The nonlinear Dirichlet problem under consideration has the form
where denotes an open ball in , is the -Laplacian operator for , is locally Lipschitzian and superlinear, and . Let denote a solution of the polar form of (1) satisfying , . The energy functional for is
The main theorem states that (1) has infinitely many radial solutions with if as (, respectively) uniformly in . The proof is based on phase plane analysis of the polar form of (1). More than half of the paper is devoted to the development of sufficient conditions, via a shooting method, for the energy hypothesis of the theorem to hold. Results of this type for were obtained by M. Struwe [Arch. Math. 39, 233-240 (1982; Zbl 0496.35034)], A. Castro and A. Kurepa [Proc. Am. Math. Soc. 101, 57-64 (1987; Zbl 0656.35048)].