연구홍보포털

논문 초록 (Abstract)

We consider the elliptic equation Delta u + K(vertical bar x vertical bar)e(u) = 0 in R-n \ {0} with n > 2 when for l > -2, K(r) behaves li ke r(l) near 0 or infinity. The asymptotic behavior of radial solutions at infinity is described by -(2 + l) log r for l > -2 and - log log r f or l = -2. When r(-l) K(r) -> c > 0 as r -> infinity and r -> 0, regula r radial solutions at oo and singular radial solutions at 0 exhibit self-similarity at infinity and 0, respectively. Singular solutio ns with the asymptotic self-similarity exist uniquely in the radial class. Moreover, for n >= 10 + 4l, separation of any two radial solutio ns with the asymptotic self-similarity may happen, while intersection of two solutions may occur for 2 < n < 10 + 4l. In particul ar, for n >= 10 + 4l with l > -2, if K (not equal 0) satisfies that r( 2)K(r) -> 0 as r -> 0 and 0 <= k(r) = r(-l) K(r) <= n-2/4(2 + l) inf(0 0, then any two radial solu tions do not intersect each other and each radial solution is linearly stable. When n >= 10 + 4l, we apply the global results to prove the uniqueness of positive radial solutions for theDirichlet problem with zero data on a ball. (C) 2015 Elsevier Inc. All rights reserved.