  Consider Vlasov-Poisson system with a fixed ion background and periodic
condition on the space variables, in any dimension d\geq2. First, we show that
for general homogeneous equilibrium and any periodic x-box, within any small
neighborhood in the Sobolev space W_{x,v}^{s,p} (p>1,s<1+(1/p)) of the steady
distribution function, there exist nontrivial travelling wave solutions (BGK
waves) with arbitrary traveling speed. This implies that nonlinear Landau
damping is not true in W^{s,p}(s<1+(1/p)) space for any homogeneous equilibria
and in any period box. The BGK waves constructed are one dimensional, that is,
depending only on one space variable. Higher dimensional BGK waves are shown to
not exist. Second, for homogeneous equilibria satisfying Penrose's linear
stability condition, we prove that there exist no nontrivial invariant
structures in the (1+|v|^{2})^{b}-weighted H_{x,v}^{s} (b>((d-1)/4), s>(3/2))
neighborhood. Since arbitrarilly small BGK waves can also be constructed near
any homogeneous equilibria in such weighted H_{x,v}^{s} (s<(3/2)) norm, this
shows that s=(3/2) is the critical regularity for the existence of nontrivial
invariant structures near stable homogeneous equilibria. These generalize our
previous results in the one dimensional case.
