Qualitative analysis to an eigenvalue problem of the Henon equation

摘要

In this paper we study the following eigenvalue problem @@@ {-Delta v = lambda C(alpha)(p(alpha) - epsilon)vertical bar x vertical bar(alpha)u(epsilon)(p alpha-epsilon-1)v in Omega, @@@ u=0 on partial derivative Omega, @@@ where Omega subset of R-N is a smooth bounded domain containing the origin, C(alpha) = (N + alpha)(N - 2), N >= 3, p(alpha) = N+2+2 alpha/N-2, alpha > 0, epsilon > 0 is a small parameter and u(epsilon) is a single peaked solution of Henon equation @@@ {-Delta u = C(alpha)vertical bar x vertical bar(alpha)u(p alpha-epsilon) in Omega, @@@ u > 0 in Omega, @@@ u=0 on partial derivative Omega, @@@ which established by Gladiali and Grossi (2012) [21]. By using various local Pohozaev identities and blow-up analysis, we prove some asymptotic behavior of the eigen values lambda(epsilon,i) and corresponding eigen functions @@@ v(epsilon,i), i = 2, ... , Sigma(1 <= k <= 2+alpha/2) (N + 2k - 2) (N + k - 3)!/(N - 2)!k! + 2 @@@ when alpha is not an even integer. As a consequence, if 0 < alpha < 2, we have that the Morse index of the single peaked solutions is N + 1, which gives an affirmative answer to a conjecture raised by Gladiali and Grossi.

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