The theory of liberal multiculturalism is based on a normative concept of liberalism leading to the implication of recognized cultural minority rights. However, there is no clear depiction of how liberal multiculturalism responds to the case of different cultural groups placing claims over one same excludable resource. In this thesis I analyze if there are any viable solutions to the uncertainties of conflicting cultural group claims within liberal multiculturalism. I use the practical case of Kven and Sami land claims to illustrate the consequences of the theory. Finding that legitimate claims to rights cannot be simply retracted, I provide two solutions involving sharing the resource in question. Both solutions resonate differently with the definition of culture and principles of liberal multiculturalism. The solutions involve cultural ownership. As we will see, this comes with certain infringements on the concept of the liberal self. I find that there is no unequivocal solution to the dilemmas described, however, I argue that collective ownership is the solution most consistent with liberal multiculturalism.