Existence and Uniqueness of Radial Solutions of Quasilinear Equations in a Ball

Abstract

We consider the boundary value problem for the quasilinear equation div(A(|Du|)Du) + f(u) = 0, u > 0, x ∈ B_R(0), u|_{∂B_R(0)} = 0, where A and f are continuous functions in (0, ∞) and f is positive in (0, 1), f(1) = 0. We prove that (1) if f is strictly decreasing, the problem has a unique classical radial solution for any real number R > 0; (2) if f is not monotonous, the problem has at least one classical radial solution for some R > 0 large enough.