The author considers the equation
, and calls this equation oscillatory if for each
there is some bounded domain G in the complement of the ball
such that the equation has a nontrivial solution in the Sobolev space
. In the paper various conditions on a(x) – in the form of pointwise or integral inequalities – are given which imply oscillatory and nonoscillatory behaviour, respectively. The proofs consist in an estimation of the quadratic form associated with the equation.