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

摘要

This paper is devoted to the study of semi-stable radial solutions u∈H ~1(B _1) of -Δu=g(u) in B _1{0}, where g∈C ~1(R) is a general nonlinearity and B _1 is the unit ball of R ~N. 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 B _1, with Dirichlet data u|?B _1=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 H ~1(B _1) solutions.