Consider a random process s solution of the stochastic partial differential equation Ls = w with L a homogeneous operator and w a multidimensional Lévy white noise. In this paper, we study the asymptotic effect of zooming in or zooming out of the process s. More precisely, we give sufficient conditions on L and w so that the rescaled versions of s converges in law to a self-similar process of order H at coarse scales and at fine scales. The parameter H depends on the homogeneity order of the operator L and the Blumenthal-Getoor indices associated to the Lévy white noise w. Finally, we apply our general results to several notorious classes of random processes and random fields.