Can the cosmological constant support a scalar field?

Static spherically symmetric gravitational equilibria of the real scalar field are discussed in the presence of the cosmological constant. We find nontrivial solutions if we take the self-interaction of the field into account while there is no such equilibria in the noninteracting case. The system has critical parameters beyond which new solutions disappear. They are determined by the ratio of the cosmological horizon to the “Compton radius” of the scalar field. We also discuss the stability of the solutions by means of a linear perturbation method and find that the number of unstable modes depends on the node number of the scalar field.