In this paper we study the problem of stability of existence of solutions of a classical variational inequality in a Hilbert space. More precisely, we extend some stability results to an infinite dimensional and non-convex setting. To do so, we use the concepts of Robinson's normal map and Kien and Yao's natural map in order to transform the initial multivalued problem into single-valued ones. Both maps being a perturbation of the identity map, these transformations allow to apply a stability result using the degree theory under some general hypotheses. Finally we discuss how those hypotheses are fulfilled by the class of prox-regular set-valued mapping. More precisely, we consider two concepts of prox-regularity for parameterized set-valued mapping, the local prox-regularity and the uniform prox-regularity, and see which one of the hypotheses of the stability results are fulfilled in both cases.