Regularity of radial minimizers and extremal solutions of semilinear elliptic equations |
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Authors: | Xavier Cabré Antonio Capella |
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Institution: | a Institució Catalana de Recerca i Estudis Avançats (ICREA) b Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada I, Diagonal 647, 08028 Barcelona, Spain |
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Abstract: | 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. |
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Keywords: | Semi-stable radial solutions Local minimizers Extremal solutions Semilinear elliptic equations Reagularity theory |
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