We consider a special class of radial solutions of semilinear equations −?u = 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 u ? H1
0 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.
CitationCabré, Xavier; Capella Kort, Antonio. “Regularity of radial minimizers and extremal solutions of semilinear elliptic equations”. Journal of functional analysis, 2006, vol. 238, núm. 2, p. 709-733.
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