  In this paper, we are concern with the multiplicity of solutions for a
p-Laplacian problem. A weaker super-quadratic assumptions is required on the
nonlinearity. Under the weaker condition we give a new proof for the infinite
solutions having a prescribed number of nodes to the problem. It turns out that
the weaker condition on nonlinearity suffices to guarantee the infinitely many
solutions. At the same time, a global characterization of the critical values
of the nodal radial solutions are given.
