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Infinitely many radially symmetric solutions for a quasilinear elliptic problem in a ball. (English) Zbl 0852.34021

The nonlinear Dirichlet problem under consideration has the form

-Δ p u=f(u)+h(|x|),xB,u| B =0,(1)

where B denotes an open ball in n , Δ p is the p-Laplacian operator for p>1, f: is locally Lipschitzian and superlinear, and hL (B,). Let y(t,a) denote a solution of the polar form of (1) satisfying y(0)=a0, y ' (0)=0 (t0). The energy functional for y(t,a) is

E(t,a)=p-1 p|y ' (t,a)| p + 0 y(t,a) f(s)ds·

The main theorem states that (1) has infinitely many radial solutions u with u(0)>0 (u(0)<0) if E(t,a)+ as a+ (a-, respectively) uniformly in t. 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 p=2 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)].

MSC:
34B15Nonlinear boundary value problems for ODE
35J65Nonlinear boundary value problems for linear elliptic equations