Regularity of radial minimizers and extremal solutions of semilinear elliptic equations

Abstract

We consider a special class of radial solutions of semilinear equations − Δ u = g ( u ) in the unit ball of R n . It is the class of semi-stable solutions, which includes local minimizers, minimal solutions, and extremal solutions. We establish sharp pointwise, L q , and W k , 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 u ∈ H 0 1 is bounded if n ⩽ 9 (for every g), and belongs to H 3 = W 3 , 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.