ON THE UNIQUENESS OF POSITIVE SOLUTIONS OF A QUASILINEAR EQUATION CONTAINING A WEIGHTED p-LAPLACIAN, THE SUPERLINEAR CASE

Abstract. We consider the quasilinear equation of the form (P) −∆pu = K(|x|)f(u), x ∈ Rn, n> p> 1, where ∆pu: = div(|∇u|p−2∇u) is the degenerate p-Laplace operator and the weight K is a positive C1 function defined in R+. We deal with the case in which f ∈ C[0,∞) has one zero at u0> 0, is non positive and not identically 0 in (0, u0), and is locally lipschitz, positive and satisfies some superlinear growth assumption in (u0,∞). We carefully study the behaviour of the solution of the corresponding initial value problem for the radial version of the quasilinear equation, as well as the behaviour of its derivative with respect to the initial value. Combining, as Cortázar, Felmer and Elgueta, comparison arguments due to Coffman and Kwong, with some separation results, we show that any zero of the solutions of the initial value problem is monotone decreasing with respect to the initial value, which leads immediately the uniqueness of positive radial ground states, and the uniqueness of positive radial solutions of the Dirichlet problem on a ball