Sharp estimates for semi-stable radial solutions of semilinear elliptic equations

This paper is devoted to the study of semi-stable radial solutions u∈H1(B1) of −Δu=g(u)inB1∖{0}, where g∈C1(R) is a general nonlinearity and B1 is the unit ball of RN. We establish sharp pointwise estimates for such solutions. As an application of these results, we obtain optimal pointwise estimates for the extremal solution and its derivatives (up to order three) of the semilinear elliptic equation −Δu=λf(u), posed in B1, with Dirichlet data u|∂B1=0, where f is a continuous, positive, nondecreasing and convex function on [0,∞) such that f(s)/s→∞ as s→∞. In addition, we provide, for N⩾10, a large family of semi-stable radially decreasing unbounded H1(B1) solutions.