Regularity of radial minimizers and extremal solutions of semilinear elliptic equations

We consider a special class of radial solutions of semilinear equations −Δu=g(u) in the unit ball of Rn. It is the class of semi-stable solutions, which includes local minimizers, minimal solutions, and extremal solutions. We establish sharp pointwise, Lq, and Wk,q estimates for semi-stable radial solutions. Our regularity results do not depend on the specific nonlinearity g. Among other results, we prove that every semi-stable radial weak solution is bounded if n⩽9 (for every g), and belongs to H3=W3,2 in all dimensions n (for every g increasing and convex). The optimal regularity results are strongly related to an explicit exponent which is larger than the critical Sobolev exponent.