Regularity of Minimizers up to Dimension 7 in Domains of Double Revolution
نویسنده
چکیده
We consider the class of semi-stable positive solutions to semilinear equations −∆u = f(u) in a bounded domain Ω ⊂ Rn of double revolution, that is, a domain invariant under rotations of the first m variables and of the last n−m variables. We assume 2 ≤ m ≤ n− 2. When the domain is convex, we establish a priori Lp and H1 0 bounds for each dimension n, with p = ∞ when n ≤ 7. These estimates lead to the boundedness of the extremal solution of −∆u = λf(u) in every convex domain of double revolution when n ≤ 7. The boundedness of extremal solutions is known when n ≤ 3 for any domain Ω, in dimensions n ≤ 4 when the domain is convex, and in dimensions n ≤ 9 in the radial case. In dimensions 5 ≤ n ≤ 9 it remains an open question.
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