
Structure of the Positive Radial Solutions for the Supercritical Neumann Problem ε2Δu − u + up = 0 in a BallStructure of the Positive Radial Solutions for the Supercritical Neumann Problem ε2Δu − u + up = 0 in a BallAA11021653 
"/Miyamoto, Yasuhito/"Miyamoto, Yasuhito
22
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3
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, pp.685

739 , 20150715 , Graduate School of Mathematical Sciences, The University of Tokyo
ISSN:13405705
NCID:AA11021653
Description
We are interested in the structure of the positive radial solutions of the supercritical Neumann problem in a unit ball ε2(U''+ N−1/r U) − U + Up = 0, 0 < r < 1, U'(1) = 0, U >0, 0 < r < 1, where N is the spatial dimension and p > pS := (N + 2)/(N − 2), N ≥ 3. We showthat there exists a sequence {ε∗n}∞n=1 (ε∗1 > ε∗2 > ···→0) such that this problem has infinitely many singular solutions {(ε∗n, U∗n)}∞n=1 ⊂ R×(C2(0, 1)∩C1(0, 1]) and that the nonconstant regular solutions consist of infinitely many smooth curves in the (ε,U(0)) plane. It is shown that each curve blows up at ε∗n and if pS < p < pJL, then each curve has infinitely many turning points around ε∗n. Here, pJL := 1 + 4/N−4−2√N−1 (N ≥ 11), ∞ (2 ≤ N ≤ 10). In particular, the problem has infinitely many solutions if ε ∈ {ε∗n}∞n=1. We also showthat there exists ¯ε > 0 such that the problem has no nonconstant regular solution if ε > ¯ε. The main technical tool is the intersection number between the regular and singular solutions.
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