ direction.
