The Islanders distinguish between those states that recognize the jurisdiction of the Court and those that don't, and between those who have ratified the Treaty on the Non-Proliferation of Nuclear Weapons and those which haven't, but include all nine—China, North Korea, France, India, Israel, Pakistan, Russia, the U.K. and the U.S.—in the suit. Their case can be read here. The Republic has itself signed the Treaty.
The Islanders have good cause for their suit. From 1946 to 1958, the United States tested 67 nuclear weapons in the islands, including the largest nuclear test it ever conducted. In 1956, the U.S. Atomic Energy Commission regarded the Marshall Islands as "by far the most contaminated place in the world." In 1952, the U.S. tested its first hydrogen bomb on the island of Elugelab, utterly destroying it. The republic is the only country in the world where the UN has authorized the use of nuclear weapons. No one can question the Islanders credentials.
The suit calls for the court to acknowledge breach of international law, namely the Non-Proliferation Treaty, in both retaining a nuclear arsenal and in acting to improve weapons systems. It also seeks a court order compelling each nation to begin disarmament negotiations within a year.
Such a little country taking on such giants in such a noble endeavour, how can you help but wish them the greatest good will. The Islanders act for all seven billion of us, and you can help by signing the petition at the Nuclear Zero website.