Uniqueness of ground states for quasilinear elliptic operators

(1.1) ∆mu+ f(u) = 0, see Proposition 1 below. Here, as usual, ∆m denotes the degenerate m-Laplace operator div(|Du|m−2Du), m > 1, and f(u) is a continuously differentiable function defined for 0 0 in B. Here, with the help of their identity, we consider the complementary problem of uniqueness of radial ground states of (1.1), that is, non-negative, non-trivial radially symmetric solutions on R such that u = u(x)→ 0 as |x| → ∞. This problem, like that considered by Erbe and Tang, has exercised many researchers in the past decade, principally in the case when f(u) is negative for small u and superlinear for large u; see in particular [CL], [Co], [CEF1], [Kw], [M], [MS], and most recently [CEF2], when m = 2, n ≥ 3; and [C2] when 1 1, c.f. the remark following equation (2.1). Our results apply in particular to the important canonical nonlinearity