Existence of Positive Weak Solutions with a Prescribed Singular Set of Semilinear Elliptic Equations
نویسندگان
چکیده
In this paper, we consider the problem of the existence of non-negative weak solution u of A u + u P = O i n ~ u = 0 on Of~ n n + 2 ~ / n 1 having a given closed set S as its singular set. We prove that when < p < _ _ and n 2 n 4 + 2 nv/'n-L--1 S is a closed subset of f2, then there are infinite many positive weak solutions with S as their singular set. Applying this method to the conformal scalar curvature equation for n > 9, we construct a weak solution n+2 n+2 u ~ L ~ (S n) ofLou + L n-2 = 0 such that S n is the singular set ofu where L 0 is the conformal Laplacian with respect to the standard metric of S n. When n = 4 or 6, this kind of solution has been constructed by Pacard.
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