No one will, as long as they have ratified the Outer Space Treaty, which all space fairing nations have. It explicitly states (all text from the State Department page):
Article II
Outer space, including the moon and other celestial bodies,
is not subject to national appropriation by claim of sovereignty, by
means of use or occupation, or by any other means.
Additionally, SpaceX is governed by the treaty as well:
Article VI
States Parties to the Treaty shall bear international
responsibility for national activities in outer space, including the
Moon and other celestial bodies, whether such activities are carried
on by governmental agencies or by non-governmental entities, and for
assuring that national activities are carried out in conformity with
the provisions set forth in the present Treaty. The activities of
non-governmental entities in outer space, including the Moon and other
celestial bodies, shall require authorization and continuing
supervision by the appropriate State Party to the Treaty. When
activities are carried on in outer space, including the Moon and other
celestial bodies, by an international organization, responsibility for
compliance with this Treaty shall be borne both by the international
organization and by the States Parties to the Treaty participating in
such organization.
However, they will retain ownership of there hardware:
Article VIII
A State Party to the Treaty on whose registry an object
launched into outer space is carried shall retain jurisdiction and
control over such object, and over any personnel thereof, while in
outer space or on a celestial body. Ownership of objects launched into
outer space, including objects landed or constructed on a celestial
body, and of their component parts, is not affected by their presence
in outer space or on a celestial body or by their return to the Earth.
Such objects or component parts found beyond the limits of the State
Party to the Treaty on whose registry they are carried shall be
returned to that State Party, which shall, upon request, furnish
identifying data prior to their return.