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It has been shown previously that an extremal Reissner-Nordstr\"om or anextremal Kerr black hole cannot be overcharged or overspun by a test particleif radiative and self-force effects are neglected. In this paper, we considerextremal charged and rotating black holes with cosmological constants. Bystudying the motion of test particles, we find the following results: Anextremal Reissner-Nordstr\"om anti-de Sitter (RN-AdS) black hole can beovercharged by a test particle but an extremal Reissner-Nordstr\"om de Sitter(RN-dS) black hole cannot be overcharged. We also show that both extrmalKerr-de-Sitter (Kerr-dS) and Kerr-anti-de-Sitter (Kerr-AdS) black holes can beoverspun by a test particle, implying a possible breakdown of the cosmiccensorship conjecture. For the Kerr-AdS case, the overspinning requires thatthe energy of the particle be negative, a reminiscent of the Penrose process.In contrast to the extremal RN and Kerr black holes, in which cases the cosmiccensorship is upheld, our results suggest some subtle relations between thecosmological constants and the comic censorship. We also discuss the effect ofradiation reaction for the Kerr-dS case and find that the magnitude of energyloss due to gravitational radiation may not be enough to prevent the violationof the cosmic censorship.

Testing cosmic censorship conjecture near extremal black holes with cosmological constants