Weak cosmic censorship with self-interacting scalar and bound on charge to mass ratio

We study the model of Einstein-Maxwell theory minimally coupling to a massive charged self-interacting scalar field, parameterized by the quartic and hexic coupling, labelled by $\lambda$ and $\beta$, respectively. In the absence of scalar field, there is a class of counterexamples to cosmic censorship. Moveover, we investigate the properties of full nonlinear solution with nonzero scalar field, and argue that, by assuming massive charged self-interacting scalar field with sufficiently large charge above one certain bound, these counterexamples can be removed. In particular, this bound on charge for self-interacting scalar field is no longer equal to the weak gravity bound for free scalar case. In the quartic case, the bounds are below free scalar case for $\lambda<0$, while above free scalar case for $\lambda>0$. Meanwhile, in the hexic case, the bounds are above free scalar case for both $\beta>0$ and $\beta<0$.