Abstract
We consider evolution inequalities of Sobolev type involving non-linearities of the form |x|sigma-N * |u|p and |x|sigma-N * |Vu|p, where * is the con-volution product in RN, p > 1 and 0 < sigma < N. For each case, we prove the existence of a critical exponent pcr(sigma, N) E (1, oo] depending on the parameter sigma and the dimension N, in the following sense: if 1 < p < pcr(sigma, N), then there is no local weak solutions; if p > pcr(sigma, N), then local weak solutions exist for some initial data.