We prove that the semilinear Dirichlet problem for a Laplace equation on a unit ball, involving the nonlinearity f(r,u)=-a(r)u+b(r)up, with a subcritical p, has a unique positive solution, provided a(r) is positive, increasing and convex, while b(r) is positive, decreasing and concave. Moreover, we prove that this solution is non-degenerate. We also present a uniqueness result in case a(r) is negative.
