Journal
JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 261, Issue 12, Pages 6668-6698Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2016.08.047
Keywords
Classification of singularity; Choquard equation; Nonlocal problem; Decay asymptotic
Categories
Funding
- NSFC [11401270, 11661045, 11271133, 11431005]
- Jiangxi Provincial Natural Science Foundation [20161ACB20007]
- Shanghai Key Laboratory of PMMP [13dz2260400]
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In this paper we classify the isolated singularities of positive solutions to Choquard equation -Delta u + u = I-alpha[u(p)]u(q) in R-N \ {0}, lim vertical bar x vertical bar ->+infinity u (x) = 0, where p > 0, q >= 1, N >= 3, alpha is an element of(0, N) and I-alpha[u(P)](x) = integral R-N u(y)(P)/vertical bar x-y vertical bar(N-alpha)dy.We show that any positive solution u is a solution of -Delta u + u = I-alpha[u(P)]u(q) + k delta(0) in R-N in the distributional sense for some k >= 0, where delta(0) is the Dirac mass at the origin. We prove the existence of singular solutions in the subcritical case: p + q < N+alpha/N-2 and p < N/N-2, q < N/N-2 and prove that either the solution u has removable singularity at the origin or satisfies lim vertical bar x vertical bar -> 0+ u(x)vertical bar x vertical bar(N-2) = C-N which is a positive constant. In the supercritical case: p + q >= N+alpha/N-2 or p >= N/N-2, or q >= N/N-2 we prove that k = 0. (C) 2016 Elsevier Inc. All rights reserved.
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