Black hole no hair conjecture in the Einstein-Maxwell-scalar system in asymptotically de Sitter spacetime

We discuss a charged black hole with a scalar field in asymptotically de Sitter spacetime in the Einstein-Maxwell-scalar-$\Lambda$ system. By imposing the spherically symmetric ansatz we find that there is no solution when the scalar field is massless or has the “convex” potential. We also investigate the double well potential as an example of the generic non-negative potential, and find the nontrivial solutions. The scalar field of these solutions must pass over the top of the potential barrier several times; i.e., they have a node number. However, these solutions have just the same number of the unstable modes as their node number in the linear perturbation analysis. This suggests that the black hole no-hair conjecture holds in this system.