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In this paper, we consider the radially symmetric spacelike solutions of a nonlinear Dirichlet problem for the prescribed mean curvature spacelike equation in a Friedmann–Lemaître–Robertson–Walker spacetime. By using a conformal change of variable, this problem can be translated an equivalent problem in the Minkowski spacetime. By using the lower and upper solution method, fixed point, a priori bounds and topological degree method, we obtain the existence, nonexistence and multiplicity of radially symmetric spacelike solutions.